26 Mixed States: Renormalization of Matrix Product Operators
Starting from the MPO, MPDO, and LPDO notation of Chapter 23, this chapter studies renormalization fixed points for density-operator chains. Following the order of the mixed-state analysis in [ CPGSV16 , Section 4 ] , the discussion first states the local renormalization condition and then records its comparison with the pure-state formalism. It next passes to zero correlation length, purification fixed points, mutual information, and saturation of the area law. The source definitions of simple tensors and Gibbs states of nearest-neighbor commuting Hamiltonians precede the simple local and commuting structures. The boundary-theory interlude of the source is outside the present MPO development. The general case and its algebraic characterization continue in Chapter 27.
26.1 Preliminaries for physical renormalization
We first collect the transfer-map, channel, and physical-closure notions used to state the mixed-state renormalization condition. The renormalization fixed point of [ CPGSV16 , Definition 4.1 ] is stated in the next section.
An MPO tensor \(M\) has transfer-map idempotence if
A linear map \(\mathcal S\) between matrix algebras is completely positive in rectangular Kraus form if there are finitely many operators \(A_i:H\to K\) such that
No trace-preservation normalization is imposed.
When the input and output matrix algebras agree, rectangular Kraus complete positivity is equivalent to the square-map notion of complete positivity.
A map in rectangular Kraus form sends positive semidefinite matrices to positive semidefinite matrices. Every trace-preserving completely positive Kraus map is a completely positive Kraus map.
The sum of two completely positive maps in rectangular Kraus form is completely positive in rectangular Kraus form.
Concatenating Kraus families for the two summands gives a Kraus family for their sum.
A linear map \(\mathcal{S}\) between matrix algebras is trace-preserving completely positive if it has a Kraus form \(\mathcal{S}(X)=\sum _i A_iXA_i^\dagger \) with \(\sum _i A_i^\dagger A_i=I\). The Kraus operators may be rectangular, so the input and output dimensions need not agree.
A completely positive map in rectangular Kraus form that preserves the matrix trace is trace-preserving completely positive.
If \((A_i)_i\) is a Kraus family, trace preservation and cyclicity give \(\operatorname{tr}((\sum _i A_i^\dagger A_i)X)=\operatorname{tr}(X)\) for every \(X\). Nondegeneracy of the trace pairing implies \(\sum _i A_i^\dagger A_i=\mathbb {1}\).
Every trace-preserving completely positive map in rectangular Kraus form preserves the matrix trace.
If \(\mathcal S(X)=\sum _a A_aXA_a^\dagger \) and \(\sum _a A_a^\dagger A_a=I\), cyclicity gives
If \(\mathcal S\) is trace-preserving completely positive and \(X\geq 0\), then \(\mathcal S(X)\geq 0\).
The identity map \(\operatorname{id}(X)=X\) is trace-preserving completely positive, with single Kraus operator \(A_0=I\).
Take \(r=1\) and \(A_0=I\); then \(\operatorname{id}(X)=IXI^\dagger =X\) and \(A_0^\dagger A_0=I\).
Let \(H\) and \(K\) be finite-dimensional complex vector spaces, and let \(V:H\to K\) satisfy \(V^\dagger V=I_H\). Then the map
is trace-preserving and completely positive. This is the general one-isometry form of the local basis change used in [ CPGSV16 , Appendix C.2, lines 1439 and 1520 ] .
Take the sole Kraus operator to be \(V\). Its resolution of the identity is precisely \(V^\dagger V=I_H\).
For a rectangular matrix \(V:H\to K\), define the linear map
For every matrix \(X\), one has \(\Phi _V(X)=VXV^\dagger \).
If \(V^\dagger V=I_H\), then \(\Phi _V\) is trace-preserving and completely positive.
For every rectangular operator \(V:H\to K\), the map \(X\mapsto VXV^\dagger \) is completely positive.
Let \(e:I\simeq J\) be an equivalence of index sets. The associated matrix reindexing is the linear map \(R_e:\mathbb {C}^{I\times I}\to \mathbb {C}^{J\times J}\) determined by
For every equivalence \(e:I\simeq J\) between finite index sets, the reindexing map \(R_e\) is trace-preserving and completely positive.
Let \(P_e:\mathbb {C}^I\to \mathbb {C}^J\) be the permutation matrix of \(e\). Then \(R_e(X)=P_eXP_e^\dagger \) and \(P_e^\dagger P_e=I_{\mathbb {C}^I}\). The claim follows from the preceding isometry lemma.
The composition of two trace-preserving completely positive maps is again trace-preserving completely positive. If \(\mathcal{S}\) has Kraus operators \(A_i\) and \(\mathcal{T}\) has Kraus operators \(B_j\), then \(\mathcal{S}\circ \mathcal{T}\) has Kraus operators \(A_iB_j\).
The Kraus form of the composite and its resolution of the identity are
The composition of two completely positive rectangular Kraus maps is completely positive. If the two families are \((A_i)_i\) and \((B_j)_j\), the composite family is \((A_iB_j)_{i,j}\).
Let \(\mathcal S:\operatorname{End}_{\mathbb {C}}(H)\to \operatorname{End}_{\mathbb {C}}(K)\) be trace-preserving and completely positive, and let \(R\) be another finite-dimensional space. Then
are trace-preserving and completely positive.
The first assertion is the tensor-extension closure theorem. For the second, let \(\tau _H:R\otimes H\simeq H\otimes R\) and \(\tau _K:K\otimes R\simeq R\otimes K\) be the canonical factor swaps. Then
where \(\mathcal R_\tau \) denotes reindexing both matrix coordinates along \(\tau \). Both reindexing maps are trace-preserving and completely positive, so closure under composition proves the claim. No choice of Kraus family is needed for this preservation statement.
Let \(I\) be a finite index set, let \(J\) be an arbitrary index set, and let \((A_a\in \mathbb {C}^{J\times I})_a\) be a finite family of matrices. Its rectangular Kraus map is
Let \((A_a:H\to K)_a\) be a finite family of rectangular operators. If \(\sum _a A_a^\dagger A_a=I_H\), then \(X\mapsto \sum _a A_aXA_a^\dagger \) is trace-preserving and completely positive.
The displayed family is already a Kraus representation, and the assumed identity is precisely its trace-preserving normalization.
Every finite rectangular Kraus family defines a completely positive map, without a resolution-of-identity assumption.
If \(e:I\simeq I'\) and \(A'_b=A_{e^{-1}(b)}\), then \(\Phi _{A'}=\Phi _A\).
Reindexing the finite sum gives \(\sum _{b\in I'}A_{e^{-1}(b)}XA_{e^{-1}(b)}^\dagger =\sum _{a\in I}A_aXA_a^\dagger \).
Let \(H=\bigoplus _k H_k\) and \(K=\bigoplus _k K_k\). For each \(k\), let \((A_{k,a}:H_k\to K_k)_a\) be a finite family of operators, and let \(\widetilde A_{k,a}:H\to K\) agree with \(A_{k,a}\) on \(H_k\) and vanish on every other summand. The orthogonally controlled Kraus map is
In particular, \(\mathcal C(X)_{kl}=0\) for \(k\ne l\). This is the sector control in the definitions of \(\mathcal T_1\) and \(\mathcal S_1\) in [ CPGSV16 , Appendix C.2, lines 1523–1535 and 1548–1555 ] .
Let \(X_{kk}:H_k\to H_k\) denote the \(k\)th diagonal block of \(X\). Then
If \(j\ne k\), then every zero-extended Kraus operator satisfies \([\widetilde A_{j,a}X\widetilde A_{j,a}^\dagger ]_{(k,b),(k,c)}=0\). Hence only the \(j=k\) terms survive, and \([\mathcal C(X)]_{kk}=\sum _a A_{k,a}X_{kk}A_{k,a}^\dagger \).
If \(k\ne l\), then \(\mathcal C(X)_{kl}=0\).
For every \((j,a)\) and \(k\ne l\), the zero-extended operator satisfies \([\widetilde A_{j,a}X\widetilde A_{j,a}^\dagger ]_{(k,b),(l,c)}=0\). Therefore \([\mathcal C(X)]_{kl}=\sum _{j,a}0=0\).
Suppose that for every \(k\) the sectorwise Kraus family resolves the identity, \(\sum _a A_{k,a}^\dagger A_{k,a}=I_{H_k}\). Then the orthogonally controlled map \(\mathcal C\) is trace-preserving and completely positive.
Each embedded operator has support in one summand, and therefore
Apply Theorem 26.1.22 to the combined rectangular Kraus family \((\widetilde A_{k,a})_{k,a}\).
For a matrix \(X\) on \(A\otimes B\), the right partial trace is the linear map from matrices on \(A\otimes B\) to matrices on \(A\) given by
After the retained and discarded subspins have been regrouped as \(A\otimes B\), this is the partial-trace ingredient of the maps \(\mathcal T_0\) and \(\mathcal S_0\) in [ CPGSV16 , Appendix C.2, lines 1521–1522 and 1547 ] .
For square matrices \(X\) on \(A\) and \(Y\) on \(B\), \(\operatorname{tr}_B(X\otimes Y)=\operatorname{tr}(Y)X\).
For all \(i,j\), \([\operatorname{tr}_B(X\otimes Y)]_{ij} =\sum _t X_{ij}Y_{tt}=\operatorname{tr}(Y)X_{ij}\).
The map \(\operatorname{tr}_B\) is trace-preserving and completely positive for arbitrary finite-dimensional spaces \(A\) and \(B\).
For each basis vector \(e_k\) of \(B\), define \(V_k:A\otimes B\to A\) by \(V_k(e_i\otimes e_\ell )=\delta _{k\ell }e_i\). Then
Theorem 26.1.22 applies.
Let \(\rho \) be a matrix on a finite-dimensional space \(B\). The state-preparation map from matrices on \(A\) to matrices on \(A\otimes B\) is
This is the elementary preparation operation used in the maps \(\mathcal T_1\) and \(\mathcal S_1\) of [ CPGSV16 , Appendix C.2, lines 1527–1533 and 1551–1555 ] .
Let \(R=\sqrt\rho \). For an orthonormal basis \((e_j)_j\) of \(B\), define rectangular operators \(A_j:A\to A\otimes B\) by \(A_j(e_a)=e_a\otimes Re_j\).
If \(\rho \succeq 0\) and \(A_j(e_a)=e_a\otimes \sqrt\rho \, e_j\), then
Put \(R=\sqrt\rho \). Since \(R\) is Hermitian and \(R^2=\rho \), the \(((a,s),(b,t))\) entry of the left-hand side is
If \(\rho \geq 0\) and \(\operatorname{tr}(\rho )=1\), then \(\sum _j A_j^\dagger A_j=I_A\).
Since \(R\) is Hermitian and \(R^2=\rho \), the diagonal entries of the sum are \(\sum _{j,t}\overline{R_{tj}}R_{tj} =\operatorname{tr}(R^2)=\operatorname{tr}(\rho )=1\), while its off-diagonal entries vanish.
If \(\rho \geq 0\), then the map \(X\mapsto X\otimes \rho \) is completely positive. No trace normalization is required.
If \(\rho \geq 0\) and \(\operatorname{tr}(\rho )=1\), then \(\mathcal P_\rho :X\mapsto X\otimes \rho \) is trace-preserving completely positive.
For an MPO tensor \(M\), a length \(N\), and a virtual operator \(X\), define the physical operator \(M_N(X)\) by
This is linear in \(X\). The displayed order chooses the cut of the cyclic virtual contraction immediately before the first site, so that \(X\) follows the last site. The one- and two-site operators are constructed in [ CPGSV16 , lines 638–654 and Definition 4.1 ] . When \(M=\mathcal K\), write these operators as \(\mathcal K_N(X)\); the cases \(N=2,3\) occur in [ CPGSV16 , Proposition C.7, lines 1510–1516 ] .
Closing the virtual legs with the identity gives the periodic MPO operator: \(M_N(1)=\rho ^{(N)}(M)\).
For every pair of physical words \(\sigma ,\tau \),
Under the canonical identifications of one-site configurations with physical indices and two-site configurations with pairs of physical indices,
Under these identifications, the length-one and length-two word evaluations are \(M^{ij}\) and \(M^{i_0j_0}M^{i_1j_1}\), respectively.
The general three-site physical closure has coefficients
For \(M=\mathcal K\), this is the operator \(\mathcal K_3(X)\) in [ CPGSV16 , Proposition C.7, lines 1510–1516 ] .
The length-three word evaluation is \(M^{i_0j_0}M^{i_1j_1}M^{i_2j_2}\).
26.2 Renormalization fixed points
The following is the mixed-state fixed-point condition of [ CPGSV16 , Definition 4.1 ] . Its two local channels extend to every longer periodic chain by acting on the first one or two sites and leaving the remaining sites unchanged.
An MPO tensor \(M\) is a renormalization fixed point if there exist two trace-preserving completely positive maps \(\mathcal{S}, \mathcal{T}\) on the physical indices such that \(\mathcal{S}[M_2(X)]=M_1(X)\) and \(\mathcal{T}[M_1(X)]=M_2(X)\) for every virtual operator \(X\), where \(M_1(X)_{ij}=\operatorname{tr}(M^{ij}X)\) and \(M_2(X)_{(i_1i_2)(j_1j_2)} =\operatorname{tr}(M^{i_1j_1}M^{i_2j_2}X)\) are the one- and two-site physical operators obtained by closing one or two tensors with \(X\). This condition is distinct from idempotence of the doubled-index transfer map for general mixed states. Its source zero-correlation-length consequence concerns the physical-trace transfer and is proved below. This is [ CPGSV16 , Definition 4.1, line 657 ] . On the open coefficient tensors the two maps have types:
![\begin{tenkzcd}[maps, species={channel},
column sep=10mm, row sep=8mm]
\tnpic[inline, physical=updown]{
\tn[mpo]{M}
} &
\tnpic[inline, physical=updown]{
\tn[mpo]{M} & \tn[mpo]{M}
} \\
\tnpic[inline, physical=updown]{
\tn[mpo]{M} & \tn[mpo]{M}
} &
\tnpic[inline, physical=updown]{
\tn[mpo]{M}
}
\tnarrow[from={(1,1)}, to={(1,2)}, species=channel]{\mathcal T}
\tnarrow[from={(2,1)}, to={(2,2)}, species=channel]{\mathcal S}
\end{tenkzcd}](tenkz_svg/tenkz-617ff3b874aab1c8.svg)
Let \(R_1\) identify an \((N+1)\)-tuple with its first coordinate and the remaining \(N\)-tuple. Let \(R_2\) identify an \((N+2)\)-tuple with its first two coordinates and the remaining \(N\)-tuple.
Let \(N\) count the sites that are left unchanged. For physical maps \(\mathcal T:M_d\to M_{d^2}\) and \(\mathcal S:M_{d^2}\to M_d\), define
using \(R_1\) and \(R_2\). Thus \(N=0\) leaves no spectator site, but the maps still act on a one-site or two-site operator; no empty ring occurs.
For a physical word \(\sigma \) of length \(N+1\),
For a physical word \(\sigma \) of length \(N+2\),
All four identities hold by definition of the product-index equivalences.
If \(\mathcal S\) and \(\mathcal T\) are trace-preserving completely positive, then so are \(\widehat{\mathcal S}_N\) and \(\widehat{\mathcal T}_N\) for every \(N\).
If \((K_j)_j\) are Kraus operators for \(\mathcal T\) with \(\sum _j K_j^\dagger K_j=I_d\), then the localized map has Kraus operators \(K_j\otimes I_{d^N}\), and
The two regroupings are unitary conjugations and preserve complete positivity and the trace. The argument for \(\mathcal S\) is identical.
Suppose that \(\mathcal S[M_2(X)]=M_1(X)\) and \(\mathcal T[M_1(X)]=M_2(X)\) for every virtual operator \(X\). Then, for every \(N\) and \(X\),
In particular, these identities hold for every periodic MPO operator. Hence the two maps supplied by a renormalization fixed point act on rings of every length as required in the proof of [ CPGSV16 , Appendix C, lines 1333–1341 ] .
For words \(u,v\) on the \(N\) spectator sites, write \(M^{u,v}\) for their virtual word evaluation and write \([Y]^{(r)}_{u,v}\) for the block of an operator \(Y\) on the first \(r\) sites. Regrouping the first site gives
Therefore
Equality of all blocks proves the refinement identity. Interchanging \(M_1\), \(M_2\) and using \(\mathcal S[M_2(Y)]=M_1(Y)\) proves the coarsening identity. Setting \(X=1\) and using \(M_N(1)=\rho ^{(N)}(M)\) (Lemma 26.1.39) gives the periodic-ring identities
26.3 Pure-state recovery inside the MPO formalism
An MPS tensor \(A=\{ A^i\} _{i=0}^{d-1}\) determines an MPO tensor by placing \(A^i\) on the diagonal in the bra and ket indices:
Equivalently, \(M^{ii}=A^i\) and \(M^{ij}=0\) for \(i\ne j\).
The transfer map of the diagonal MPO associated to \(A\) agrees with the original MPS transfer map: \(\mathcal{E}_{A^{\mathrm{MPO}}}=\mathcal{E}_A\).
In the double sum defining \(\mathcal{E}_{A^{\mathrm{MPO}}}\), all off-diagonal terms vanish because \(M^{ij}=0\) for \(i\ne j\), while the diagonal terms are exactly \(A^iX(A^i)^\dagger \). Thus only the sum over \(i\) remains, which is the defining formula for \(\mathcal{E}_A\).
For an MPS tensor \(A\), the diagonal MPO embedding has idempotent transfer map if and only if \(A\) is an RFP:
Both conditions assert idempotence of the corresponding transfer map. Theorem 26.3.2 identifies those transfer maps, so the two predicates are equivalent.
For an MPS tensor \(A\), the diagonal MPO embedding has idempotent transfer map if and only if \(A\) has zero correlation length.
26.4 Zero correlation length
The physical-trace transfer of an MPO tensor \(M\) is the virtual matrix obtained by contracting the ket and bra physical legs of one tensor:
This is the transfer object appearing in the zero-correlation-length condition of [ CPGSV16 , Definition 4.2, lines 736–741 ] .
An MPO tensor has source zero correlation length when, for some real number \(\lambda {\gt}0\), the following conditions hold:
The nonzero condition excludes the degenerate zero transfer, and the positive scalar makes the condition invariant under rescaling of \(M\).
If \(\mathcal T_M\ne 0\) and \(\mathcal T_M^2=\mathcal T_M\), then \(M\) has source zero correlation length.
This is the preceding definition with \(\lambda =1\).
![\begin{tenkz}[physical=updown, trace=physical]
\tn[mpo]{M}
\end{tenkz}](tenkz_svg/tenkz-e1ab2eb94906b37e.svg)
\( = \) ![\begin{tenkz}[physical=updown, trace=physical]
\tn[mpo]{M} & \tn[mpo]{M}
\end{tenkz}](tenkz_svg/tenkz-1f492d0db29d6b46.svg)
This is the normalized identity in [ CPGSV16 , Definition 4.2, lines 736–741 ] ; Definition 26.4.2 permits the positive factor \(\lambda \).
If \(M\) has source zero correlation length, then there exists \(\lambda {\gt}0\) such that \(\lambda ^{-1}\mathcal T_M\) is idempotent.
From source zero correlation length, choose \(\lambda {\gt}0\) with \(\mathcal T_M^2=\lambda \, \mathcal T_M\). Then
An MPO tensor \(M\) satisfies the doubled-index transfer condition when its completely positive transfer map is idempotent:
This is the condition obtained from the doubled-index MPS view. It is not the physical-trace condition of Definition 26.4.2.
An MPO tensor satisfies the doubled-index transfer condition if and only if its doubled-index MPS view is a renormalization fixed point.
By Lemma 23.2.2, the transfer map of \(M\) agrees with the transfer map of its doubled-index MPS view. Thus \(\mathcal{E}_M\circ \mathcal{E}_M=\mathcal{E}_M\) holds exactly when the transfer map of that doubled-index MPS tensor is idempotent, which is the RFP condition.
If \(M\) is a renormalization fixed point via trace-preserving maps, then \(\mathcal T_M^2=\mathcal T_M\). This proves the zero-correlation-length component of implication (i)\(\Rightarrow \)(ii) in [ CPGSV16 , Theorem 4.9, lines 851–893 ] . The corresponding calculation appears in the proposition “RFP implies ZCL and SAL” in Appendix C, lines 1333–1340; the part establishing saturation of the area law is separate.
For every virtual matrix \(X\), the one-site and two-site closures satisfy
Write \(\Phi \) for the one-to-two trace-preserving map \(\mathcal T\) of Definition 26.2.1; this is distinct from the physical-trace transfer \(\mathcal T_M\). Then \(\Phi [M_1(X)]=M_2(X)\), and hence
Nondegeneracy of the trace pairing now gives \(\mathcal T_M^2=\mathcal T_M\).
If \(M\) is a renormalization fixed point via trace-preserving maps and \(\mathcal T_M\ne 0\), then \(M\) has source zero correlation length.
Apply the preceding idempotence theorem and take the positive scalar in Definition 26.4.2 to be \(1\).
MPO transfer-map idempotence is exactly the doubled-index transfer condition of Definition 26.4.5.
Both conditions assert \(\mathcal{E}_M\circ \mathcal{E}_M=\mathcal{E}_M\); they are definitionally identical.
MPO transfer-map idempotence is equivalent to the pure-state RFP condition for the doubled-index MPS tensor.
Unfold transfer-map idempotence to its definitional equivalent, the doubled-index transfer condition, and apply Theorem 26.4.6.
26.5 Purification fixed points
Given matrices \(A^{(i,k)}\) with a spin index \(i\) and an ancillary index \(k\), the associated MPS tensor on the product physical space is \(\widehat A^{(i,k)}=A^{(i,k)}\). This is the tensor whose finite-chain vector appears in the purification formula of [ CPGSV16 , lines 744–751 ] .
For a purifying spin–ancilla tensor \(A\), the reduced spin density matrix is defined by
This is the coefficient form of the ancillary trace in [ CPGSV16 , eq. (4.7), line 751 ] .
For a matrix \(X\) on the spin–ancilla space, the ancillary trace map is
This is the one-site trace over ancillary degrees of freedom appearing in [ CPGSV16 , lines 751 and 761–764 ] .
For every ancillary dimension, the one-site ancillary trace map is trace-preserving and completely positive.
This is Lemma 26.1.31 with \(A\) the spin space and \(B\) the ancillary space.
An MPO tensor \(M\) satisfies the global purification equation with a purifying tensor \(A\) if \(\rho ^{(N)}(M)=\rho ^{(N)}_p(A)\) for every positive length \(N\). Equivalently, for every positive length \(N\),
as in [ CPGSV16 , line 751 ] .
For every ancillary dimension, the spin reduction obtained by tracing the ancillary index is trace-preserving and completely positive.
An MPO tensor \(M\) has a bare global purification RFP witness if it satisfies the global purification equation for some purifying spin–ancilla tensor \(A\), and \(\widehat A\) is a pure-state renormalization fixed point: \(\mathcal{E}_{\widehat A}\circ \mathcal{E}_{\widehat A}=\mathcal{E}_{\widehat A}\). This records the displayed conditions of Definition 4.3 in [ CPGSV16 , lines 756–764 ] . It does not include the nondegeneracy implicit when the subsequent theorem speaks of density operators.
An MPO tensor satisfies the bare global purification RFP predicate if it has a positive-length global purification RFP witness. Equivalently, it admits a purifying tensor \(A\) such that, for every positive length \(N\),
and \(\widehat A\) is a pure-state renormalization fixed point. These are the displayed conditions of Definition 4.3 in [ CPGSV16 , lines 756–764 ] ; the nondegenerate tensor-level predicate is Definition 26.5.13.
A purification RFP is viewed together with trace-preserving spin reduction by applying Corollary 26.5.7 to the ancillary trace map appearing in its purification equation. This records the structure obtained after tracing the ancillary indices in [ CPGSV16 , lines 761–764 ] , without adding a separate hypothesis to Definition 4.3.
Recording the trace-preserving ancillary reduction does not change the purification RFP condition: \(\mathrm{PRFP}_{\mathrm{tp}}(M)\Longleftrightarrow \mathrm{PRFP}(M)\).
Immediate from the equivalence in Definition 26.5.10.
An MPO tensor \(M\) satisfies the local purification RFP condition if it is an LPDO whose purifying tensor \(A\), viewed as a matrix product state tensor on the combined spin–ancilla index, is a pure-state renormalization fixed point. Thus
This local condition is motivated by the purification tensor formula [ CPGSV16 , lines 744–747 ] , but it is a one-site tensor-level condition rather than the global finite-chain definition in Definition 26.5.5.
An MPO tensor is a nondegenerate purification renormalization fixed point if it is the local ancillary contraction of a pure-state renormalization fixed point and \(\mathcal T_M=\sum _i M^{ii}\ne 0\). The nonzero condition records the nondegeneracy implicit in the density operators of [ CPGSV16 , lines 744–786 ] and excludes the zero purifying tensor.
Every tensor satisfying the local purification RFP condition generates matrix product density operators.
Dropping the renormalization-fixed-point condition on the purifying tensor leaves the local purification structure, so \(M\) is an LPDO; the conclusion follows from Theorem 23.3.3.
There is an MPO tensor satisfying the local purification RFP condition for which the literal transfer-map idempotence \(\mathcal{E}_M\circ \mathcal{E}_M=\mathcal{E}_M\) fails.
Take the diagonal purification at \(d=d_K=2\) and \(D=D'=1\) with amplitudes \(A=[\tfrac {1}{\sqrt2},0,0,\tfrac {1}{\sqrt2}]\). The purifying tensor is a pure-state renormalization fixed point, since \(\sum |A|^2=1\) makes its transfer map the identity. The ancilla contraction \(M^{ij}=\sum _k A^{(i,k)} \overline{A^{(j,k)}}\) gives the scalar entries \(M^{00}=M^{11}=\tfrac 12\) (off-diagonal \(0\)), so the induced transfer map is \(\tfrac 12\cdot \operatorname{id}\) and \(\mathcal{E}_M\circ \mathcal{E}_M=\tfrac 14\cdot \operatorname{id}\neq \mathcal{E}_M\). The trace contraction has dropped the leading eigenvalue from \(1\) to \(\tfrac 12\); the literal zero-correlation-length condition is therefore strictly stronger than the source’s normalized one.
For the maximally mixed witness \(M_{\mathrm w}\), closing the ket and bra physical legs gives \(\mathcal T_{M_{\mathrm w}} =M_{\mathrm w}^{00}+M_{\mathrm w}^{11} =\tfrac 12+\tfrac 12=1\).
This is the direct computation of the two nonzero diagonal entries of the witness tensor.
The maximally mixed witness \(M_{\mathrm w}\) has source zero correlation length.
Lemma 26.5.16 gives \(\mathcal T_{M_{\mathrm w}}=1\). The identity is nonzero and idempotent, so source zero correlation length holds with \(\lambda =1\).
Suppose that \(M\) is the ancillary contraction of a fixed spin–ancilla tensor \(A\) through a bond identification \(e\):
Then the purification tensor \(\widehat A\) is a pure-state renormalization fixed point if and only if \(\mathcal T_M^2=\mathcal T_M\). This is the normalized tensor-level equivalence underlying the purification theorem in [ CPGSV16 , lines 744–786 ] .
Let \(K'\) be the transfer matrix of \(\widehat A\), let \(s\) interchange the two Kronecker factors, and let \(K\) denote the same Kronecker sum in the physical ordering. The local contraction gives
Since both reindexings are bijective and commute with matrix multiplication,
Injectivity of the transfer-matrix representation gives
If an MPO tensor \(M\) is the ancilla contraction of a pure-state renormalization fixed point, then its physical-trace transfer \(\mathcal T_M=\sum _i M^{ii}\) is idempotent: \(\mathcal T_M\mathcal T_M=\mathcal T_M\).
Apply the forward implication of Theorem 26.5.18 to the local purifying tensor and bond identification.
An MPO tensor satisfies the local purification RFP condition if and only if it has a local purification and its physical-trace transfer is idempotent:
Choose the local purifying tensor and bond identification. The equivalence is Theorem 26.5.18 for that fixed purification.
An MPO tensor is a nondegenerate purification renormalization fixed point if and only if it is a local purification density operator (Definition 23.3.2) and
This is the normalized, nonzero structural characterization used in the forward PRFP-to-ZCL implication of [ CPGSV16 , lines 744–786 ] .
Combine the preceding equivalence with the nonzero clause in Definition 26.5.13.
Let \(M\) be the ancilla contraction of a pure-state renormalization fixed point, satisfying the local purification renormalization fixed point condition. If the physical-trace transfer \(\mathcal T_M=\sum _i M^{ii}\) is nonzero, then \(M\) has source zero correlation length.
Every nondegenerate purification renormalization fixed point has source zero correlation length.
Let \(M\) be the ancilla contraction of a purifying spin–ancilla tensor \(A\) through a bond identification \(e\), so that
Then at every length \(N\) the matrix product operator equals the ancillary trace of the pure spin–ancilla matrix product state, \(\rho ^{(N)}(M)=\rho ^{(N)}_p(A)\). This is the coefficient form of the purification equation of [ CPGSV16 , line 751 ] .
Applying the mixed-product property \((A\otimes B)(C\otimes D)=(AC)\otimes (BD)\) repeatedly, the length-\(N\) product of the contracted tensor entries distributes over the ancillary index sum:
Tracing this sum and using \(\operatorname{tr}(X\otimes \overline Y)=\operatorname{tr}(X) \overline{\operatorname{tr}(Y)}\) gives the coefficient of the ancillary trace at every \(\sigma ,\tau \). The equation holds at every length, so no positive-length restriction is needed.
Every MPO tensor satisfying the local purification renormalization fixed point condition is a purification renormalization fixed point in the sense of Definition 4.3 of [ CPGSV16 , lines 756–764 ] .
The local condition supplies a purifying tensor \(A\) that is a pure-state renormalization fixed point and whose ancilla contraction is \(M\). By Theorem 26.5.24 this contraction satisfies the global purification equation at every positive length, so the same \(A\) is a purification RFP witness.
Every nondegenerate purification renormalization fixed point satisfies the positive-length global purification predicate of Definition 4.3 in [ CPGSV16 , lines 744–758 ] .
Forget the nonzero condition and apply Theorem 26.5.25.
There is an MPO tensor satisfying the bare positive-length global purification RFP predicate which does not have source zero correlation length.
Take the zero tensor at spin, ancillary, and bond dimensions one, purified by the zero spin–ancilla tensor. Its transfer map is idempotent and every positive-length density operator is zero, so the global purification equation holds. Its physical-trace transfer is zero, which is excluded by source zero correlation length.
There is an MPO tensor which is an MPDO, satisfies the positive-length global purification RFP predicate, and has nonzero physical-trace transfer, but does not have source zero correlation length.
Take one physical state and bond dimension three, with the only tensor entry
Since \(\operatorname{tr}(Q^N)=1\) for every \(N{\gt}0\), all positive-length MPOs equal the scalar density operator \(1\) and are purified by the scalar pure-state RFP tensor \(A=1\). At length zero the MPO is the positive scalar \(3\), so the tensor is an MPDO at every length. Its physical-trace transfer is the nonzero matrix \(Q\). However, \(Q^2=\operatorname{diag}(1,0,0)\) is not a positive scalar multiple of \(Q\), because the \((2,3)\) entry is lost on squaring. Thus source ZCL fails. The global equation cannot detect this nilpotent bond sector.
26.6 Saturation of the area law
For a fixed system size \(N\), a block length \(L\le N\), and an MPO tensor \(M\) such that \(\rho ^{(N)}(M)\) is positive semidefinite, let
denote the normalized state and the reduced state of the first \(L\) of \(N\) spins. The \(L\)-block entropy is \(S_L^{(N)}(M)=S(\rho _L^{(N)}(M))\).
For the same fixed system size \(N\), block length \(L\le N\), and positive semidefiniteness of \(\rho ^{(N)}(M)\), the mutual information between a block of \(L\) spins and the rest is \(I_L=S_L^{(N)}(M)+S_{N-L}^{(N)}(M)-S_N^{(N)}(M)\).
An MPO tensor \(M\) that generates an MPDO and whose positive-length finite-chain operators \(\rho ^{(N)}(M)\) all have nonzero trace (so the normalized states are well-defined) saturates the area law if \(I_L=I_{L+1}\) for all \(1\le L{\lt}\lfloor N/2\rfloor \) and all \(N\); that is, the mutual information is constant, \(I_1=I_2=\cdots =I_{\lfloor N/2\rfloor }\). This is [ CPGSV16 , Definition 4.6, line 811 ] .
Let \(a,n\in \mathbb N\), and let \(f\) be defined on the integer interval \([0,n]\). If \(f(m)=f(m+1)\) for every \(a\le m{\lt}n\), then \(f(i)=f(j)\) for all \(i,j\in [a,n]\).
Compose the successive equalities from the smaller index to the larger.
If an MPO tensor \(M\) saturates the area law, then the mutual informations coincide, \(I_L=I_{L'}\), for all \(1\le L,L'\le \lfloor N/2\rfloor \).
The defining condition supplies \(I_m=I_{m+1}\) throughout the interval \(1\le m{\lt}\lfloor N/2\rfloor \). Apply Lemma 26.6.4.
Let the \(N\)-site operator generated by \(M\) be positive semidefinite, and let three consecutive regions have lengths \(a\), \(b\), and \(c\), with \(a+b+c\leq N\). After reindexing the reduced state of the first \(a+b+c\) sites as a tripartite state on \(A\otimes B\otimes C\), its four entropies in strong subadditivity are
Write \(\rho ^{(k)}\) for the first-\(k\)-site reduced state, with \(\cong \) denoting equality up to a permutation of basis indices. Reindexing a matrix does not change its von Neumann entropy, giving the first equality \(S(ABC)=S_{a+b+c}\) immediately. Tracing out \(C\) leaves the first \(a+b\) sites, \(\operatorname{tr}_C(\rho _{ABC})\cong \rho ^{(a+b)}\), so \(S(AB)=S_{a+b}\). Tracing out \(A\) and \(C\) leaves the middle \(b\) sites, while tracing out \(A\) leaves the final \(b+c\) sites; translation invariance gives
Hence \(S(B)=S_b\) and \(S(BC)=S_{b+c}\).
Let \(M\) generate an MPDO whose finite-chain operator \(\rho ^{(N)}(M)\) is positive semidefinite with nonzero trace. For every block length \(L\) with \(2L+1\le N\), one has \(I_L\le I_{L+1}\). The implication \(1\le L{\lt}\lfloor N/2\rfloor \Longrightarrow 2L+1\le N\) shows that this contains the range of [ CPGSV16 , Proposition 4.5, line 801 ] .
Apply strong subadditivity to the reduced state of the first \(N-L\) spins, split into three contiguous segments of lengths \(1\), \(L\), and \(N-2L-1\). Tracing the appropriate segments yields the four block entropies \(S_{N-L}\), \(S_L\), \(S_{L+1}\), and \(S_{N-L-1}\), where the cyclic invariance of \(\rho ^{(N)}(M)\) identifies the entropy of a contiguous block with that of any block of the same length. Strong subadditivity then gives \(S_{N-L}+S_L\le S_{L+1}+S_{N-L-1}\), which rearranges to \(I_L\le I_{L+1}\) after subtracting \(S_N\).
Let \(M\) have physical dimension two and bond dimension three, with
The tensor in Definition 26.6.8 generates the positive operators
Their traces are \(2+(-1)^N\). The all-zero diagonal entry of the normalized one-site reduced state is therefore one for odd \(N\) and \(1/3\) for positive even \(N\). In particular, the normalized periodic one-site reduced states do not converge as \(N\to \infty \).
Thus normalized periodic marginals need not converge under the hypotheses of [ CPGSV16 , Proposition 4.5, lines 801–806 ] . The mutual information of this same family is computed in docs/paper-gaps/cpgsv17_mpdo_mutual_information_bound.tex.
A contraction around the virtual loop vanishes unless the bra and ket configurations agree and are both constant. The all-zero configuration receives weight one. The all-one configuration receives the sum of the two virtual-sector weights, namely \(1+(-1)^N\). This gives the displayed diagonal form. Its coefficients are non-negative: the second coefficient is zero for odd \(N\) and two for even \(N\). Hence every positive-length operator is positive semidefinite, and summing its two possible diagonal entries gives the trace \(2+(-1)^N\).
After normalization, the all-zero weight is consequently one at odd lengths and \(1/3\) at positive even lengths. Tracing out all but the first site does not change this entry, since among the configurations beginning with zero only the all-zero configuration has nonzero weight. The two subsequences of one-site marginals therefore have distinct all-zero entries, so the sequence cannot converge.
Fix \(L\geq 1\). For every \(N{\gt}L\), the tensor in Definition 26.6.8 satisfies
Consequently the sequence \(K\mapsto I_L^{(L+K+1)}\) does not converge, giving the counterexample described in [ CPGSV16 , Proposition 4.5, lines 801–806 ] .
At odd \(N\), the normalized state and both of its nonempty marginals are pure, so all three terms in \(I_L^{(N)}=S_L+S_{N-L}-S_N\) vanish. At even \(N\), the full state and both marginals have the same two nonzero eigenvalues \(1/3\) and \(2/3\). Each of their entropies is therefore \(h_2(1/3)\), and hence \(I_L^{(N)}=h_2(1/3)+h_2(1/3)-h_2(1/3)=h_2(1/3)\). The binary entropy is strictly positive at \(1/3\). Thus the odd and even subsequences have distinct constant values.
For a linear map \(\Psi _B\) and a bipartite matrix \(\rho _{AB}\), define
where \(\Sigma \) denotes the canonical exchange of the two tensor factors.
For linear maps \(\Phi _A\) and \(\Psi _B\), set
If \(\Phi _A\) is trace-preserving completely positive and \(\rho _{AB}\geq 0\), then \((\Phi _A\otimes \operatorname{id}_B)(\rho )\geq 0\).
If \(\Psi _B\) is trace-preserving completely positive and \(\rho _{AB}\geq 0\), then \((\operatorname{id}_A\otimes \Psi _B)(\rho )\geq 0\).
If \(\Phi _A\) and \(\Psi _B\) are trace-preserving completely positive and \(\rho _{AB}\geq 0\), then \((\Phi _A\otimes \Psi _B)(\rho )\geq 0\).
Let \(\rho _{AB}\) be a bipartite density operator and let \(\Phi _A\) be a trace-preserving completely positive map whose input and output matrix algebras may have different dimensions. Then \(I(A':B)_{(\Phi _A\otimes \operatorname{id}_B)(\rho )}\leq I(A:B)_\rho \).
Choose a rectangular Stinespring isometry for \(\Phi _A\). Conjugating by \(W=V_A\otimes \operatorname{id}_B\) gives \(\omega _{A'EB}=W\rho _{AB}W^\dagger \). Since \(W=V_A\otimes \operatorname{id}_B\) and \(V_A^\dagger V_A=\operatorname{id}_A\), its marginals satisfy
Entropy is preserved by the isometry \(W\), and also by \(V_A\) on the first marginal. Therefore
Hence \(I(A'E:B)_\omega =I(A:B)_\rho \). The defining Stinespring identity is \(\operatorname{tr}_E(\omega _{A'EB})=(\Phi _A\otimes \operatorname{id}_B)(\rho _{AB})\). Strong subadditivity in the form \(I(R:A')\leq I(R:A'E)\) proves the stated inequality with \(R=B\).
Let \(\rho _{AB}\) be a bipartite density operator and let \(\Psi _B\) be a trace-preserving completely positive map whose input and output matrix algebras may have different dimensions. Then \(I(A:B')_{(\operatorname{id}_A\otimes \Psi _B)(\rho )}\leq I(A:B)_\rho \).
Exchange the two tensor factors and apply Theorem 26.6.16.
Let \(\rho _{AB}\) be a bipartite density operator and let \(\Phi _A\) and \(\Psi _B\) be trace-preserving completely positive maps whose input and output matrix algebras may have different dimensions. Then \(I(A':B')_{(\Phi _A\otimes \Psi _B)(\rho )}\leq I(A:B)_\rho \).
Apply the two one-sided inequalities successively.
For \(N=L+K\), split a physical configuration into consecutive intervals of lengths \(L\) and \(K\), flatten each interval to a finite index, and denote the resulting bipartite form of the normalized periodic MPO by \(\sigma ^{(N)}_{L:K}\). The flattened local maps on the two factors are denoted by \(\widehat{\mathcal T}_L\) and \(\widehat{\mathcal S}_K\).
Across the consecutive cut after the first \(L\) sites, the two marginals of \(\sigma ^{(N)}_{L:N-L}\) are the reduced states on \(L\) and \(N-L\) sites, up to the standard finite-index identifications. Consequently, \(I_L(M)\) is the ordinary bipartite mutual information of \(\sigma ^{(N)}_{L:N-L}\).
The first marginal is the defining partial trace over the last \(N-L\) sites. Periodic translation invariance identifies the second marginal with the reduced state on the first \(N-L\) sites. Entropy is invariant under the finite-index identifications, and retaining the full chain leaves the normalized state unchanged.
Suppose that \(N=(a+1)+(b+2)=(a+2)+(b+1)\). The local closure equations imply
Fixing the word on the second interval leaves a physical closure on the first interval, whose virtual boundary is the fixed word evaluation. Apply the appropriate global closure identity to this slice. Fixing the resulting word on the first interval gives the closure on the second interval. Cyclicity of the virtual trace identifies the two cuts of the periodic contraction. The reverse identity is the same argument with \(\mathcal S\) and \(\mathcal T\) interchanged.
Let \(M\) be a renormalization fixed point. If its \(N\)-site operator is positive semidefinite and has nonzero trace, then \(I(\sigma ^{(N)}_{a+1:b+2}) =I(\sigma ^{(N)}_{a+2:b+1})\). This is the channel calculation in [ CPGSV16 , Appendix C, lines 1333–1341 ] . For the forward implication proved here, the required nonzero trace follows from normalized BNT-refined horizontal form, the MPDO condition, and the renormalization maps; simplicity is not used. This horizontal hypothesis is stronger than literal CPSV canonical form.
Apply mutual-information data processing to the forward pair of local channels and to the reverse pair. The two transfer identities turn the resulting inequalities into opposite inequalities between the displayed mutual informations.
Let \(M\) be in normalized BNT-refined horizontal form, generate positive semidefinite ring operators, and satisfy the local renormalization fixed-point equations. This horizontal hypothesis is stronger than literal CPSV canonical form. Then \(\operatorname{tr}\rho ^{(N)}(M)\neq 0\) for every \(N\geq 1\).
Normalized BNT-refined horizontal form gives a nonzero sector compression for one positive length. Positivity makes its trace nonzero. The fixed-point equations make the physical-trace transfer idempotent, so the trace of its positive powers is independent of the positive exponent.
Let \(M\) be a matrix product density operator in normalized BNT-refined horizontal form which satisfies the local renormalization fixed-point equations of Definition 4.1. This horizontal hypothesis is stronger than literal CPSV canonical form. Then \(M\) saturates the area law: \(I_L(M)=I_{L+1}(M)\) whenever \(1\leq L{\lt}\lfloor N/2\rfloor \). This is the normalized BNT-refined specialization of the forward implication of Proposition propsimple in Appendix C of [ CPGSV16 ] . Simplicity is among the ambient hypotheses of Theorem 4.9 but is not used in this implication.
Write the cut as \((a+1):(b+2)\). The local channels transfer one site to obtain \((a+2):(b+1)\), and data processing in both directions gives equality of the two bipartite mutual informations. The consecutive-cut identification turns this into \(I_L=I_{L+1}\). The preceding lemma supplies the normalization at every positive chain length.
Let \(M\) be a matrix product density operator in normalized BNT-refined horizontal form which satisfies the local renormalization fixed-point equations of Definition 4.1. This horizontal hypothesis is stronger than literal CPSV canonical form. Then \(M\) has source zero correlation length and saturates the area law. This is the normalized-BNT-refined specialization of Proposition propsimple in Appendix C of [ CPGSV16 ] , lines 1333–1341.
The preceding nonvanishing lemma at length one rules out a zero physical-trace transfer. Trace preservation of the refinement map then gives source zero correlation length, while Theorem 26.6.24 gives saturation of the area law.
For a normalized positive semidefinite finite-chain operator, every block entropy is non-negative: \(0\le S_m\).
The reduced state of a block is a density matrix, and the von Neumann entropy of a density matrix is non-negative.
Let \(M\) generate an MPDO whose finite-chain operator \(\rho ^{(N)}(M)\) is positive semidefinite with nonzero trace. For every block length \(L\le N\), one has \(0\le I_L\).
Apply strong subadditivity with a trivial middle subsystem (equivalently, subadditivity) to the bipartition of the chain into the first \(L\) and last \(N-L\) spins: \(S_N+S_0\le S_L+S_{N-L}\). Since \(S_0\ge 0\) (Lemma 26.6.26), rearranging gives \(I_L=S_L+S_{N-L}-S_N\ge S_0\ge 0\).
For every block length \(L\le N\), the mutual information of the first \(L\) spins equals that of their complement: \(I_L=I_{N-L}\).
Both sides equal \(S_L+S_{N-L}-S_N\) by the symmetric definition \(I_L=S_L+S_{N-L}-S_N\), using \(N-(N-L)=L\).
A positive semidefinite matrix of trace one has positive rank.
If its rank were zero, every eigenvalue would vanish, contradicting that their sum is the trace, which equals one.
A positive semidefinite matrix of trace one and rank at most one has zero von Neumann entropy.
Its rank is positive by Lemma 26.6.29, hence it is exactly one. The rank bound gives \(S\le \log 1=0\), while entropy is non-negative.
For a normalized positive semidefinite finite-chain operator, the block entropy of the empty block vanishes: \(S_0=0\).
The reduced state of zero spins is one-dimensional, so its rank is at most one. Lemma 26.6.30 gives \(S_0=0\).
For a normalized positive semidefinite finite-chain operator, the mutual information of the empty block vanishes: \(I_0=0\).
By definition, \(I_0=S_0+S_N-S_N=S_0\), which is zero by Lemma 26.6.31.
For a block of \(L\) spin–ancilla pairs, regroup
and trace the ancillary factor. Denote this map by \(\operatorname{Tr}_{\mathrm{anc},L}\). It is the blockwise analog of the one-site ancillary trace in Definition 26.5.3.
The map \(\operatorname{Tr}_{\mathrm{anc},L}\) is trace-preserving and completely positive.
The reindexing equivalence is a channel, as is the partial trace over the ancillary factor. Their composition is therefore trace-preserving and completely positive.
If \(X\) is an operator on a spin–ancilla block, then for \(i,j\in [d]^L\),
For \(i,j\in [d]^L\) and \(x,y\in [d]^K\), applying the ancillary traces to both blocks across the cut \(L\mid K\) gives
After regrouping the spin and ancillary factors, the definition of the partial trace gives the first coefficient identity. Applying this identity to both tensor factors gives the displayed iterated sum; the two finite sums may be interchanged.
Fix a local-purification representation
with purifying MPS tensor \(\widetilde A\) of physical dimension \(d d_K\) and bond dimension \(D'\). For every \(N=L+K\), tracing all ancillary degrees of freedom gives
Define the normalized-form purifying operator by
where the inverse scalar is taken with the convention \(0^{-1}=0\). After reindexing across the cut \(L\mid K\),
This identity also holds when the common trace vanishes, in which case both normalized-form operators are zero.
This is the finite-chain channel identity from the purification preceding equation (4) of [ WVHC08 ] , under the explicit local purification of [ CPGSV16 , Section 4.3 ] .
The coefficient formula for one block gives
Trace preservation gives equality of the two normalizing traces. Splitting \(\kappa \) into its left and right restrictions changes the last sum into the two independent sums in Theorem 26.6.35, which proves the normalized identity across \(L\mid K\).
Let \(M\) be an LPDO. For every decomposition \(N=L+K\), there exist an ancillary dimension \(d_K\), a purifying bond dimension \(D'\), and a local-purification family \(A\). Let \(\widetilde A\) be its associated purifying MPS tensor. Then
Choose a local-purification representation of \(M\) and apply Theorem 26.6.36 to its purifying tensor.
Let \(M\) be an LPDO. There exist an ancillary dimension \(d_K\), a purifying bond dimension \(D'\), and a local-purification family \(A\), all independent of the cut, such that for every decomposition \(N=L+K\),
where \(\widetilde A\) is the purifying MPS tensor associated with \(A\).
Choose one local-purification representation of \(M\). For an arbitrary decomposition \(N=L+K\), apply Theorem 26.6.36 to the same purifying tensor.
Let \(M\) be an MPO tensor whose generated \(N\)-site operator is positive semidefinite. Let \(A\) be an MPS tensor of bond dimension \(D'\geq 1\) whose \(N\)-site vector is nonzero. Suppose that, across the cut \(L\mid (N-L)\), the normalized state generated by \(M\) is obtained from the normalized pure state of \(A\) by trace-preserving completely positive maps \(\Phi _L\) and \(\Phi _R\) on the two blocks:
Then \(I_L(M)\leq 4\log D'\).
Write \(\sigma _A^{(N)}\) for the normalized pure state of \(A\) in the \(L\mid (N-L)\) bipartition. Then
The equality is the channel-image hypothesis, the first inequality is data processing, and the last inequality is Proposition 26.6.77.
Fix a local-purification representation with purifying MPS bond dimension \(D'\). If the \(N\)-site operator is positive semidefinite and its purifying pure state is nonzero, then \(I_L(M)\leq 4\log D'\) for every \(L\leq N\).
Consequently, if \(M\) is an LPDO, \(N\geq 1\), and \(\operatorname{tr}\rho ^{(N)}(M)\neq 0\), then its local-purification representation supplies a bond dimension \(D'\) satisfying this estimate. The quantity \(D'\) is the bond dimension of the purifying MPS, not the MPO bond dimension \(D\).
By Theorem 26.6.36,
The first inequality is data processing and the second is the pure-state boundary estimate. Nonvanishing of the purifying pure state implies \(D'{\gt}0\). For an LPDO, positivity follows from Theorem 23.3.3, while the trace identity in Theorem 26.6.36 transfers the nonzero-trace hypothesis to the purifying pure state.
Split a periodic chain into consecutive blocks of lengths \(L\) and \(R\). For words \(x\in \{ 0,\ldots ,d-1\} ^L\) and \(y\in \{ 0,\ldots ,d-1\} ^R\), set
Then the ordinary complex rank of \(P\) satisfies \(\operatorname{rank}_{\mathbb {C}}P\le D^2\).
For each word, write \(A_x=M^{x_0x_0}\cdots M^{x_{L-1}x_{L-1}}\) and \(B_y=M^{y_0y_0}\cdots M^{y_{R-1}y_{R-1}}\). The identity
factors \(P\) through the \(D^2\)-dimensional space indexed by pairs \((a,b)\).
Let \(K\) be a field, let \(A\) be a finite matrix over \(K\), and let \(a\in K\) be nonzero. Then \(\operatorname{rank}_{K}(aA)=\operatorname{rank}_{K}A\).
Multiplication by \(a\) identifies the column spaces, with inverse given by multiplication by \(a^{-1}\).
Let \(K\subseteq L\) be a faithful field extension, let \(\iota :K\hookrightarrow L\) be the inclusion, and let \(A\) be a finite matrix over \(K\). Then \(\operatorname{rank}_{L}\! \left((\iota (A_{ij}))_{ij}\right)=\operatorname{rank}_{K}A\).
For every finite selection of columns, the selected columns are linearly independent over \(K\) if and only if their scalar extensions are linearly independent over \(L\). Selecting a basis from each column space proves the two rank inequalities.
Let \(X\) and \(Y\) be finite sets. A matrix \(P=(P_{x,y})_{x\in X,y\in Y}\) is a joint probability distribution if \(P_{x,y}\geq 0\) for every \(x\in X\) and \(y\in Y\), and
Its row and column marginals are respectively
For \(t{\gt}0\), set \(h(t)=-t\log t\), and set \(h(0)=0\). The entropy of a probability distribution \(a=(a_z)_{z\in Z}\) on a finite set is \(H(a)=\sum _{z\in Z}h(a_z)\). For a joint probability distribution \(P\) with row and column marginals \(p\) and \(q\), put
Let \(p=(p_z)_{z\in Z}\) be a probability distribution on a finite set \(Z\). Define \(h(t)=-t\log t\) for \(t{\gt}0\) and \(h(0)=0\). Then
Let \(k=\lvert \{ z\in Z:p_z\neq 0\} \rvert \). Jensen’s inequality for the concave function \(h\), with uniform weights on the support, gives
Multiplication by \(k\) proves the result, since values outside the support contribute \(h(0)=0\).
Let \(X\) and \(Y\) be finite sets and let \(P=(P_{x,y})_{x\in X,y\in Y}\) be a joint probability distribution. With natural logarithms, \(I(X:Y)_P\leq \log \operatorname{rank}_{\mathbb {R}}P\). Equivalently, with logarithms to base two, \(2^{I(X:Y)_P}\leq \operatorname{rank}_{\mathbb {R}}P\).
This is Theorem 4.1 of [ RV17 ] . If the number of nonzero rows exceeds the ordinary real rank, choose a nontrivial real linear relation among those rows. Nonnegativity forces the relation to have coefficients of both signs. Rescaling each row by \(1+\varepsilon \beta _x\) gives two endpoint distributions. For every \(y\in Y\), the row relation gives
Thus each endpoint preserves every column marginal, removes at least one row, and has no larger rank. Write \(h(t)=-t\log t\) for \(t{\gt}0\), with \(h(0)=0\), and put \(m=\min _x\beta _x{\lt}0{\lt}M=\max _x\beta _x\). Then
If \(S(\beta )\geq 0\), choose \(\varepsilon =-m^{-1}\geq 0\); otherwise choose \(\varepsilon =-M^{-1}\leq 0\). In either case \(\varepsilon S(\beta )\geq 0\), so the selected endpoint has mutual information no smaller than the original distribution. Repeating the construction leaves at most \(\operatorname{rank}_{\mathbb {R}}P\) nonzero rows. Finally, \(I(X:Y)\leq H(X)\), and the entropy of a distribution supported on \(k\) points is at most \(\log k\).
Let \(D\geq 1\), and let \(W\) be a real diagonal coefficient matrix across a periodic cut of an MPO with bond dimension \(D\). Suppose its scalar extension to \(\mathbb {C}\) is the cut matrix \(P^{\mathbb {C}}\) from Theorem 26.6.41. If \(Z{\gt}0\) and \(P=Z^{-1}W\) is a joint probability distribution, then \(I(X:Y)_P\leq 2\log D\).
Normalization does not change real rank by Theorem 26.6.42. Faithful scalar extension does not change rank by Theorem 26.6.43; hence Theorem 26.6.41 gives \(\operatorname{rank}_{\mathbb {R}}P\leq D^2\). Theorem 26.6.47 gives \(I(X:Y)_P\leq \log \operatorname{rank}_{\mathbb {R}}P\), and monotonicity of the logarithm gives the result.
For a chain of length \(N=L+R\), call \(\rho ^{(N)}(M)\) diagonal if \(\rho ^{(N)}(M)_{u,v}=0\) whenever \(u\ne v\). Call \(M\) globally diagonal if \(\rho ^{(N)}(M)\) is diagonal for every \(N{\gt}0\). For configurations \(x\) and \(y\) on the two consecutive blocks, define
For every MPO tensor \(M\) and cut lengths \(L,R\), the total diagonal coefficient mass satisfies \(Z=\operatorname{Re}\operatorname{tr}\rho ^{(L+R)}(M)\).
Pairing each configuration \(\sigma \) on \(L+R\) sites with its unique block split \((x,y)\) on \(L\) and \(R\) sites gives
Let \(D\geq 1\). Suppose that \(\rho ^{(L+R)}(M)\) is diagonal and positive semidefinite, and that \(Z{\gt}0\). Then \(W_{x,y}\geq 0\), the scalar extension of \(W\) to \(\mathbb {C}\) is the diagonal cut matrix of Theorem 26.6.41, and \(\widehat P\) is a joint probability distribution. Its classical mutual information satisfies \(I(X:Y)_{\widehat P}\leq 2\log D\).
Positive semidefiniteness makes every diagonal entry real and non-negative, so \(W_{x,y}\geq 0\) and its scalar extension recovers the complex diagonal cut matrix. Since \(Z\) is the sum of the entries of \(W\), \(\sum _{x,y}\widehat P_{x,y} =Z^{-1}\sum _{x,y}W_{x,y}=1\). Thus \(\widehat P\) is a joint probability distribution. Diagonality says that these coefficients constitute the full classical finite-chain state, and Theorem 26.6.48 gives the bound.
Let \(D\geq 1\), and let \(M\) generate diagonal positive semidefinite operators at every positive chain length. If \(L+R{\gt}0\) and the trace at length \(L+R\) is positive, then the classical mutual information of its normalized diagonal coefficients satisfies \(I(X:Y)\leq 2\log D\).
Global diagonality and positivity give the fixed-length hypotheses of Theorem 26.6.51. Since the chain operator is diagonal,
so the trace positivity hypothesis supplies the required normalization.
For a bipartite density operator \(\rho \in M_{d_A}(\mathbb {C})\otimes M_{d_B}(\mathbb {C})\), its operator-Schmidt rank is the least integer \(r\) for which there are matrices \(A_t\in M_{d_A}(\mathbb {C})\) and \(B_t\in M_{d_B}(\mathbb {C})\) satisfying
No Hermiticity or positivity condition is imposed on the factors. This is the definition in [ DlCDN19 , Equation (1) ] ; the same formula defines the rank of an arbitrary complex bipartite matrix.
Let \(M\) be an MPO tensor of operator bond dimension \(D\), and let \(N=L+K\). Opening the two virtual bonds at the consecutive cut gives
and
For left-block words \(x,x'\) and right-block words \(y,y'\), the factors are
and
This is the identity \(\operatorname{tr}(AB)=\sum _{a,b}A_{ab}B_{ba}\), with the normalization scalar absorbed into the left factor. The algebraic decomposition does not use positivity. Since \(0^{-1}=0\), it also remains valid when the trace vanishes; a nonzero trace is needed only to interpret \(\sigma ^{(N)}_{L:K}\) as the normalized physical state.
This is the two-virtual-boundary algebraic step toward the finite mutual-information estimate in [ CPGSV16 , Proposition 4.5 ] . It does not by itself prove \(I_L\leq 4\log D\): such a conclusion would require an entropy bound in terms of the operator-Schmidt rank.
Split each closed virtual word into its left and right products. Expanding the trace over the two exposed virtual indices gives the displayed sum of \(D^2\) product matrices. The minimality property of the operator-Schmidt rank then bounds it by the number of displayed terms.
Write \(\rho _{ij}\in M_{d_B}(\mathbb {C})\) for the blocks determined by a basis of the first factor, and define \(\mathcal R_\rho (X)=\sum _{i,j}X_{ij}\rho _{ij}\). Then
The range of \(\mathcal R_\rho \) is the span of the blocks. If \(\rho =\sum _{t=1}^{r}A_t\otimes B_t\), every block belongs to \(\operatorname{span}\{ B_1,\ldots ,B_r\} \), so the range dimension is at most \(r\). Conversely, choose a basis \(B_1,\ldots ,B_s\) of the block span and expand each block as \(\rho _{ij}=\sum _t(A_t)_{ij}B_t\). This gives \(\rho =\sum _{t=1}^{s}A_t\otimes B_t\), where \(s=\dim \operatorname{range}\mathcal R_\rho \).
Let \(\rho \geq 0\) be a bipartite complex matrix whose first marginal is faithful. Choose an eigenbasis in which \(\operatorname{tr}_B\rho =\operatorname{diag}(p_1,\ldots ,p_{d_A})\) with every \(p_i{\gt}0\), and define the linear map \(\Phi _\rho \) on matrix units by
Then \(\Phi _\rho \) is completely positive and trace preserving, \(\dim \operatorname{range}\Phi _\rho =\operatorname {OSR}(\rho )\), and application of \(\Phi _\rho \) to the first half of \(\sum _{i,j}\sqrt{p_ip_j}\, E_{ij}\otimes E_{ij}\), followed by restoring the order of the two factors, reconstructs \(\rho \).
Strict positivity of the \(p_i\) makes the entrywise input scaling invertible, so it does not change the range. Positivity of \(\rho \) gives a Kraus representation of \(\Phi _\rho \), while the displayed marginal equation gives trace preservation. Substitution on matrix units proves the reconstruction identity. This is the finite-dimensional Choi representation [ Cho75 ] with the input marginal absorbed into the canonical purification.
The proof of Proposition 4.5 of [ CPGSV16 ] invokes the uniform estimate \(I_L\leq 4\log D\) for every MPDO. Its cited argument, however, treats mixed tensor-network states obtained from local completely positive maps [ WVHC08 ] . After purifying those maps, data processing and the pure-state boundary estimate give
A periodic one-dimensional interval has two boundary bonds, and hence the last expression is \(4\log D\). A general positive MPO need not admit this local purification.
By Theorem 26.6.55, the two virtual bonds give an operator-Schmidt decomposition with at most \(D^2\) terms, but they do not bound the ordinary rank of either marginal. For example, a bond-one tensor can generate \(q^{\otimes N}\) for a full-rank one-site density matrix \(q\). Thus its \(L\)-site marginal has rank \(\operatorname{rank}(q)^L\), although its mutual information vanishes. The unrestricted estimate therefore requires a direct operator-Schmidt argument not supplied by the cited proof.
Diagonal operators cannot violate the finite-chain estimate. Indeed, Theorem 26.6.51 bounds by \(2\log D\) the classical mutual information of the normalized diagonal coefficients whenever the finite-chain operator is diagonal and positive semidefinite and has positive total mass. Thus any counterexample must have genuine quantum coherences. Theorem 26.6.56 identifies the faithful-marginal problem with a channel whose linear range has dimension \(\operatorname {OSR}(\rho )\); it does not supply the desired mutual-information inequality. The unrestricted quantum estimate remains open. The same example as Theorem 26.6.9 also makes the iterated mutual-information limit false; see docs/paper-gaps/cpgsv17_mpdo_mutual_information_bound.tex. The channel-image implication is Theorem 26.6.39, and the pure-state instance is Proposition 26.6.77 below. The locally purified case is Theorem 26.6.40.
For an MPS tensor \(A\), a system size \(N\), and a block length \(L\le N\), let
denote the normalized state and the reduced state of the first \(L\) spins. The \(L\)-block entropy is \(S_L^{(N)}(A)=S(\rho _L^{(N)}(A))\). The tensor saturates the area law if \(S_L^{(N)}(A)=S_{L+1}^{(N)}(A)\) for all \(1\le L{\lt}\lfloor N/2\rfloor \) and all \(N\). This is [ CPGSV16 , Definition 3.13, line 600 ] .
For an MPS tensor \(A\) on bond space \(\mathbb {C}^D\), define the associated MPO tensor \(M\) on bond space \(\mathbb {C}^{D^2}\) by \(M^{ij}=A^i\otimes \overline{A^j}\). This is the single-ancilla purification tensor of [ CPGSV16 ] .
For an MPS tensor \(A\), the pure-state operator \(|V^{(N)}(A)\rangle \! \langle V^{(N)}(A)|\) equals the MPDO generated by the tensor \(M^{ij}=A^i\otimes \overline{A^j}\), acting on a bond space of dimension \(D^2\):
This is the single-ancilla case of the purification picture of [ CPGSV16 ] ; it lets the MPDO block-entropy theory apply to pure-state block entropies.
The word evaluation of the doubled tensor is the Kronecker product of the word evaluation of \(A\) and its conjugate, \(M^{\sigma ,\tau }=A^\sigma \otimes \overline{A^\tau }\), so its trace factors as \(\operatorname{tr}(M^{\sigma ,\tau }) =\operatorname{tr}(A^\sigma ) \overline{\operatorname{tr}(A^\tau )} =\overline{V^{(N)}(A)_\tau }\, V^{(N)}(A)_\sigma \), the corresponding matrix element of \(|V^{(N)}(A)\rangle \! \langle V^{(N)}(A)|\).
For an MPS tensor \(A\), system size \(N\), and block length \(L\le N\), the MPDO block entropy of the tensor \(M^{ij}=A^i\otimes \overline{A^j}\) agrees with the pure-state block entropy: \(S_L^{(N)}(M)=S_L^{(N)}(A)\).
Lemma 26.6.60 gives \(\rho ^{(N)}(M)=|V^{(N)}(A)\rangle \! \langle V^{(N)}(A)|\), hence
Therefore the reduced block states agree:
Thus \(S_L^{(N)}(M) =S(\rho _L^{(N)}(M)) =S(\rho _L^{(N)}(A)) =S_L^{(N)}(A)\).
For an MPS tensor \(A\) and a block length \(L\le N\), the block entropies of a block and its complement coincide: \(S_L^{(N)}(A)=S_{N-L}^{(N)}(A)\).
The reduced state of the first \(L\) spins is \(\rho _L\propto WW^\dagger \), where \(W\) is the matrix of amplitudes \(\langle u\, w|V^{(N)}(A)\rangle \) with \(u\) the first \(L\) spins and \(w\) the remaining \(N-L\). By cyclic invariance of the amplitudes, the reduced state of the complement is \(\propto \overline{W^\dagger W}\). Cyclic invariance of the entropy, \(S(WW^\dagger )=S(W^\dagger W)\), together with invariance of the entropy under entrywise conjugation of a Hermitian matrix, \(S(\overline{W^\dagger W})=S(W^\dagger W)\), gives \(S_L^{(N)}(A)=S_{N-L}^{(N)}(A)\).
Let \(A\) be an MPS tensor with \(V^{(N)}(A)\neq 0\). For every block length \(L\) with \(2L+1\le N\), covering \(1\le L{\lt}\lfloor N/2\rfloor \), \(S_L^{(N)}(A)\le S_{L+1}^{(N)}(A)\). This is the pure-state area law of [ CPGSV16 , Section 3, line 599 ] .
Apply the block-entropy form of strong subadditivity, the inequality underlying Proposition 26.6.7, to the purification MPDO of \(A\), giving \(S_{N-L}+S_L\le S_{L+1}+S_{N-L-1}\). By the Schmidt symmetry \(S_{N-L}=S_L\) and \(S_{N-L-1}=S_{L+1}\), this reduces to \(S_L\le S_{L+1}\).
Let \(A\) be an MPS tensor of bond dimension \(D\). For a length-\(\ell \) configuration \(\sigma \) and a length-\(m\) configuration \(\tau \), write the operator-Schmidt factors as
The \(D^2\times D^2\) Gram matrix of the left factor equals the \(\ell \)-fold transfer map evaluated on a matrix unit:
where the sum runs over length-\(\ell \) configurations \(\sigma \). The Gram of the right factor \(R_m\) of an \(m\)-spin complement satisfies the bond-swapped identity
where the sum runs over length-\(m\) configurations \(\tau \).
The left Gram entry is \(\sum _\sigma \overline{(A^\sigma )_{a_1a_2}}(A^\sigma )_{b_1b_2}\) by definition of \(L_\ell \). Expanding
and reading off the \((b_1,a_1)\) entry gives the same sum, since
The right-factor formula is the same expansion with the two boundary bonds interchanged: the \((p,q)\) entry of \(R_mR_m^\dagger \) is \(\sum _\tau (A^\tau )_{p_2p_1}\overline{(A^\tau )_{q_2q_1}}\), which is the \((p_2,q_2)\) entry of \(\mathcal{E}_A^{\, m}(|p_1\rangle \! \langle q_1|)\).
Let \(A\) be a renormalization fixed point. For every \(L\ge 1\) and \(m\ge 1\), the operator-Schmidt Grams of the \(L\)-block and of the \(m\)-site complement are the one-site Grams:
By definition, a renormalization fixed point satisfies \(\mathcal{E}_A^{\, 2}=\mathcal{E}_A\), so \(\mathcal{E}_A^{\, k}=\mathcal{E}_A\) for every \(k\ge 1\). Applying Lemma 26.6.64 to the left and right Gram entries gives the two displayed identities.
Let \(A\) be an MPS tensor, let \(L\le N\), put \(m=N-L\), and write \(c=(\operatorname{tr}\rho ^{(N)})^{-1}\). Then the pure block entropy is the charpoly-root entropy sum of the \(D^2\times D^2\) bond environment:
The roots are counted with algebraic multiplicity.
The reduced block state has the exact factorization
Cyclic invariance of the charpoly-root entropy sum gives
which is the displayed formula.
If an MPS tensor \(A\) is a renormalization fixed point, then it saturates the area law: for every chain length \(N\) and every block length \(1\le L{\lt}\lfloor N/2\rfloor \), the pure-state block entropy is constant in the block size, \(S_L^{(N)}(A)=S_{L+1}^{(N)}(A)\). The source statement is [ CPGSV16 , Proposition 3.14, lines 606–608 ] . It assumes canonical form; the theorem above is the stronger pure-state implication obtained from transfer-map idempotence alone.
Write \(c=(\operatorname{tr}\rho ^{(N)})^{-1}\). Lemma 26.6.66 computes the entropy from the bond environment:
Since \(1\le L{\lt}\lfloor N/2\rfloor \), the four lengths \(L\), \(L+1\), \(N-L\), and \(N-(L+1)\) are positive. The RFP Gram-collapse lemma gives
Therefore the two bond environments agree:
The scalar \(c\) depends only on \(N\), so the two charpoly-root entropy sums are equal. Hence \(S_L^{(N)}(A)=S_{L+1}^{(N)}(A)\).
If an MPS tensor \(A\) is a renormalization fixed point, then all block entropies in the range \(1\le L\le \lfloor N/2\rfloor \) coincide:
This is the explicit constant-entropy chain of the area-law saturation [ CPGSV16 , Definition 3.13, line 600 ] , under the fixed-point hypothesis.
For an MPS tensor \(A\) of bond dimension \(D\) and a block length \(L\le N\), the reduced state \(\rho _L^{(N)}(A)\) of the first \(L\) spins has rank at most \(D^2\).
The reduced state is \(\rho _L\propto WW^\dagger \), where the amplitude matrix
factors through the two bond pairs cut by the block boundary. Writing the trace as a sum over the bond index pair \((a,b)\in \{ 1,\ldots ,D\} ^2\) exhibits \(W=L'R'\) with \(L'\) having \(D^2\) columns, so \(\operatorname{rank}\rho _L\leq \operatorname{rank}W\leq D^2\).
For an MPS tensor \(A\) with \(V^{(N)}(A)\neq 0\), the reduced state \(\rho _L^{(N)}(A)\) of the first \(L\) spins is a density matrix: it is positive semidefinite and has unit trace.
Positive semidefiniteness is preserved by the partial trace of the positive semidefinite normalized pure state, and the partial trace preserves the trace, which the normalization fixes to one.
Let \(A\) be an MPS tensor of bond dimension \(D\geq 1\) with \(V^{(N)}(A)\neq 0\). For every block length \(L\le N\), \(S_L^{(N)}(A)\leq 2\log D\), uniformly in \(L\) and \(N\). This is the area law of [ CPGSV16 , Section 3, line 599 ] : the block entropy is bounded by a constant independent of the block size.
For an MPS tensor \(A\) with \(V^{(N)}(A)\neq 0\) and every block length \(L\le N\), the pure-state block entropy is non-negative: \(0\le S_L^{(N)}(A)\).
The reduced state \(\rho _L^{(N)}(A)\) is a density matrix (Lemma 26.6.70), and the von Neumann entropy of a density matrix is non-negative.
For an MPS tensor \(A\) with \(V^{(N)}(A)\neq 0\), the entropy of the whole chain vanishes: \(S_N^{(N)}(A)=0\).
After reindexing the full block, the normalized operator is the rank-one pure state
A pure state has von Neumann entropy zero.
For an MPS tensor \(A\) with \(V^{(N)}(A)\neq 0\), the pure-state block entropy of the empty block vanishes: \(S_0^{(N)}(A)=0\).
By the Schmidt symmetry \(S_0^{(N)}(A)=S_N^{(N)}(A)\), and the entropy of the full system vanishes for a pure state (Lemma 26.6.73).
If an MPS tensor \(A\) saturates the area law, then the block entropies coincide for all \(1\le L,L'\le \lfloor N/2\rfloor \): \(S_L^{(N)}(A)=S_{L'}^{(N)}(A)\).
The defining condition supplies \(S_m^{(N)}(A)=S_{m+1}^{(N)}(A)\) throughout the interval \(1\le m{\lt}\lfloor N/2\rfloor \). Apply Lemma 26.6.4.
For an MPS tensor \(A\) with \(V^{(N)}(A)\neq 0\), the mutual information of the purification MPDO equals twice the pure-state block entropy: \(I_L=2S_L^{(N)}(A)\).
Let \(A\) be an MPS tensor of bond dimension \(D\geq 1\) with \(V^{(N)}(A)\neq 0\). For the purification MPDO of \(A\), whose generated operator is the pure state \(|V^{(N)}(A)\rangle \! \langle V^{(N)}(A)|\), the mutual information between a block of \(L\) spins and the rest satisfies \(I_L\leq 4\log D\), uniformly in \(L\) and \(N\).
For a pure state, \(I_L=2S_L^{(N)}(A)\), since the global entropy \(S_N\) vanishes and a block and its complement share the same Schmidt spectrum. The block-entropy bound \(S_L^{(N)}(A)\leq 2\log D\) (Proposition 26.6.71) then gives \(I_L\leq 4\log D\).
26.7 Simple tensors
For a doubled-index tensor \(B\) with physical dimension \(d^{2}\), the physical-trace transfer is the bond matrix obtained by closing the ket leg against the bra leg of one tensor:
This is the transfer object of the zero-correlation-length condition of [ CPGSV16 , Definition 4.2, lines 735–739 ] , applied to one element of a basis of normal tensors.
The physical-trace transfer of the doubled-index view of an MPO tensor is the physical-trace transfer of the tensor itself.
If \(M^{\mathrm{dbl}}\) is the doubled-index view of \(M\), then \((M^{\mathrm{dbl}})^{(i,j)}=M^{ij}\), and hence \(\mathcal T_{M^{\mathrm{dbl}}} =\sum _i(M^{\mathrm{dbl}})^{ii} =\sum _i M^{ii} =\mathcal T_M\).
For a sector decomposition with basis elements \(A_{j}\) and weights \(\mu _{j,q}\), the physical-trace transfer of the assembled tensor is the block direct sum of the weighted transfers of the copies:
The diagonal physical sum distributes over the block direct sum:
First block a positive number of physical sites and write the blocked doubled-index tensor in canonical form over a basis of normal tensors, as prescribed at line 815 of [ CPGSV16 ] . A tensor generating MPDOs is simple if one such canonical form has no nilpotent basis element [ CPGSV16 , line 822 ] . A basis element is nilpotent when tracing \(R\le D\) sites annihilates its density operators for all large chain lengths [ CPGSV16 , line 819 ] ; this is nilpotency of its physical-trace transfer matrix.
A tensor generating MPDOs is in simple canonical form if it has a basis-of-normal-tensors decomposition whose basis elements all have non-nilpotent physical-trace transfer. This records the simple horizontal-canonical-form data of the already-blocked tensor \(K\) fixed in [ CPGSV16 , Appendix C.2, line 1628 ] .
A simple tensor already in the chosen blocked canonical-form setting is in horizontal canonical form.
The canonical-form data in the simplicity hypothesis—the sector decomposition, its basis-of-normal-tensors property, and the block-diagonal gauge—give a horizontal canonical form. Positivity and non-nilpotency are not needed for this conclusion.
26.8 Gibbs states of nearest-neighbor commuting Hamiltonians
A density operator on a periodic chain is a Gibbs state of a nearest-neighbor commuting Hamiltonian if
where the one-site spaces belonging to distinct labels \(x\) are orthogonal and \([h^{(x)},\tau _1(h^{(x)})]=0\). With the projector-limit convention of [ CPGSV16 , Definition 4.8, lines 829–850 ] , this is equivalently
with commuting neighboring translates. The orthogonal sum and the multiplicities are part of the definition. Definition 4.8 decomposes the global Hilbert space into the uniform product sectors and requires the associated one-site spaces to be mutually orthogonal; it does not impose a resolution of a larger one-site identity by the sector projections.
The following auxiliary single-bond presentation does not make the sector labels or multiplicities explicit. Its bond acts on the full two-site space. It gives the one-sector case of Definition 26.8.1; the converse comparison is not used.
For a fixed chain length \(N \ge 2\), this consists of one positive semidefinite two-site operator \(B\ge 0\) whose translated copies commute pairwise on the periodic \(N\)-site chain: \([B_{i,i+1},B_{j,j+1}]=0\).
An MPO tensor \(M\) has the single-bond commuting-form property if for every \(N \ge 2\) there is a two-site operator \(B\) as in Definition 26.8.2 and a constant \(c{\gt}0\) such that
The bond in this chainwise condition may depend on \(N\). It is therefore weaker than the single translation-invariant bond family used in the proposition at [ CPGSV16 , Appendix C.2, lines 1597–1619 ] .
This condition asks, for every chain length \(N\geq 2\), for one positive bond on the full two-site space and one positive normalization such that
This auxiliary condition contains no explicit sector labels or multiplicities.
The single-bond product predicate is equivalent to the existence of single-bond commuting-form data on every finite chain. This is an equivalence between two presentations of the same condition.
Expanding the two definitions gives the same identity \(\rho ^{(N)}(M)=c\prod _{i=0}^{N-1}B_{i,i+1}\) with \(c{\gt}0\).
This is the conjunction of the single-bond condition with the doubled-index transfer identity \(\mathcal{E}_M\circ \mathcal{E}_M=\mathcal{E}_M\). The source physical-trace condition is Definition 26.4.2.
The single-bond condition with doubled-index transfer idempotence is equivalent to the conjunction of the single-bond commuting-form property with that same idempotence condition.
Expand the single-bond conjunction and apply Theorem 26.8.5.
A single positive commuting bond is the one-sector instance of the explicit GSNNCH decomposition, with sector projection equal to the identity and multiplicity one.
Take one sector, its one-site projection to be the identity, its multiplicity to be one, and its bond to be the given bond.
Let \(P\) be an orthogonal one-site projection and let \(K\) be an MPO tensor fixed by compression with \(P\) on its two physical indices. A physical support restriction consists of a dimension \(e\) and an isometry \(V:\mathbb {C}^e\longrightarrow \mathbb {C}^d\) such that
for which the restricted tensor \(K_P=V^*KV\) is injective and its inclusion into the ambient physical space recovers \(K\) exactly.
Suppose that the source BNT projectors have been constructed after the common physical blocking, and let \(\mathcal K_s\) be the normal representative with its common copy weight absorbed. Then \(\mathcal K_s\) admits a physical support restriction to the range of \(P_s\). In particular, for some isometry \(V_s\),
and the restricted tensor \(V_s^*\mathcal K_sV_s\) is injective.
The projector identity \(P_s\mathcal K_sP_s=\mathcal K_s\) gives an isometric inclusion of its range. The simultaneous one-site spanning condition makes each normal representative injective, and multiplication by its nonzero common weight preserves injectivity. Write \(K_{P_s}=V_s^*\mathcal K_sV_s\). Each ambient matrix slice satisfies
Hence
The left-hand side is the full virtual matrix algebra, so \(K_{P_s}\) is injective. The two isometry identities give the exact reconstruction.
If \(K\) satisfies saturation of the area law and \(K=V K_P V^*\) is a physical support restriction, then the injective restricted tensor \(K_P\) also satisfies saturation of the area law.
On a chain of every positive length, the sitewise inclusion \(V^{\otimes N}\) is an isometry and
The trace is therefore unchanged. After a contiguous marginal is taken, the isometry on the traced-out sites disappears, while the isometry on the retained sites remains. Thus every block marginal of \(K\) is an isometric conjugate of the corresponding marginal of \(K_P\). Their nonzero spectra, and hence their entropies and mutual informations, coincide. The consecutive equalities defining SAL for \(K\) consequently give those for \(K_P\).
Let \(K=V K_P V^*\) be a physical support restriction of an MPO tensor satisfying SAL. Proposition C.8 applied to the injective tensor \(K_P\) gives a positive two-site bond \(B_P\) whose cyclic translates commute pairwise on every periodic chain of length \(N\geq 2\). Its lift \(B=V^{\otimes 2}B_P(V^{\otimes 2})^*\) is positive, its cyclic translates commute pairwise on every such ambient chain, and \((P\otimes P)B(P\otimes P)=B\). Moreover, for every \(N\geq 2\) one has the exact identity
The preceding theorem gives SAL for the injective restricted tensor, so Proposition C.8 supplies its positive bond. Since \(V^*V=\mathbb {1}\), positivity is preserved under conjugation by \(V^{\otimes 2}\). Also, \(V^{\otimes 2}(V^{\otimes 2})^*=P\otimes P\); multiplying the lifted bond by this projection on either side therefore leaves it unchanged.
It remains to prove commutativity after the lift. On a three-site chain, isometric conjugation carries the product of two adjacent restricted bonds to the product of their lifted bonds. The restricted product is Hermitian because its positive factors commute. Its conjugate is therefore Hermitian; since the lifted factors are Hermitian, their product equals the product in the reverse order. The two-site periodic chain is handled by the same argument with the two oppositely oriented bonds. Locality and cyclic translation then give pairwise commutativity for every \(N\geq 2\).
Finally, isometric conjugation preserves every nonempty product of restricted bonds. The image of each restricted bond is the corresponding lifted bond multiplied by the projection onto the range of \(V^{\otimes N}\). The support projections contributed by the first \(N-1\) bonds cover all sites and leave the full commuting product fixed. Hence the range projection disappears, and the restricted finite-chain product identity becomes the asserted ambient identity, still with normalization scalar one.
Let \(\{ K_x\} _{x=0}^{g-1}\) be MPO tensors on a common one-site space. An orthogonally supported family of commuting sector products consists of pairwise orthogonal one-site projections \(P_x\) and positive two-site operators \(B^{(x)}\) such that
and, for every \(N\geq 2\),
Together with natural multiplicities \(n_x\), these objects determine the corresponding finite-chain sector decomposition in Definition 26.8.1.
Suppose that the BNT physical projectors \(P_x\) have been selected and that every common-weight-absorbed representative \(\mathcal K_x\) satisfies SAL. Then there are positive two-site bonds \(B^{(x)}\) such that the projections \(P_x\) are pairwise orthogonal,
all cyclic translates of each \(B^{(x)}\) commute pairwise, and, for every \(N\geq 2\),
Thus the absorbed BNT representatives have an orthogonally supported family of commuting sector products with normalization scalar one.
Restrict \(\mathcal K_x\) isometrically to the range of \(P_x\). The restricted tensor is injective and satisfies SAL, so Proposition C.8 gives its positive commuting bond product. Apply Theorem 26.8.12 in every sector. The lifted bond is supported on \(P_x\otimes P_x\) and realizes \(\mathcal K_x\) exactly. The BNT projector construction gives \(P_xP_y=0\) for \(x\ne y\), which completes the sector family.
Suppose that \(\{ K_x\} _{x=0}^{g-1}\) has an orthogonally supported family of commuting sector products and that, for every \(N\geq 2\),
Then \(M\) has the explicit GSNNCH sector form with outer sectors \(x\) and natural multiplicities \(n_x\).
For each \(N\geq 2\), the represented unnormalized sector sum is
Thus the positive proportionality constant in Definition 26.8.1 is one.
Let \(\{ K_x\} _{x=0}^{g-1}\) be a nonempty finite family of MPO tensors, let \(n_x\) be positive natural numbers, and let \(P_x\) be pairwise orthogonal one-site projections. Suppose that, for every \(N{\gt}0\),
For \(N\geq 2\) and \(1\leq L\leq N\), write \(\widehat\rho _{x,N,L}\) for the normalized marginal of \(\rho ^{(N)}(K_x)\) on the first \(L\) sites. Assume that the sector label is visible on the first retained site:
If every \(K_x\) satisfies SAL, then \(M\) satisfies SAL.
This is the conditional last inference in the proposition at [ CPGSV16 , Appendix C.2, lines 1801–1808 ] . It assumes sectorwise SAL. The printed proposition states the conclusion from the GSNNCH form alone, but its proof invokes the single-sector proposition at lines 1597–1619, whose hypotheses include ZCL. The theorem here does not supply that omitted sectorwise implication.
For fixed \(N\), put
Positivity of the chain operators, positivity of \(n_x\), and nonemptiness of the family give \(p_x{\gt}0\) and a nonzero normalization for \(M\). Every nonempty normalized marginal decomposes as
Put \(Q_x=P_x\otimes \mathbb {1}^{\otimes (L-1)}\). The support identity gives \(Q_x\widehat\rho _{x,N,L}=\widehat\rho _{x,N,L}\), while \(P_xP_y=0\) for \(x\ne y\) gives \(Q_xQ_y=0\). Thus the summands have pairwise annihilating supports. Hence
Substitution in \(I_L=S_L+S_{N-L}-S_N\) gives
The first sum is independent of \(L\), while sectorwise SAL gives \(I_L(K_x)=I_{L+1}(K_x)\) for every \(x\). Therefore \(I_L(M)=I_{L+1}(M)\) throughout the SAL range.
Suppose that \(K_x\) is the periodic product of translates of a positive two-site operator \(B_x\), and that \(B_x\) is supported on \(P_x\otimes P_x\). If \(N\geq 2\) and \(1\leq L\leq N\), then the normalized marginal of \(K_x\) on the first \(L\) sites satisfies
The first translated bond is fixed by multiplication by \(P_x\) on its first tensor factor. Hence the full periodic product is fixed by the corresponding one-site action. Taking the partial trace over the final \(N-L\) sites preserves this identity.
Let \(K_x\) be a nonempty finite family, let an orthogonally supported commuting sector family realize the tensors \(K_x\), and let \(n_x\) be positive natural numbers. Suppose that, for every \(N{\gt}0\),
If every \(K_x\) satisfies SAL, then \(M\) satisfies SAL.
The preceding lemma supplies the marginal-support hypothesis of Theorem 26.8.16; the remaining hypotheses are precisely the assumed outer-sector equality, positivity of the multiplicities, and sectorwise SAL.
Let \(M\) have a nonempty basis-of-normal-tensors decomposition with positive copy numbers \(n_x\) and copy-independent weights \(\mu _x\). Put \(K_x=\mu _xA_x\), and suppose that the decomposition generates the same positive-length matrix product vectors as \(M\). If an independently supplied orthogonally supported commuting sector family realizes the tensors \(K_x\), and every \(K_x\) satisfies SAL, then \(M\) satisfies SAL.
The positive-length canonical-form identity gives \(\rho ^{(N)}(M)=\sum _x n_x\rho ^{(N)}(K_x)\). Apply Corollary 26.8.18.
Let \(M\) be an MPO tensor and let its BNT canonical form have normal representatives \(A_x\), copy numbers \(n_x\), and copy-independent weights \(\mu _x\). Put \(\mathcal K_x=\mu _xA_x\), and assume that this canonical form generates the same matrix product vectors as \(M\) at every positive length. If the representatives \(\mathcal K_x\) have orthogonally supported commuting sector products, then
This gives the explicit GSNNCH form with the BNT copy numbers \(n_x\) as its natural multiplicities.
Apply Theorem 26.8.15 to the positive-length BNT decomposition. Its coefficients are exactly the copy numbers \(n_x\) after the common weights have been absorbed.
Let \(M\) be an MPO tensor and let its BNT canonical form have normal representatives \(A_x\), copy numbers \(n_x\), and copy-independent weights \(\mu _x\). Put \(\mathcal K_x=\mu _xA_x\), and assume that this canonical form generates the same matrix product vectors as \(M\) at every positive length. Equivalently, for every \(N{\gt}0\),
Under the BNT projector-selection hypotheses, if every \(\mathcal K_x\) satisfies SAL, then \(M\) has the source GSNNCH form. Its outer sectors are the BNT sectors and its natural multiplicities are the copy numbers \(n_x\).
Let \(B\) act on \(L\) consecutive spins of the periodic \(N\)-site chain, embedded at base site \(i\), and let \(r\) denote the cyclic shift of sites. Reindexing both chain configurations by \(r\) carries the embedded operator to its translate at the shifted base site:
Entrywise from the definitions: the shift carries the window at \(i\) to the window at \(r(i)\), preserves agreement of two configurations outside the window, and preserves the extracted window values.
For every sector \(x\) of an explicit GSNNCH decomposition and every finite chain length \(N\geq 2\), all cyclic translates of the sector bond commute: \([\tau _i(B^{(x)}),\tau _j(B^{(x)})]=0\).
Translates with disjoint windows commute because they act on disjoint sites, and two length-two windows overlap only when their starting sites coincide or are cyclic neighbors. For the neighboring pairs, applying the transport identity of Lemma 26.8.22 \(p\) times to the neighboring commutation \([\tau _0(B^{(x)}),\tau _1(B^{(x)})]=0\) of Definition 26.8.1 gives \([\tau _p(B^{(x)}),\tau _{p+1}(B^{(x)})]=0\) for every \(p\), which covers all overlapping pairs.
An operator embedded into the periodic chain from a positive semidefinite local factor is positive semidefinite; in particular every translate \(\tau _i(B^{(x)})\) of a sector bond is positive semidefinite.
In the coordinates given by the window-plus-complement splitting of the chain, the embedded operator decomposes as
where the identity acts on the cyclic complement of the window. A tensor product of positive semidefinite operators is positive semidefinite, and \(B^{(x)}\geq 0\) by Definition 26.8.1.
Each sector product \(\prod _{j=1}^N\tau _j(B^{(x)})\) of an explicit GSNNCH decomposition is positive semidefinite; hence so are the represented operator
and every chain operator equal to a positive multiple of it.
A product of pairwise commuting positive semidefinite matrices is positive semidefinite: for two commuting factors \(A\) and \(B\), the square root of \(A\) commutes with \(B\), so \(AB=\sqrt A\, B\, \sqrt A\geq 0\), as congruence by \(\sqrt A\) preserves positive semidefiniteness; iterate along the product. Each sector product is such a product by Lemmas 26.8.24 and 26.8.23, and the sum with natural coefficients preserves positivity.
Each sector product of an explicit GSNNCH decomposition is invariant under the cyclic shift of the periodic chain, where spin \(N+1\) is identified with the first; hence so are the represented operator and every chain operator equal to a positive multiple of it [ CPGSV16 , lines 838–842 ] .
If a finite-chain operator of an MPO tensor has the explicit sector form of Definition 26.8.1 and nonzero trace, then its trace normalization has the sector form as well.
By Lemma 26.8.25 the operator is positive semidefinite, so its nonzero trace is positive and the normalization multiplies the sector sum by a positive constant.
If every finite-chain operator of an MPO tensor has the explicit sector form of Definition 26.8.1 and a nonzero trace, then the normalized finite-chain states are density operators with that sector form, so the tensor generates Gibbs states of a nearest-neighbor commuting Hamiltonian.
26.9 Two-site and positive-length physical blocking
The two-site block is the virtual contraction of two copies of \(K\):
![\tnpic[physical=updown, tensor style=box]{%
\tn[mpo, up={$(i_1,i_2)$}, down={$(j_1,j_2)$}]{M}}](tenkz_svg/tenkz-2c1b3ed7fd5dd5b0.svg)
\(=\) ![\tnpic[physical=updown, tensor style=box]{%
\tn[mpo, up=$i_1$, down=$j_1$]{K} &
\tn[mpo, up=$i_2$, down=$j_2$]{K}}](tenkz_svg/tenkz-2184581ea5a0f2e8.svg)
The two ket indices and the two bra indices are grouped into the blocked physical indices, as in [ CPGSV16 , Theorem 4.9, lines 851–856 ] .
For a positive integer \(L\), group \(L\) adjacent physical sites and denote the resulting local tensor by \(M^{[L]}\). It has a length-\(L\) ket word \(I=(i_1,\ldots ,i_L)\) and a length-\(L\) bra word \(J=(j_1,\ldots ,j_L)\) as its physical indices, and
Thus the ket and bra words are grouped separately. This is the MPO version of the physical blocking used for the basis of normal tensors in [ CPGSV16 , lines 317–345 ] . It is a local tensor, not the closed-chain operator \(O_L(M)\) of [ CPGSV16 , lines 962–967 ] .
Let \(L{\gt}0\). Pairing the ket and bra letters site by site gives the canonical identification \(\{ 0,\ldots ,d^L-1\} ^2\simeq \{ 0,\ldots ,d^2-1\} ^L\). Under this identification, the doubled-index MPS tensor of \(M^{[L]}\) is the physical reindexing of the \(L\)-blocked doubled-index MPS tensor of \(M\). In particular, one is injective if and only if the other is.
Decode the blocked ket and bra words, pair their letters at each site, and compare the two ordered matrix products. Physical reindexing by a bijection preserves the span of the tensor matrices.
For \(L{\gt}0\) and two MPO tensors \(M,N\) with the same physical dimension, let juxtaposition denote the MPO tensor product obtained by contracting the intermediate physical index. Then \((MN)^{[L]}=M^{[L]}N^{[L]}\).
Expand the product tensor and sum over its length-\(L\) intermediate physical word. Reindexing this word by one blocked physical index gives the product of the two blocked tensors.
Let \(L\in \mathbb {N}\), and let
concatenate the \(N\) blocked words. For blocked configurations \(\sigma \) and \(\tau \),
The algebraic identity holds for every \(L\in \mathbb {N}\). For \(L\geq 1\), positivity of the original nonempty chain of length \(NL\) shows that blocking \(L\) adjacent physical sites of an MPDO again gives an MPDO. This is the closed-chain identification implicit when the one-site tensor and its two-site blocking are placed in vertical canonical form in [ CPGSV16 , Appendix C.4, lines 1952–2017 ] .
Write \(\sigma _k=(\sigma _{k,0},\ldots ,\sigma _{k,L-1})\) and \(\tau _k=(\tau _{k,0},\ldots ,\tau _{k,L-1})\). For each \(k\), \((M^{[L]})^{\sigma _k,\tau _k} =\prod _{j=0}^{L-1}M^{\sigma _{k,j},\tau _{k,j}}\). Taking both products in increasing index order gives
Taking the virtual trace gives the equality in the theorem. The map \(\Phi _{N,L}\) is a bijection, and simultaneous reindexing of the rows and columns preserves positive semidefiniteness.
For an MPO tensor \(K\), its two-site blocking \(K^{[2]}\) has physical indices \((i_0,i_1)\) and \((j_0,j_1)\) and matrices
This is the blocking used in [ CPGSV16 , Theorem 4.9, lines 851–856 ] .
Let \(\varphi :\{ 0,\ldots ,d^2-1\} \to \{ 0,\ldots ,d-1\} ^2\) be the canonical bijection. Then \((K^{[2]})^{i,j} =(K^{[L]})^{\varphi (i),\varphi (j)}\bigm |_{L=2}\). Consequently, for every nonempty chain length \(N\), the closed MPO of \(K^{[2]}\) is obtained from the closed MPO of the general length-two block by applying \(\varphi \) independently at every ket and bra site. In particular, if \(K\) generates matrix product density operators, then so does \(K^{[2]}\).
Decode a blocked index as an ordered pair. The two local tensors are then the same product \(K^{i_0j_0}K^{i_1j_1}\). Applying this equality at every site gives the closed-chain reindexing, and simultaneous row and column reindexing preserves positive semidefiniteness.
If an MPO tensor \(K\) is in normalized BNT-refined horizontal form, then its two-site blocking \(K^{[2]}\) is again in that form.
Block every normal representative in the horizontal sector decomposition by two sites and square each copy weight. Positive blocking preserves irreducibility, left-canonical normalization, eventual linear independence of the representative states, and inequivalence of distinct representatives. Normalized self-overlap is preserved by \(O_{A_j^{[2]}A_j^{[2]}}(N)=O_{A_jA_j}(2N)\longrightarrow 1\). Hence the blocked sector decomposition is again a BNT canonical form. If \(\varphi \) is the canonical bijection between a pair of ket–bra indices and a length-two word in the doubled alphabet, then \((K^{[2]})^{\mathrm{MPS}}=\varphi ^*\! \left(\operatorname {block}_2(K^{\mathrm{MPS}})\right)\). Thus \(\varphi \) only relabels the physical basis. Finally, if \(X\) is the original block-diagonal virtual gauge and \(S\) its sector tensor, word evaluation gives \(((K^{[2]})^{\mathrm{MPS}})_i=X(\varphi ^*(S^{[2]}))_iX^{-1}\). Therefore this blocked decomposition and the same copy gauges witness normalized BNT-refined horizontal form for \(K^{[2]}\).
Let \(K\) be an MPO tensor in normalized BNT-refined horizontal form that generates matrix product density operators. Then both \(K\) and \(K^{[2]}\) satisfy the vertical canonical-form conditions of Definition 23.4.6. This is the initial canonical-form step in [ CPGSV16 , Appendix C.4, lines 1951–1956 ] .
The preceding blocking results show that \(K^{[2]}\) is in normalized BNT-refined horizontal form and generates matrix product density operators. Apply Theorem 23.4.100 first to \(K\) and then to \(K^{[2]}\).
Let the vertical sectors be labelled by \(\alpha \), with simple matrix algebra \(M_{d_\alpha }\) and positive diagonal multiplicity matrix
Write \(m_\alpha =\operatorname{tr}(\mu _\alpha )\). The normalized embedding and the left partial trace are
On a weighted sector, \(\widetilde R_\mu (\lambda _\alpha \mu _\alpha \otimes X_\alpha ) =\lambda _\alpha m_\alpha X_\alpha \). Thus the partial trace is the canonical extension to the full block matrix space of the inverse map displayed in [ CPGSV16 , Appendix C.4, lines 1957–1971 ] .
Let \((d_\alpha )_\alpha \) and \((r_\alpha )_\alpha \) be finite families of dimensions with \(r_\alpha {\gt}0\), and suppose that every diagonal entry of \(\mu _\alpha \) is positive. Then the normalized embedding
is a direct-sum Kraus map. Without any condition on the multiplicities, the map
is a direct-sum Kraus map. Its restriction to the weighted sectors is the retraction in [ CPGSV16 , Appendix C.4, lines 1961–1970 ] .
Exchange the two coordinates within each sector by \(w(\alpha ,q,i)=(\alpha ,i,q)\). Let \(R_w\) be the corresponding matrix reindexing. For the normalized embedding, set \(\rho _\alpha =m_\alpha ^{-1}\mu _\alpha \). The hypotheses give \(\rho _\alpha \succeq 0\) and \(\operatorname{tr}(\rho _\alpha )=1\). Let \(\mathcal P\) be the orthogonally controlled preparation of these density matrices. Thus
The preparation Kraus operators resolve the identity in each sector, so \(\mathcal P\) is trace-preserving and completely positive. Reindexing by \(w^{-1}\) changes \(X_\alpha \otimes \rho _\alpha \) into \(\rho _\alpha \otimes X_\alpha \). Consequently, the full-matrix extension satisfies
This proves complete positivity of the normalized embedding.
Now let \(\mathcal C_{\operatorname{tr}}\) be the controlled partial trace over the \(\mathbb {C}^{r_\alpha }\) factors. For a matrix \(Y\) on \(\bigoplus _\alpha (\mathbb {C}^{r_\alpha }\otimes \mathbb {C}^{d_\alpha })\), the full-matrix extension satisfies
The diagonal case is the ordinary right partial trace after applying \(R_w\), and the off-diagonal case vanishes under the orthogonal sector control. Hence \(\widehat{\widetilde R}=\mathcal C_{\operatorname{tr}}\circ R_w\). Equivalence reindexing and the controlled dependent partial trace are trace-preserving completely positive maps, so their composition has a Kraus representation.
Suppose that every multiplicity space is nonzero and every diagonal entry of \(\mu _\alpha \) is positive. Then \(m_\alpha \ne 0\) for every \(\alpha \), and \(\widetilde R_\mu \circ R_\mu =\operatorname{id}\).
Positivity of the diagonal entries and nonemptiness of the multiplicity space give \(m_\alpha =\sum _q\mu _{\alpha ,q}{\gt}0\). The Kronecker-product identity for the left partial trace then gives
For vertical BNT tensors \(M_\alpha \) with simple matrix algebras \(M_{d_\alpha }\) and a family of complex scalars \((c_\alpha )_\alpha \), define
For each \(L\geq 0\), let
Write \(V_m\) for the specialization obtained by taking \(c_\alpha =m_\alpha =\operatorname{tr}(\mu _\alpha )\).
Let \(M=(M_{ab})_{a,b=0}^{D-1}\) be a matrix-product tensor whose physical index is identified with the ordered pair \((a,b)\). Then \(M(E_{ba})=M_{ab}\).
If \(r\in \{ 0,\ldots ,D^2-1\} \) encodes the ordered pair \((\lfloor r/D\rfloor ,r\bmod D)\), then \(M\! \left(E_{(r\bmod D),\lfloor r/D\rfloor }\right)=M_r\).
For scalars \(c_\alpha \ne 0\), the map
is a linear equivalence.
For a word \(w=((a_1,b_1),\ldots ,(a_L,b_L))\),
Fix \(L\geq 0\) and suppose that every \(c_\alpha \) is nonzero. If the simultaneous word tuples \((M_\alpha ^w)_\alpha \) of length \(L\) span \(\prod _\alpha M_{d_\alpha }\), then \(C_L(V_c)=\prod _\alpha M_{d_\alpha }\).
Suppose that \((M_\alpha )_\alpha \) is a basis of normal tensors and that \((c_\alpha )_\alpha \) is any family of nonzero complex scalars. Then there is a positive integer \(L\) such that \(C_L(V_c)=\prod _\alpha M_{d_\alpha }\).
Choose \(L{\gt}0\) for which the simultaneous word tuples \((M_\alpha ^w)_\alpha \) span \(\prod _\alpha M_{d_\alpha }\). Since every \(c_\alpha \) is nonzero, so is \(c_\alpha ^L\). Apply Lemma 26.9.17.
Suppose that \((M_\alpha )_\alpha \) is a basis of normal tensors, every multiplicity space is nonzero, and every diagonal entry of \(\mu _\alpha \) is positive. Then there is a positive integer \(L\) such that \(C_L(V_m)=\prod _\alpha M_{d_\alpha }=:\mathcal A_1\). This is the spanning assertion used in [ CPGSV16 , Appendix C.4, lines 1980–1990 ] .
Since every multiplicity space is nonzero and every diagonal entry is positive, \(m_\alpha =\sum _{q=0}^{\operatorname{mult}(\alpha )-1}\mu _{\alpha ,q}{\gt}0\). Apply the preceding theorem to the nonzero family \(c_\alpha =m_\alpha =\operatorname{tr}(\mu _\alpha )\).
Let \(F\) be a linear endomorphism of \(\prod _\alpha M_{d_\alpha }\). Suppose that \(C_L(V_c)\) is the whole sector algebra and, for every choice of bond matrices \(X_1,\ldots ,X_L\),
Then \(F\) is the identity.
Since the displayed products span the domain, every \(Y\in \prod _\alpha M_{d_\alpha }\) has the form \(Y=\sum _i\lambda _iV_c(X_1^{(i)})\cdots V_c(X_L^{(i)})\). Hence
Thus \(F\) is the identity.
Let \(F\) be a linear endomorphism of \(\mathcal A=\prod _\alpha M_{d_\alpha }\), and write \(\mathcal F=\{ X\in \mathcal A:F(X)=X\} \). For \(L\geq 0\), define
This is the product space used in the dimension argument of [ CPGSV16 , Appendix C.4, lines 1980–1993 ] .
Let \(\mathcal A=\prod _\alpha M_{d_\alpha }\), and let \(F:\mathcal A\to \mathcal A\) be positive and trace preserving. Suppose that the trace adjoint \(F^*\) satisfies the Schwarz inequality and that \(F\) fixes a family \(\rho =(\rho _\alpha )_\alpha \) for which every \(\rho _\alpha \) is positive definite. If \(\mathcal F=\operatorname{Fix}(F)\), then, for every \(L{\gt}0\), \(\dim C_L(\mathcal F)\leq \dim \mathcal F\).
Extend \(F\) to the full matrix algebra by diagonal compression followed by block-diagonal embedding, and choose a numbered basis of the ambient space. The extension is positive and trace preserving, its trace adjoint satisfies the Schwarz inequality, and the embedded family \(\rho \) is a positive definite fixed point. The fixed points therefore have the form
where \(\operatorname{tr}(\sigma _k)=1\). The parametrization \((X_k)_k\mapsto \bigoplus _k\sigma _k\otimes X_k\) is injective, since its left partial trace is \((X_k)_k\). Every product of \(L\) fixed points lies in the range of
Thus the span of these products has dimension at most \(\sum _k h_k^2\), which is the dimension of the fixed-point space of the extension. Writing \(\mathcal E\) for the fixed-point space of the full-matrix extension, the injective block-diagonal embedding and diagonal compression give the dimension chain
Let \(F\) be a linear endomorphism of \(\mathcal A=\prod _\alpha M_{d_\alpha }\). Suppose that the following three conditions hold, with the second required for every bond matrix \(X\):
Then \(F=\operatorname{id}_{\mathcal A}\).
This is the finite-dimensional conclusion in [ CPGSV16 , Appendix C.4, lines 1980–1993 ] . It does not assume that the products \(V_c(X_1)\cdots V_c(X_L)\) are fixed.
Since every \(V_c(X)\) belongs to \(\mathcal F\), every product occurring in \(C_L(V_c)\) belongs to \(C_L(\mathcal F)\). Hence
so \(C_L(\mathcal F)=\mathcal A\). Therefore
Thus \(\mathcal F=\mathcal A\), and \(F\) is the identity.
Let \(F\) be a positive trace-preserving endomorphism of \(\mathcal A=\prod _\alpha M_{d_\alpha }\). Suppose that each \(V_j\) is fixed by \(F\) and, for some \(L{\gt}0\), the identity belongs to
Then there is a fixed family \(\rho =(\rho _\alpha )_\alpha \) such that every \(\rho _\alpha \) is positive definite.
This is a local finite-product consequence used to formalize the fixed-point argument in [ CPGSV16 , Appendix C.4, lines 1980–1993 ] ; CPSV16 applies Wolf’s density-block description directly rather than stating this intermediate result.
Extend \(F\) to the full matrix algebra by diagonal compression followed by block-diagonal embedding. This extension is positive and trace preserving. Block-diagonal embedding preserves the identity and every product \(V_{j_1}\cdots V_{j_L}\), so Theorem 26.9.62 gives a positive-definite fixed point \(\widehat\rho \) of the extension. Fixed points of the extension are block diagonal. Thus
Every diagonal block \(\rho _\alpha \) of the positive-definite matrix \(\widehat\rho \) is positive definite.
Let \(F\) be a positive trace-preserving endomorphism of \(\mathcal A=\prod _\alpha M_{d_\alpha }\) whose trace adjoint satisfies the Schwarz inequality. Suppose that, for some \(L{\gt}0\), \(C_L(V_c)=\mathcal A\) and \(F(V_c(X))=V_c(X)\) for every bond matrix \(X\). Then \(F=\operatorname{id}_{\mathcal A}\). No product \(V_c(X_1)\cdots V_c(X_L)\) is assumed to be fixed.
This is the implication used for each transported square composite in [ CPGSV16 , Appendix C.4, lines 1980–1995 ] .
The fixed contractions and their positive-length product span give a positive-definite fixed family \(\rho =(\rho _\alpha )_\alpha \). Hence Theorem 26.9.22 gives \(\dim C_L(\mathcal F)\leq \dim \mathcal F\), where \(\mathcal F=\operatorname{Fix}(F)\). Since \(C_L(V_c)=\mathcal A\) and \(V_c(X)\in \mathcal F\), the preceding theorem gives \(F=\operatorname{id}_{\mathcal A}\).
Retain the coordinates \((\alpha ,a,i)\), where \(\alpha \) labels a simple sector, \(a\) labels a diagonal entry of \(\mu _\alpha \), and \(0\leq i{\lt}d_\alpha \). The family \((Y_\alpha )_\alpha \) is represented on this space by \(\bigoplus _\alpha Y_\alpha \). Write \(\iota \) for this inclusion, \(\pi \) for extraction of the diagonal summands, and \(P=\iota \pi \). Composing \(\iota \) and \(\pi \) with \(R_\mu \) and \(\widetilde R_\mu \) defines their retained-coordinate forms.
The diagonal-summand extraction is a left inverse of the inclusion: \(\pi \iota =\operatorname{id}\).
Restricting \(\bigoplus _\beta Y_\beta \) to the rows and columns with sector label \(\alpha \) gives \(Y_\alpha \).
For every retained-coordinate matrix \(Z\), \(P(Z)_{(\alpha ,a,i),(\alpha ,b,j)} =Z_{(\alpha ,a,i),(\alpha ,b,j)}\).
Both indices lie in the same extracted diagonal summand.
If \(\alpha \neq \beta \), then \(P(Z)_{(\alpha ,a,i),(\beta ,b,j)}=0\).
The two indices belong to distinct diagonal summands.
The projection onto the vertical-sector diagonal summands is idempotent: \(P^2=P\).
The identity \(\pi \iota =\operatorname{id}\) gives \(P^2=\iota \pi \iota \pi =\iota \pi \).
Suppose that every multiplicity space is nonzero and every diagonal entry of \(\mu _\alpha \) is positive. Then the retained-coordinate partial trace is a left inverse of the normalized retained-coordinate embedding.
First extract the diagonal summands, then apply \(\widetilde R_\mu R_\mu =\operatorname{id}\).
For every horizontal bond matrix \(X\), contraction of \(\bigoplus _\alpha \mu _\alpha \otimes M_\alpha \) gives \(\bigoplus _\alpha \mu _\alpha \otimes M_\alpha (X)\).
Flattening the index triple \((\alpha ,a,i)\) gives the entry formula
Let \(A\) be a tensor whose physical label is a pair \((a,b)\) of horizontal bond indices. For a bond matrix \(X\), define
The order of the entries of \(X\) is chosen so that \(A(X)_{ij}=\operatorname{tr}(M^{ij}X)\) when \(A_{ab,ij}=M^{ij}_{ab}\).
Let \(L\) and \(K\) be fixed matrices of compatible sizes. Then \(\left((a,b)\mapsto L A_{ab}K\right)(X)=L A(X)K\).
Distributivity of matrix multiplication over finite sums gives
For a weighted block-diagonal tensor \(A=\bigoplus _\alpha \mu _\alpha A_\alpha \), one has \(A(X)=\bigoplus _\alpha \mu _\alpha A_\alpha (X)\).
Since each tensor letter is block diagonal,
For the vertically viewed tensor \(\widetilde M_{ab,ij}=M^{ij}_{ab}\), bond-matrix contraction is the one-site physical closure: \(\widetilde M(X)=M(X)\).
For every pair of physical indices,
The contraction of the assembled vertical tensor is the weighted block diagonal of the sector contractions:
Applying the preceding lemma to the block form \(\bigoplus _\alpha \mu _\alpha \otimes M_\alpha \) of the assembled tensor gives
Let \(B_{ab}\) be an arbitrary vertical tensor. If \(U\widetilde M_{ab}U^\dagger =B_{ab}\) for every pair \((a,b)\), then, for every bond matrix \(X\), \(U M(X)U^\dagger =B(X)\).
By the definition of bond-matrix contraction,
Let \(B_{ab}\) be an arbitrary vertical tensor. If \(\widetilde M_{ab}=U^\dagger B_{ab}U\) for every pair \((a,b)\), then, for every bond matrix \(X\), \(M(X)=U^\dagger B(X)U\).
Contracting the letterwise identity gives
Suppose that
is a one-site vertical canonical-form identity. Contracting against an arbitrary bond matrix \(X\) gives
If the reverse letterwise identity is also given, then
These are the contracted canonical-form identities used in [ CPGSV16 , Appendix C.4, lines 1955–1979 ] .
Since \(A(X)_{ij}=\sum _{a,b}X_{ba}A_{ab,ij}\) and \(\widetilde M_{ab,ij}=M^{ij}_{ab}\), one has \(M(X)=\sum _{a,b}X_{ba}\widetilde M_{ab}\). Distributivity of fixed left and right multiplication over finite sums gives
Substituting the forward hypothesis into this equality gives
Applied blockwise, the same calculation gives
If the reverse letterwise identity is given, the same calculation yields
For a virtual matrix \(X\), the four-site physical closure of \(K\) is the operator \(K_4(X)\) whose matrix coefficients are
The canonical four-coordinate regrouping and its inverse satisfy
Both identities follow from the definition of the canonical right-associated regrouping.
Under the canonical right-associated identification of four-site configurations with quadruples of physical indices,
Expand the general four-site closure in the right-associated coordinates.
After identifying one blocked physical index with two original indices, \((K^{[2]})_1(X)=K_2(X)\).
Expand one blocked tensor matrix and decode its two physical indices.
Let \(U_1\) and \(U_2\) be the physical coordinate changes in vertical canonical forms of \(M\) and its two-site blocking, respectively. Let \(J\) be the canonical relabelling from matrices indexed by pairs of one-site physical indices to matrices indexed by one blocked physical index. For the refinement and coarse-graining maps \(T\) and \(S\), define
These are the physical maps after absorbing the two vertical coordinate changes, as in [ CPGSV16 , Appendix C.4, lines 1955–1979 ] . They are intermediate maps on the retained physical coordinate spaces; the normalized sector maps \(R_1,R_2,\widetilde R_1,\widetilde R_2\) have not yet been composed with them.
For a bond matrix \(X\), write \(\mathcal M^{(1)}(X)\) and \(\mathcal M^{(2)}(X)\) for the one-site and two-site physical closures, respectively, and write
for the contracted one-site and two-site vertical canonical forms. If
then
These are the physical-coordinate identities which, after composition with \(R_1,R_2,\widetilde R_1,\widetilde R_2\), give the two displayed identities for \(\widetilde T\) and \(\widetilde S\) in [ CPGSV16 , Appendix C.4, lines 1955–1979 ] .
The reverse one-site canonical-form identity gives \(U_1^\dagger B_1(X)U_1=\mathcal M^{(1)}(X)\). Applying \(T\), relabelling its two-site output by \(J\), and using the forward blocked canonical-form identity gives
The reverse blocked canonical-form identity and the equation for \(S\) give, in the other direction,
Define maps between the one-site and two-site simple-sector algebras by
The partial trace and diagonal-summand extraction define these maps on the full matrix algebras. Their agreement with the normalized maps in [ CPGSV16 , Appendix C.4, lines 1972–1979 ] is asserted on the weighted block-diagonal image. At this stage there is no assertion that the image is preserved, that the two maps are inverse, completely positive, or trace preserving, or that the sector labels have been matched. The final renormalization fixed-point conclusion is separate.
A vertical-sector family \(X=(X_\alpha )_\alpha \) is positive when every \(X_\alpha \) is positive semidefinite. Its total sector trace is
Suppose that every multiplicity space is nonzero and every diagonal entry of \(\mu _\alpha \) is positive. The normalized embedding \(R_\mu \) sends a positive sector family to a positive retained matrix and satisfies \(\operatorname{tr}(R_\mu (X))=\operatorname {Tr}_{\mathrm{sec}}(X)\). Conversely, the sectorwise partial trace sends a positive retained matrix to a positive sector family and satisfies \(\operatorname {Tr}_{\mathrm{sec}}(\widetilde R_\mu (Z))=\operatorname{tr}(Z)\). These trace identities follow by summing the traces of the diagonal sectors; deleting the off-diagonal sector entries does not change the trace.
A positive diagonal weight matrix has positive trace, and its Kronecker product with each positive \(X_\alpha \) is positive. Partial trace preserves positivity. The two trace formulas follow from \(\operatorname{tr}(A\otimes B)=\operatorname{tr}(A)\operatorname{tr}(B)\) and the normalization by \(\operatorname{tr}(\mu _\alpha )\).
Suppose that every multiplicity space is nonzero and every diagonal entry of \(\mu _\alpha \) is positive. For \(m_\alpha =\operatorname{tr}(\mu _\alpha )\),
In each sector,
For \(m_\alpha =\operatorname{tr}(\mu _\alpha )\),
Taking the partial trace over the multiplicity factor gives
Let \(U:\mathbb {C}^n\to \mathbb {C}^r\) satisfy \(UU^\dagger =1\). If \(Z\geq 0\), then
For every \(W\in \mathbb {C}^{r\times r}\), one also has \(\operatorname{tr}(U^\dagger WU)=\operatorname{tr}(W)\).
Put \(P=U^\dagger U\). The matrices \(P\) and \(1-P\) are orthogonal projections, and
The last trace vanishes precisely when \((1-P)Z=0\). The formula for \(W\) follows by cyclicity of trace and \(UU^\dagger =1\).
Let \(Z_T(X)\) be the matrix obtained by applying the normalized one-site embedding, expanding by \(U_1^\dagger \), applying \(T\), and relabelling the two-site indices, before compression by \(U_2\). Define \(Z_S(Y)\) in the reverse direction, before compression by \(U_1\).
Suppose that the physical maps are completely positive and trace preserving and \(U_iU_i^\dagger =1\). For positive sector families \(X\in \mathcal A_1\) and \(Y\in \mathcal A_2\), the matrices \(Z_T(X)\) and \(Z_S(Y)\) are positive and
After the final compression and sectorwise partial trace,
Positivity follows successively from the normalized embedding, expansion by an adjoint, the physical map, relabelling, and partial trace. The trace identities follow from the preceding two lemmas and trace preservation of the physical maps.
Suppose that the multiplicity spaces are nonzero, their diagonal weights are positive, the physical maps \(T\) and \(S\) are completely positive and trace preserving, and the vertical coordinate maps satisfy \(U_iU_i^\dagger =1\). Then \(\widetilde T\) and \(\widetilde S\) preserve pointwise positive semidefiniteness. For pointwise positive families \(X\in \mathcal A_1\) and \(Y\in \mathcal A_2\), their total sector traces obey
The trace loss vanishes precisely when
respectively.
This result concerns positivity and trace on positive elements. It does not assert a complete-positivity structure for the finite products, a trace-preserving extension to their full ambient matrix spaces, or an inverse relation between the two transported maps.
The normalized diagonal embedding and the sectorwise partial trace preserve positivity. Before the final compression, trace preservation of the physical map and \(U_iU_i^\dagger =1\) give the original sector trace. For a positive matrix \(Z\) and a coisometry \(U\), put \(P=U^\dagger U\). Then
Since \(1-P\) is an orthogonal projection, equality is equivalent to \(\operatorname{tr}((1-P)Z)=0\), hence to \((1-P)Z=0\).
For a bond matrix \(X\), set \(V_1(X)=\bigoplus _\alpha m_\alpha M_\alpha (X)\). Suppose that both vertical canonical forms satisfy their exact reconstruction identities, the one-site multiplicity spaces are nonzero, their diagonal weights are positive, \(T(\mathcal M^{(1)}(X))=\mathcal M^{(2)}(X)\), and \(U_2U_2^\dagger =1\). Then \(U_2^\dagger U_2 Z_T(V_1(X))=Z_T(V_1(X))\). This is the refinement support identity used in [ CPGSV16 , Appendix C.4, lines 1955–1980 ] .
Let \(\mathcal R^{-1}\) relabel a matrix indexed by two one-site indices as a matrix indexed by the blocked two-site index. Then
Here the first equality is the normalized embedding, the second and last are the two reconstruction identities, and the middle equality is the refinement identity. The equality before the last is precisely the blocked-index relabelling theorem. Therefore
For a bond matrix \(X\), set \(V_2(X)=\bigoplus _\beta n_\beta A_\beta (X)\). Suppose that both vertical canonical forms satisfy their exact reconstruction identities, the two-site multiplicity spaces are nonzero, their diagonal weights are positive, \(S(\mathcal M^{(2)}(X))=\mathcal M^{(1)}(X)\), and \(U_1U_1^\dagger =1\). Then \(U_1^\dagger U_1 Z_S(V_2(X))=Z_S(V_2(X))\). This is the coarse-graining support identity used in [ CPGSV16 , Appendix C.4, lines 1955–1980 ] .
Let \(\mathcal R\) decode a matrix indexed by the blocked two-site index as a matrix indexed by two one-site indices. Then
Here the first equality is the normalized embedding, the next equality is the blocked-index relabelling theorem, the middle equality is the coarse-graining identity, and the last is the one-site reconstruction. Therefore
Under the hypotheses of Lemma 26.9.56, suppose in addition that \(U_1U_1^\dagger =1\) and that \(T\) is completely positive and trace preserving. Then \(\operatorname {Tr}_{\mathrm{sec}}(\widetilde T(V_1(X)))=\operatorname {Tr}_{\mathrm{sec}}(V_1(X))\).
Cyclicity of trace and the preceding support identity give
Under the hypotheses of Lemma 26.9.57, suppose in addition that \(U_2U_2^\dagger =1\) and that \(S\) is completely positive and trace preserving. Then \(\operatorname {Tr}_{\mathrm{sec}}(\widetilde S(V_2(X)))=\operatorname {Tr}_{\mathrm{sec}}(V_2(X))\).
Cyclicity of trace and the preceding support identity give
Suppose that every multiplicity space for \(\mu _\alpha \) and \(\nu _\beta \) is nonzero, every diagonal entry of these two matrices is positive, and
Writing \(m_\alpha =\operatorname{tr}(\mu _\alpha )\) and \(n_\beta =\operatorname{tr}(\nu _\beta )\), one has
In particular, take \(X_\alpha =M_\alpha (X)\) and \(Y_\beta =A_\beta (X)\) from the contractions of the one-site and two-site vertical canonical forms. The physical closure identities for \(T\) and \(S\) then imply the two hypotheses above, so the same conclusions hold for these source-generated sector operators.
Since \(\widetilde T=\widetilde R_2T_{\mathrm v}R_1\), one has
Likewise, \(\widetilde S=\widetilde R_1S_{\mathrm v}R_2\) gives
For every bond matrix \(X\), set
Under the hypotheses of the preceding theorem, the transported composites satisfy
Thus the one-site and two-site contraction families lie in the respective fixed-point spaces, as in Appendix C.4, lines 1974–1980. These equations do not assert that either composite is the identity on the whole sector algebra; that conclusion requires the fixed-point structure argument of Appendix C.4, lines 1980–1993.
The preceding theorem gives \(\widetilde T(V_1(X))=V_2(X)\) and \(\widetilde S(V_2(X))=V_1(X)\). Hence
Let \(\mathcal M\) be a finite full matrix algebra over \(\mathbb C\), and let \(F:\mathcal M\to \mathcal M\) be positive and trace nonincreasing. Suppose that a family \((V_j)_{j\in J}\) consists of fixed points of \(F\) and that, for some \(L{\gt}0\), the identity belongs to the linear span of the products \(V_{j_1}\cdots V_{j_L}\) in \(\mathcal M\). Then \(F\) has a positive-definite fixed point. This is the local full-matrix intermediate implication used in [ CPGSV16 , Appendix C.4, lines 1980–1995 ] ; it is not stated there as a separate theorem.
Boundedness of the forward orbits gives the mean-ergodic projection \(P\). Set \(\rho =P(\mathbf1)\) and let \(Q\) be the support projection of \(\rho \). For every \(j\), the maximal-support property of \(\rho \) gives \(QV_jQ=V_j\). Since \(Q\) is a projection, this implies \(QV_j=V_j=V_jQ\). By induction, for every \(k\geq 1\),
The product-span hypothesis and linearity then give \(Q\mathbf1Q=\mathbf1\). Since \(Q^2=Q\), one has \(Q=Q^2=Q\mathbf1Q=\mathbf1\). Thus \(\rho \) is positive definite, and the construction of \(P\) gives \(F(\rho )=\rho \).
Let \(\mathcal M\) be a finite full matrix algebra over \(\mathbb C\), and let \(F:\mathcal M\to \mathcal M\) be positive and trace nonincreasing. Suppose that \(F(V_j)=V_j\) for every \(j\in J\). Put \(\mathcal V=\operatorname{span}_{\mathbb {C}}\{ V_j:j\in J\} \) and, in the notation of [ CPGSV16 , Appendix C.4 ] , define
If \(\mathbf1\in C_L(\mathcal V)\) for some \(L{\gt}0\), then \(F\) is trace preserving.
Multilinearity gives
By Theorem 26.9.62, there is a positive-definite matrix \(\rho \) such that \(F(\rho )=\rho \). Let \(F^*\) be the trace-pairing adjoint and set \(\Delta =\mathbf1-F^*(\mathbf1)\). For every positive semidefinite \(X\), trace nonincrease gives \(\operatorname{tr}(\Delta X)=\operatorname{tr}(X)-\operatorname{tr}(F(X))\geq 0\). Positivity of \(F^*\) and self-duality of the positive semidefinite cone therefore give \(\Delta \geq 0\). The fixed-point equation gives \(\operatorname{tr}(\rho \Delta )=0\), so positive definiteness of \(\rho \) forces \(\Delta =0\). Thus \(F^*(\mathbf1)=\mathbf1\), which is equivalent to trace preservation.
Let \(\mathcal A=\bigoplus _{\alpha \in I}\mathcal M_{d_\alpha \times d_\alpha }\), let \(\iota :\mathcal A\to \operatorname{End}(\bigoplus _\alpha \mathbb C^{d_\alpha })\) be the block-diagonal embedding, and let \(\pi \) be diagonal compression. For a linear map \(F:\mathcal A\to \mathcal A\), set \(\widehat F=\iota \circ F\circ \pi \). Then:
if \(F\) is trace nonincreasing on positive elements of \(\mathcal A\), then \(\widehat F\) is trace nonincreasing;
if \(\widehat F\) is trace preserving, then \(F\) preserves the total trace on \(\mathcal A\).
If \(Y\geq 0\), then \(\pi (Y)\geq 0\), and the trace identities for diagonal compression and block-diagonal embedding give
This proves the first assertion. For \(X\in \mathcal A\), the identity \(\pi \circ \iota =\operatorname{id}_{\mathcal A}\) gives, whenever \(\widehat F\) is trace preserving,
Let \(\mathcal A=\bigoplus _{\alpha \in I}\mathcal M_{d_\alpha \times d_\alpha }\) be a finite sum of full matrix algebras, with total trace \(\operatorname{tr}_{\mathcal A}(X)=\sum _{\alpha \in I}\operatorname{tr}(X_\alpha )\). Let \(F:\mathcal A\to \mathcal A\) be positive and trace nonincreasing. Suppose that \(F(V_j)=V_j\) for every \(j\in J\). With \(\mathcal V=\operatorname{span}_{\mathbb {C}}\{ V_j:j\in J\} \), put
If \(\mathbf1_{\mathcal A}\in C_L(\mathcal V)\) for some \(L{\gt}0\), then \(F\) preserves the total trace on \(\mathcal A\).
Multilinearity first gives
Let \(\iota :\mathcal A\to \operatorname{End}(\bigoplus _\alpha \mathbb C^{d_\alpha })\) be the block-diagonal embedding and let \(\widehat F\) be the canonical full-matrix extension. These maps satisfy
The extension is positive and trace nonincreasing. The matrices \(\iota (V_j)\) are fixed by \(\widehat F\), and the displayed identities carry the product-span hypothesis to the full matrix algebra. Theorem 26.9.63 makes \(\widehat F\) trace preserving. Restricting to block-diagonal matrices proves preservation of \(\operatorname{tr}_{\mathcal A}\) by \(F\).
Let \(\mathcal A\) and \(\mathcal B\) be finite sums of full matrix algebras, and let \(T:\mathcal A\to \mathcal B\) and \(S:\mathcal B\to \mathcal A\) be linear maps. Suppose that \(T\) is positive, both maps are trace nonincreasing, and \(S\circ T\) is trace preserving. Then \(T\) is trace preserving.
For every positive \(X\in \mathcal A\), positivity of \(T\) and the two trace-nonincreasing inequalities give
Hence the middle term equals the endpoints. Equality extends from the positive cone to Hermitian matrices by positive and negative parts. For arbitrary \(X\), put
Both \(H_1\) and \(H_2\) are Hermitian, so linearity proves trace preservation for every \(X\).
Suppose that every one-site and two-site multiplicity space is nonzero and every corresponding diagonal weight is positive. Let \(\widetilde M^{(1)}\) and \(\widetilde M^{(2)}\) be the vertical tensors of \(M\) and its two-site blocking, and let \(B_1\) and \(B_2\) be their assembled one-site and two-site canonical forms. Assume that both canonical forms are bases of normal tensors and that, for every vertical letter \(\ell \),
Assume in addition that the physical maps \(T\) and \(S\) are completely positive and trace preserving and that, for every horizontal bond matrix \(Z\), the two physical-coordinate identities hold:
Then each transported map and both square composites preserve the appropriate total sector trace. For the individual maps,
For the square composites,
Neither of the stronger range inclusions is assumed:
Each transported map is positive and trace nonincreasing, and hence so are the square composites \(\widetilde S\widetilde T\) and \(\widetilde T\widetilde S\). The fixed-generator lemma and the product-spanning theorem give the hypotheses of Theorem 26.9.65 for both composites. Thus both preserve the corresponding total trace.
For positive \(X\in \mathcal A_1\), one has
Theorem 26.9.66 therefore makes \(\widetilde T\) trace preserving. Applying the same theorem to \(\widetilde S\) and \(\widetilde T\widetilde S\) proves the assertion for \(\widetilde S\).
If the physical refinement map \(T\) is completely positive, then the transported map \(T_{\mathrm v}\) in vertical canonical coordinates is completely positive.
By Definition 26.9.45,
The two single-operator conjugations and the reindexing \(J\) are completely positive, so the assertion follows by composition.
If the physical coarse-graining map \(S\) is completely positive, then the transported map \(S_{\mathrm v}\) in vertical canonical coordinates is completely positive.
By Definition 26.9.45,
Again, single-operator conjugation, reindexing, and composition preserve complete positivity.
Let \(\iota \) be block-diagonal inclusion of the weighted vertical-sector block family into its full matrix algebra, and let \(W\) be the canonical reindexing to retained physical coordinates. Then the retained-coordinate inclusion is \(\iota _{\mathrm v}=W\circ \iota \).
This is the defining relation between the direct-sum and retained coordinates.
Let \(\pi \) be diagonal-block extraction to the weighted vertical-sector block family, and let \(W\) be the canonical reindexing to retained physical coordinates. Then the retained-coordinate extraction is \(\pi _{\mathrm v}=\pi \circ W^{-1}\).
This is the inverse defining relation between the retained and direct-sum coordinates.
Write \(\widehat R_1\) for normalized state preparation, \(\widehat{\widetilde R}_2\) for controlled partial trace, and \(W_i\) for retained-coordinate reindexing. The full-matrix extension of the transported refinement map is
Substitute the preceding inclusion and extraction identities into \(\widetilde T=\widetilde R_2T_{\mathrm v}R_1\).
Write \(\widehat R_2\) for normalized state preparation, \(\widehat{\widetilde R}_1\) for controlled partial trace, and \(W_i\) for retained-coordinate reindexing. The full-matrix extension of the transported coarse-graining map is
Substitute the preceding inclusion and extraction identities into \(\widetilde S=\widetilde R_1S_{\mathrm v}R_2\).
Under the hypotheses of Theorem 26.9.67, the maps
are completely positive. The trace adjoints of the two square composites satisfy the Schwarz inequality in every simple summand.
Let \(\iota _i\) and \(\pi _i\) be block-diagonal inclusion and diagonal-block extraction, respectively, and let \(W_i\) denote the canonical reindexing from the weighted vertical-sector coordinates to the retained physical coordinates. The full-matrix extension of the transported refinement map has the factorization (??). Here \(\widehat R_1\) is normalized state preparation, \(\widehat{\widetilde R}_2\) is controlled partial trace, and
Every factor is completely positive. The analogous identities
and (??) prove complete positivity of \(\widetilde S\). These factorizations are exact on the full matrix algebras: \(\pi _iW_i^{-1}\) is precisely extraction of the retained diagonal sector blocks. Thus no range inclusion is used.
Composition gives complete positivity of \(\widetilde S\widetilde T\) and \(\widetilde T\widetilde S\). The preceding theorem gives trace preservation of these two composites. Their trace adjoints are therefore unital completely positive maps, and the Kadison–Schwarz inequality applies in each simple summand.
Under precisely the hypotheses of Theorem 26.9.67, the transported maps satisfy
Thus \(\widetilde T\) and \(\widetilde S\) are mutually inverse. This is the identity-composition conclusion of [ CPGSV16 , Appendix C.4, lines 1974–1995 ] .
For each square composite \(F\), positivity follows from positivity of the two transported maps. Theorem 26.9.67 and Theorem 26.9.74 give trace preservation and the Schwarz inequality for \(F^*\), respectively. Lemma 26.9.61 gives \(F(V_i(X))=V_i(X)\) for every bond matrix \(X\). Theorem 26.9.19 gives a positive integer \(L_i\) such that \(C_{L_i}(V_i)=\mathcal A_i\). Applying Theorem 26.9.25 first to \(F=\widetilde S\circ \widetilde T\) and then to \(F=\widetilde T\circ \widetilde S\) proves (??) and (??).
Under the hypotheses of Theorem 26.9.75, write
There are an equivalence \(\sigma \) of the two sector index sets, equalities \(D^{(1)}_\alpha =D^{(2)}_{\sigma (\alpha )}\), and unitaries \(V_\alpha \in M_{D^{(2)}_{\sigma (\alpha )}}(\mathbb {C})\) such that
where the nonzero entry on the right is in position \(\sigma (\alpha )\), and
where the entries on the left and right are in positions \(\sigma (\alpha )\) and \(\alpha \), respectively. These identities hold for every \(X\in M_{D^{(1)}_\alpha }(\mathbb {C})\) and \(Y\in M_{D^{(2)}_{\sigma (\alpha )}}(\mathbb {C})\). Here \(\iota _\alpha \) is the reindexing induced by the displayed dimension equality. These are the mutually inverse unitary formulas of [ CPGSV16 , Appendix C.4, line 1997 ] . They make no assertion about the multiplicities, weights, or coefficient identity at lines 2001–2008.
The two basis-of-normal-tensors decompositions have positive matrix dimensions. The transported maps are completely positive and preserve the total block trace, and the preceding theorem gives both inverse laws. Apply Theorem 25.7.9 to this pair. Its equivalence of summands, dimension equalities, and unitary matrices give the two displayed formulas without changing the chosen relabelling.
Under the hypotheses and with the notation of Theorem 26.9.76, let \(\mu _\alpha \) and \(\nu _\beta \) be the positive diagonal multiplicity matrices in the one-site and two-site vertical canonical forms, and set \(m_\alpha =\operatorname{tr}(\mu _\alpha )\) and \(n_\beta =\operatorname{tr}(\nu _\beta )\). For every horizontal bond matrix \(X\) and every sector \(\alpha \),
Consequently, for every tensor letter \(a\),
This is the coefficient identity in [ CPGSV16 , Appendix C.4, lines 2001–2008 ] . It compares the traces of the multiplicity matrices; it does not assert equality of their dimensions or of their individual diagonal entries.
Decompose the weighted contraction family into its individual matrix summands. At the summand paired with \(\alpha \), the unitary formula gives
Positivity of \(\nu _{\sigma (\alpha )}\) makes \(n_{\sigma (\alpha )}\) nonzero, so division gives the first identity. Taking \(X\) to be a matrix unit gives the tensor-letter identity.
Under the hypotheses and with the sector correspondence of Theorem 26.9.77, let \(B\) be the vertical reading of the two-site blocking, and write \(O_L(A)=\operatorname{Tr}_{\mathrm{bond}}(A_{i_1}\cdots A_{i_L})\) for the length-\(L\) closed-chain operator. If \(L{\gt}0\), then the same equivalence \(\sigma \), dimension identifications, and unitaries appearing in the tensor-letter identity satisfy
These are the two representations in [ CPGSV16 , Appendix C.4, lines 2011–2018 ] .
Local fix (blocked coefficient exponent): CPSV16 Appendix C.4, line 2013 prints \(m_\gamma ^L/n_\gamma \). The line-2008 tensor scaling and line 2040 give \((m_\gamma /n_\gamma )^L\). This is documented in docs/paper-gaps/cpsv16_blocked_operator_trace_ratio_exponent.tex.
The two-site vertical canonical form first gives \(O_L(B)=\sum _\delta \operatorname{tr}(\nu _\delta ^L) O_L(A^{(2)}_\delta )\). Reindex this sum by \(\delta =\sigma (\alpha )\). The tensor-letter identity from the preceding theorem multiplies every letter of \(A^{(1)}_\alpha \) by \(m_\alpha /n_{\sigma (\alpha )}\). Along a chain of length \(L\) this scalar therefore occurs to the power \(L\); unitary conjugation and the bond-index identification leave the closing trace unchanged. This proves the first formula.
For the second formula, the vertical reading of a two-site block is the product tensor of two copies of the one-site vertical reading. Closed chains turn this tensor product into multiplication of operators. Expand each copy by the one-site vertical canonical form, \(O_L(\widetilde M)=\sum _\alpha \operatorname{tr}(\mu _\alpha ^L)O_L(A^{(1)}_\alpha )\), and distribute the product of the two finite sums.
Suppose that the vertical reading of \(M\) is reconstructed from a retained tensor \(A\) by a coisometry \(U\). The two-site blocking is then reconstructed from the product of two copies of \(A\) by the Kronecker square of \(U\). Moreover, if \(M\) is in normalized BNT-refined horizontal form and generates positive operators, this retained product tensor has invariant-projector closure and has no nontrivial periodic vectors. No full-support assumption is imposed on the retained bond space.
The Kronecker square of a coisometry is again a coisometry. Expanding a blocked vertical letter gives the product of the two one-site reconstructions and hence the asserted exact compression formula. Normalized BNT-refined horizontal form and positivity are preserved by two-site blocking. Invariant-projector closure descends to an exact reducing compression, while a periodic vector in the compression would embed isometrically as a periodic vector of the blocked vertical tensor.
In the retained coordinates, the product tensor is the orthogonal direct sum over ordered pairs of BNT copies. Its summand indexed by \(((\alpha ,q),(\beta ,r))\) is \(\mu _{\alpha ,q}\mu _{\beta ,r}(M_\alpha M_\beta )\). Every such summand has invariant-projector closure and has no nontrivial periodic vectors. This includes zero-dimensional summands and requires no nonvanishing assumption on the displayed scalar.
The canonical inclusion of a dependent direct-sum block is an isometry. After transporting this inclusion through the retained-coordinate equivalence, it intertwines the chosen summand with the full retained product tensor on both sides. Its range projection therefore commutes with every tensor letter. Invariant-projector closure descends by exact compression, and any periodic vector of the summand would embed as a periodic vector of the full retained product.
For every ordered pair of retained BNT copies, the weighted product tensor has an exact orthogonal decomposition into positive multiples of normal tensors. The corresponding corner isometries are retained, intertwine in both directions, and reconstruct every tensor letter. The active family is allowed to be empty; in particular, no normal block is introduced for a zero-dimensional or identically zero copy-pair tensor.
Apply the canonical-form sufficient condition to the invariant-projector closure and periodic-exclusion properties of the preceding theorem. Its spectral construction discards precisely the corners on which every letter vanishes and retains the isometry of every nonzero corner.
A retained-product spectral family records, simultaneously for every ordered pair of retained BNT copies, all nonzero normal corners, their positive spectral coefficients, and their local isometries. Its active labels form the dependent union of the local corner families; this union is enumerated by a single finite type. The local isometries and their inclusions remain part of the data, and their range projections can be formed when needed.
Suppose that the one-site vertical tensor has the stated exact coisometric reconstruction, that \(M\) is in normalized BNT-refined horizontal form, and that \(M\) generates positive operators. Then simultaneous retained-product spectral data exist. In copy coordinates, the assembled vertical tensor is the direct sum of the weighted simple blocks. The canonical inclusion of each copy is an isometry, intertwines in both directions, and its compression selects exactly that weighted block. Likewise, the canonical inclusion of a copy pair, followed by its local corner isometry, is an isometry into the retained product bond space and intertwines in both directions. Composing once more with the adjoint of the squared vertical coisometry gives an isometry into the bond space of the blocked vertical tensor. Under the exact coisometric reconstruction, these ambient inclusions intertwine each active weighted corner with the blocked vertical tensor in both directions. In particular, if \(J_j\) is the ambient inclusion of an active corner, then \(J_j^\dagger \mathcal V(M^{(2)})^iJ_j=\lambda _jA_j^i\). These explicit composite inclusions retain the range projections required for the sector-weight comparison.
In copy coordinates the assembled tensor is block diagonal. The canonical dependent-sum inclusion is an isometry, and the two block-diagonal intertwining identities give its forward, adjoint, and compression formulas. Choose the exact normal-corner decomposition of every copy-pair tensor. The canonical copy-pair inclusions are isometries and select the weighted diagonal blocks of the retained product tensor. Composition with each local isometry gives the retained inclusion and both intertwining identities. Finally, write the blocked vertical tensor as \(T=W^\dagger CW\) and the ambient inclusion as \(J=W^\dagger R\). The coisometry identity \(WW^\dagger =1\) transports both retained intertwining identities from \(C\) to \(T\); the compression identity then follows from \(J^\dagger J=1\). For any blocked BNT label, choose its first multiplicity copy. Composing its canonical inclusion with the adjoint reconstruction coisometry gives an isometric reference corner of the blocked vertical tensor. Transport along an equality of bond dimensions preserves its isometry and exact compression formula.
Enumerate the dependent family of active corners by a single finite label set. The resulting coefficients are positive, the blocks are normal, and the inclusion ranges are pairwise orthogonal. For every tensor letter,
Each compression \(W_j^\dagger C^iW_j\) is \(\lambda _jA_j^i\), and \(C\) has the same positive-length closed chains as the finite direct sum \(\bigoplus _j\lambda _jA_j\).
First reconstruct the retained product as the sum of its canonical copy-pair blocks. Substitute the exact local spectral reconstruction in every pair, distribute the two inclusion maps across the finite sums, and identify the iterated sum with the dependent active-label sum. Finally transport this sum through the active-label enumeration. Isometry, orthogonality, intertwining, and compression are inherited by composition; the closed-chain equality follows from the exact isometric reconstruction. This is the active-corner decomposition in [ CPGSV16 , Appendix C.4, lines 2020–2029 ] .
Suppose that the one-site vertical tensor has the stated exact coisometric reconstruction, and let \((A^{(2)}_\gamma )_\gamma \) be a basis of normal tensors for the blocked vertical tensor. For every active product corner \(A_j\) there are a label \(\gamma (j)\), an equality of bond dimensions, an invertible matrix \(X_j\), and a scalar \(\zeta _j\) of modulus one such that, after transporting the blocked tensor along the dimension equality,
The choices may be made simultaneously. No converse coverage or surjectivity of \(j\mapsto \gamma (j)\) is asserted.
Scope restriction (active product BNT): The conclusion gives only one-way coverage of active product corners. It does not assert that every blocked BNT label occurs. This is documented in docs/paper-gaps/cpsv16_bnt_uniqueness_zero_coefficient.tex.
Local fix (per-pair support): Appendix C.4, lines 2011–2029 of [ CPGSV16 ] compares a sum over all retained copy pairs and does not isolate a fixed pair. Here fixed-pair support follows from the exact squared reconstruction and the canonical direct-sum corner; a zero pair contributes an empty active family. See docs/paper-gaps/cpsv16_figure11_per_pair_support.tex.
The squared coisometric reconstruction gives equality of all positive-length closed chains between the blocked vertical tensor and the retained product. Transport the blocked basis of normal tensors along this equality. Proposition 2.7, applied to the flattened positive normal decomposition, then covers each active corner by a blocked BNT tensor up to an invertible gauge and a unit-modulus phase. Classical choice makes these comparisons simultaneous.
Assume that every blocked BNT label has positive multiplicity and that the blocked vertical tensor has the exact coisometric reconstruction \(T=U_2^\dagger C_2U_2\). Let \(E_{\gamma ,0}\) be the canonical inclusion of the first copy of label \(\gamma \), and put \(F_\gamma =U_2^\dagger E_{\gamma ,0}\). Then \(F_\gamma \) is an isometry, intertwines \(T\) with the distinguished weighted BNT copy in both directions, and compresses \(T\) to that copy exactly. For an active product corner covered by \(\gamma \), transport \(F_\gamma \) along the stated equality of bond dimensions. The transported map remains an isometry and obeys the corresponding transported compression identity.
The copy inclusion is isometric and selects one diagonal block of the assembled tensor. Inserting \(T=U_2^\dagger C_2U_2\) and \(U_2U_2^\dagger =1\) gives both intertwining identities and compression. Transport along equality of bond dimensions preserves these equations.
Suppose that \(M\) is in normalized BNT-refined horizontal form and generates positive operators. Assume exact coisometric reconstructions of its one-site and blocked vertical tensors, positive multiplicities and weights for the blocked BNT, and normality of its BNT representatives. For every active product corner \(j\), one has \(0{\lt}\lambda _j\zeta _j\).
Compare two isometric corners of the blocked vertical tensor carrying the same normal representative. The distinguished blocked copy has positive weight by hypothesis. The active corner has weight \(\lambda _j\zeta _j\) by its exact compression and gauge–phase identity. This scalar is nonzero because \(\lambda _j{\gt}0\) and \(|\zeta _j|=1\). Normality gives a nonzero range-projection corner; the BNT-refined horizontal hypothesis separates it at some finite chain length. The sector-weight comparison with the distinguished reference corner then yields \(\lambda _j\zeta _j{\gt}0\).
Under the hypotheses of the preceding positivity theorem, for every active product corner \(j\) there is a real number \(\omega _j{\gt}0\) such that
Apply the Figure 8 comparison to the active corner and the distinguished blocked reference corner. Both coefficients are positive, so horizontal canonicality and positivity identify their Gram-dressed normal tensors. Normal-tensor rigidity then makes \(X_j^\dagger X_j\) a positive real multiple of the identity. Division by the positive square root gives a unitary matrix.
An original-label corner family for the retained product records, for every active corner of each copy pair, an original one-site BNT label, a positive coefficient, and an isometric inclusion into the raw product bond space. Distinct corners belonging to one copy pair have orthogonal ranges. Each inclusion intertwines the raw product tensor with the stated positive multiple of its original BNT representative, and the resulting corners reconstruct every tensor letter exactly.
Local fix (Figure-11 fixed-pair support): A copy pair with no active corner is represented by an empty family; no unsupported corner is inserted. See docs/paper-gaps/cpsv16_figure11_per_pair_support.tex.
Let the active product corners be covered by blocked BNT representatives, let the blocked representatives be identified with the original BNT by the trace-ratio unitary conjugacies, and suppose that the active gauge–phase coefficients are positive. If \(X_j^\dagger X_j=\omega _j\mathbb {1}\) with \(\omega _j{\gt}0\), then every retained copy pair has an original-label corner family. Its coefficient at an active corner is
which is positive. The normalized inclusions are isometries with pairwise orthogonal ranges, intertwine the raw product tensor with these positive multiples of the original BNT representatives, and reconstruct every raw product letter exactly.
Local fix (Figure-11 fixed-pair support): Empty active families remain empty. Exact weighted reconstruction and positivity of the outer copy weight imply that the corresponding raw product vanishes. See docs/paper-gaps/cpsv16_figure11_per_pair_support.tex.
Transport each local inclusion first by the normalized active comparison gauge and then by the unitary identifying the blocked representative with its original one-site label. The first normalization preserves isometry and orthogonality; the second unitary transport does likewise. The two conjugacy equations give the stated coefficient and intertwining identity. Positivity follows from the active gauge–phase coefficient, the two positive multiplicity traces, and the positive outer copy weight. Conjugating each corner term through the two transports recovers its local spectral term. Summing these identities gives the weighted reconstruction, and cancellation of the nonzero outer weight gives the raw reconstruction.
Given an original-label corner family, choose the distinguished retained copy of every BNT label. For fixed labels \(\alpha ,\beta \), let the multiplicity of \(\gamma \) be the number of active corners of the distinguished copy pair whose original label is \(\gamma \), and put the corresponding positive corner coefficients on the diagonal of \(\chi _{\alpha ,\beta ,\gamma }\). The block row of the adjoints of the corner inclusions, regrouped by these label fibers, defines a fusion coisometry. It satisfies
together with the exact reconstruction
Empty label fibers contribute zero-dimensional diagonal blocks. This is the decomposition of CPSV16, Appendix C.4, lines 2020–2029.
Scope restriction (active product BNT): Only active product corners are retained. A label absent from the distinguished copy pair has zero fusion multiplicity. See docs/paper-gaps/cpsv16_bnt_uniqueness_zero_coefficient.tex.
Local fix (Figure-11 fixed-pair support): The distinguished copy pair may have an empty active family; no unsupported corner is inserted. See docs/paper-gaps/cpsv16_figure11_per_pair_support.tex.
Local fix (Figure-11 fusion coisometry): The retained-row map is a coisometry onto the active direct sum. Exact reconstruction includes any common zero corner discarded by the forward map. See docs/paper-gaps/cpsv16_figure11_fusion_coisometry.tex.
For each distinguished copy pair, form the block row from the adjoints of its pairwise orthogonal isometric corner inclusions. The block-row identity makes this map a coisometry. Its forward conjugation is block diagonal over the active corners, while the corner reconstruction gives the reverse identity. Regrouping the active-corner sum by the original label turns each label fiber into the diagonal matrix whose entries are precisely the positive corner coefficients. These identities are valid without separate cases when a fiber or the whole active family is empty.
Under the one-site and two-site vertical basis-of-normal-tensors decompositions and the two completely positive, trace-preserving physical-closure transport identities, assume that the one-site tensor is in normalized BNT-refined horizontal form and defines a matrix product density operator. Then there are positive diagonal multiplicity matrices \(\chi _{\alpha ,\beta ,\gamma }\) and matrices \(U_{\alpha ,\beta }\) such that every diagonal entry is positive and
while the exact reconstruction is
This is the positive fusion decomposition of CPSV16, Appendix C.4, lines 2020–2029.
Scope restriction (active product BNT): Only active product corners occur. A label absent from a fixed product pair has zero fusion multiplicity. See docs/paper-gaps/cpsv16_bnt_uniqueness_zero_coefficient.tex.
Local fix (Figure-11 fixed-pair support): A fixed product pair may have no active corner, and no unsupported sector is inserted. See docs/paper-gaps/cpsv16_figure11_per_pair_support.tex.
Local fix (Figure-11 fusion coisometry): The retained-row map is a coisometry onto the active direct sum, and its adjoint gives the exact reconstruction. See docs/paper-gaps/cpsv16_figure11_fusion_coisometry.tex.
First decompose every retained copy-pair tensor into its active normal corners and compare these corners with the two-site vertical basis. Positivity of matrix product density operators makes every resulting scalar positive, while the Gram comparison normalizes each gauge to a unitary. The inverse vertical-sector transport then returns the corners to the original one-site labels. Their orthogonal isometric inclusions form the rows of \(U_{\alpha ,\beta }\). Grouping these rows by the original label gives the positive diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\); corner intertwining and corner reconstruction give the two displayed identities.
Let \(M\) be in normalized BNT-refined horizontal form, generate matrix product density operators, and satisfy the renormalization fixed-point condition of Definition 26.2.1. Then one may choose a vertical basis of normal tensors \(\{ M_\alpha \} _\alpha \) for which there are positive diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\) and matrices \(U_{\alpha ,\beta }\). Every diagonal entry of \(\chi _{\alpha ,\beta ,\gamma }\) is positive, and
Thus the renormalization fixed-point and matrix-product-density-operator assumptions, together with normalized BNT-refined horizontal form, imply the positive fusion decomposition of CPSV16, Appendix C.4, lines 2020–2029.
Scope restriction (active product BNT): Only active product corners occur. A label absent from a fixed product pair has zero fusion multiplicity. See docs/paper-gaps/cpsv16_bnt_uniqueness_zero_coefficient.tex.
Local fix (Figure-11 fixed-pair support): A fixed product pair may have no active corner, and no unsupported sector is inserted. See docs/paper-gaps/cpsv16_figure11_per_pair_support.tex.
Local fix (Figure-11 fusion coisometry): The retained-row map is a coisometry onto the active direct sum, and its adjoint gives the exact reconstruction. See docs/paper-gaps/cpsv16_figure11_fusion_coisometry.tex.
Apply the proved normalized BNT-refined specialization of Proposition 4.13 to \(M\) and to its two-site blocking. In each case, retain the CPSV basis-of-normal-tensors assertion, the positive multiplicity matrices, and both coisometric decomposition identities. Unpack the two completely positive, trace-preserving maps in the renormalization fixed-point condition. The preceding transported-sector theorem then supplies the positive diagonal family, the row coisometries, the forward fusion identity, and exact reconstruction for the chosen one-site vertical basis.
After identifying two blocked physical indices with four original indices, one has \((K^{[2]})_2(X)=K_4(X)\).
Expand the two blocked tensor matrices and reassociate their product.
26.10 Simple local structure from SAL and ZCL
26.10.1 Simple canonical-form weights
Suppose a tensor with zero correlation length is gauge-equivalent, through block-diagonal copy gauges, to an assembled sector decomposition. There is a scalar \(\lambda {\gt}0\) such that every copy satisfies
This is the block equation in [ CPGSV16 , Appendix C.2, lines 1652–1657 ] .
Write \(\mathcal T_M=G\mathcal T_SG^{-1}\), where \(G\) is the induced block-diagonal gauge. The equation \(\mathcal T_M^2=\lambda \mathcal T_M\) therefore becomes \(\mathcal T_S^2=\lambda \mathcal T_S\) after conjugation by \(G^{-1}\). Restriction to the \((j,q)\)-th diagonal block gives (??).
Let a tensor with zero correlation length be gauge-equivalent, through block-diagonal copy gauges, to an assembled sector decomposition whose basis elements all have non-nilpotent physical-trace transfer. Then the weights do not depend on the copy index: \(\mu _{j,q}=\mu _{j}\) [ CPGSV16 , Appendix C.2, lines 1646–1661 ] .
Zero correlation length states \(\mathcal T_M^{2}=\lambda \mathcal T_M\) with \(\lambda {\gt}0\). If \(\mathcal T_M=X\mathcal T_SX^{-1}\) is the block-diagonal gauge to the transfer of the assembled sector decomposition, then cancellation of \(X\) and \(X^{-1}\) gives \(\mathcal T_S^{2}=\lambda \mathcal T_S\). By Lemmas 26.7.2 and 26.7.3, the identity descends to every copy:
The left side does not depend on \(q\), and \(\mathcal T_{A_j}\ne 0\) because it is not nilpotent, so \(\lambda /\mu _{j,q}\) is independent of \(q\), which gives \(\mu _{j,q}=\mu _{j,q'}\).
A simple canonical-form tensor (Definition 26.7.5) with zero correlation length has non-nilpotent basis transfers and weights independent of the copy index.
Apply Theorem 26.10.1.2 to the witness supplied by the simple canonical-form predicate.
Suppose the weights over each representative do not depend on the copy index. Choose the common weight \(\mu _j\) from any copy over \(j\) and absorb it into the representative:
This is the convention of [ CPGSV16 , Appendix C.2, equation CFK, lines 1660–1665 ] .
For every copy \(q\) over \(j\), one has \(\mu _{j,q}=\mu _j\).
Compare the \(q\)-th copy with the distinguished copy used to define \(\mu _j\).
Let \(r_j\) be the number of horizontal copies of representative \(j\). Then its coefficient on a chain of length \(N\) is
Thus the natural multiplicity is the horizontal copy count \(r_j\).
Every term in the copy sum equals \(\mu _j^N\), and there are \(r_j\) terms.
The common weight is nonzero, and hence for every chain length \(N\),
This is the coefficient nonvanishing used in the choice \(m_j=q_j\widetilde m_j\) in [ CPGSV16 , Appendix C.2, lines 1714–1718 ] . It does not assert nonvanishing of the independent virtual tail contraction \(\widetilde m_j\).
The distinguished copy weight is nonzero by the sector data. Since \(r_j{\gt}0\), both factors in \(r_j\mu _j^N\) are nonzero.
Suppose the weights over each representative are independent of the copy index, and none of the physical-trace transfers \(\mathcal B_j=\sum _i A_j^{ii}\) is nilpotent. There is one positive tail length \(L\) such that, for every representative \(A_j\), some virtual indices \(\alpha ,\beta \) satisfy \(\widetilde m_j=[\mathcal B_j^L]_{\alpha ,\beta }\ne 0\). At total chain length \(L+3\), the corresponding copy coefficient is \(q_j=\sum _q\mu _{j,q}^{L+3}\ne 0\). Thus the three-site closing matrix satisfies
This is the traced-tail selection made in [ CPGSV16 , Appendix C.2, lines 1714–1718 ] .
Since \(\mathcal B_j\) is not nilpotent, every positive power of \(\mathcal B_j\) is nonzero; in particular, one common positive power has a nonzero entry for every \(j\). The preceding theorem supplies the nonzero coefficient at length \(L+3\). Therefore its product with the selected entry of \(\mathcal B_j^L\) is nonzero.
After absorbing the common weight, the matrix product vector of the assembled tensor is
This is equation CFK of [ CPGSV16 , Appendix C.2, lines 1660–1665 ] ; the same multiplicities occur in equation sigmaNKj of [ CPGSV16 , Appendix C.2, lines 1756–1759 ] .
Expand the assembled tensor by representatives, use Theorem 26.10.1.6, and absorb \(\mu _j^N\) into the length-\(N\) matrix product vector of \(A_j\).
Let \((\mathcal K_s)_s\) be a finite family of MPO tensors, with \(\mathcal K_s^{ij}\in M_{D_s}(\mathbb {C})\), and let \(R_s\in M_{D_s}(\mathbb {C})\). A three-site operator is a family closure when
Given a simultaneous left inverse \(C\) as in Theorem 27.3.74, contracting \(C\) against the first and third physical sites of \(\rho \) defines the outer inverse contraction
The closure is the sector-sum form of the displayed equation for \(\sigma _3^{(N)}\) in [ CPGSV16 , Appendix C.2, lines 1343–1348 ] , used with the block-canonical tensor in the Case II calculation at [ CPGSV16 , Appendix C.2, lines 1726–1732 ] . The outer contraction is the sector-family form of the Case I contraction in [ CPGSV16 , Appendix C.2, lines 1415–1438 ] ; the same construction is used with sector projectors at lines 1719–1725.
Suppose that the doubled-index tensor has the horizontal BNT representation \(S\), the copy weights over each representative agree, the physical-trace transfers \(\mathcal B_j=\sum _iA_j^{ii}\) are not nilpotent, and the original tensor satisfies SAL. After identifying length-three configurations with ordered triples, the normalized three-site marginal of the four-site state is the family closure
where
Every \(\widehat R_j\) is nonzero. More generally, a tail of length \(L\) gives \(\widehat R_j=\operatorname{tr}(\rho ^{(L+3)})^{-1}q_j\mathcal B_j^L\). This is the normalized traced-tail expansion used in [ CPGSV16 , Appendix C.2, lines 1714–1732 ] .
Equality of the positive-length matrix-product-vector families expands every matrix entry of the original operator as the sum of the BNT entries with coefficients \(q_j\). Summing the diagonal physical configurations of the tail gives \(\mathcal B_j^L\). The global normalization multiplies every closing matrix by \(\operatorname{tr}(\rho ^{(L+3)})^{-1}\). At \(L=1\), SAL makes this scalar nonzero, copy independence makes \(q_j\) nonzero, and nonnilpotence makes \(\mathcal B_j\) nonzero.
Let \(x=(s,\alpha _1,\beta _1)\) and \(y=(t,\alpha _3,\beta _3)\). For a three-site family closure and a simultaneous left inverse, the outer inverse contraction vanishes when \(s\ne t\). When \(s=t\), it is \((\mathcal K_s^{i_2j_2})_{\beta _1,\alpha _3} (R_s)_{\beta _3,\alpha _1}\). In particular, the contraction retains precisely one entry of the middle physical slice and one entry of the closing matrix, with the orientation shown above. This is the sector-family form of the Case I contraction in [ CPGSV16 , Appendix C.2, lines 1415–1438 ] . The basis-of-normal-tensors inverse is constructed at lines 1666–1676 and the resulting contraction is used in the Case II calculation at lines 1719–1732 of the same appendix. The closing entry \(R_s(\beta _3,\alpha _1)\) is the corrected tail-index orientation of the display at lines 1422–1438, recorded in docs/paper-gaps/cpgsv17_mpdo_sal_zcl_eta_local_structure.tex.
Expand the family closure and interchange the finite sector and physical sums. For \(x=(x_s,x_\alpha ,x_\beta )\), the inverse identity is
Hence distinct labels give zero. For equal labels, the remaining trace is
Under the one-letter product-algebra spanning hypothesis, there is a simultaneous left inverse for which every three-site family closure has the preceding diagonal-sector contraction and vanishing off-diagonal contractions.
Choose the simultaneous left inverse from Theorem 27.3.74 and apply Theorem 26.10.1.12.
26.10.2 Markov decomposition and inverse-map sector factorization
For the simple MPDO case in Appendix C.2 of [ CPGSV16 ] , the strong-area-law hypothesis is used locally through the equality case of strong subadditivity for a normalized three-site reduced state.
For a three-site density operator \(\rho _{ABC}\), the local \(\eta \)-structure is the quantum Markov decomposition on the middle subsystem \(B\) produced by equality in strong subadditivity. Concretely, after a unitary change of basis on \(B\), one has a finite direct-sum decomposition
Let \(B=\bigoplus _k(b_1^{(k)}\otimes b_2^{(k)})\) be the middle-site decomposition, and let \(E_k\) be the coordinate projection onto its \(k\)-th summand. In the original physical basis, define \(Q_k=U_B^*E_kU_B\). These are the projectors introduced in [ CPGSV16 , Appendix C.2, lines 1364–1368 ] ; the unitary implements the basis change described in [ CPGSV16 , Appendix C.2, lines 1437–1440 ] .
Each \(Q_k\) is an orthogonal projection, distinct summands are mutually orthogonal, and the family resolves the identity:
The first two assertions follow from the orthogonal direct-sum decomposition in [ CPGSV16 , Appendix C.2, lines 1364–1368 ] ; the resolution of the identity is stated in [ CPGSV16 , Appendix C.2, lines 1693–1697 ] .
The coordinate projections satisfy
Reindexing the direct sum and conjugating by \(U_B\) preserve these identities. Hence each \(Q_k=U_B^*E_kU_B\) is a star projection and therefore an orthogonal projection, while \(\sum _k Q_k=U_B^*(\sum _kE_k)U_B=\mathbb {1}\). Finally, multiplying the resolution on both sides by \(Q_k\) gives
The \(\ell =k\) term already equals \(Q_k\), so every term with \(\ell \ne k\) vanishes. Thus \((Q_\ell Q_k)^*(Q_\ell Q_k)=0\), hence \(Q_\ell Q_k=0\). Taking adjoints gives \(Q_kQ_\ell =0\) for \(\ell \ne k\).
Let \(C_x\) and \(C_y\) be two rows of a simultaneous left inverse, and choose a Hayashi decomposition of the three-site operator. In its \(k\)-th Markov sector, set
These are the two factors obtained in the projected contraction of [ CPGSV16 , Appendix C.2, lines 1719–1725 ] .
Let \(\Gamma _{x,y}\) denote the outer inverse contraction. In the basis of the chosen Hayashi decomposition,
In particular, the probability \(p_k\) remains explicit. This identity requires neither a nonvanishing condition on \(p_k\) nor assumptions of saturated area law or zero correlation length.
Expand the two matrix products and the outer inverse contraction. The Hayashi block decomposition makes the middle-site contraction zero for \(k\ne k'\). For \(k=k'\), the remaining four outer sums separate into the product defining \(A_x^{(k)}\) and \(B_y^{(k)}\), with the common factor \(p_k\).
Let \(x=(s,\alpha _1,\beta _1)\) and \(y=(t,\alpha _3,\beta _3)\). Comparing the two expressions for the same outer inverse contraction gives
Here \(\kappa ^{(s)}_{\beta _1,\alpha _3}(i,j) ={(\mathcal K_s^{ij})}_{\beta _1,\alpha _3}\). Thus both the normal-sector and Markov-sector off-diagonal entries are retained in the statement, rather than being suppressed by a prior choice of equal labels. This is the full entrywise identity used in [ CPGSV16 , Appendix C.2, lines 1719–1732 ] .
The closing entry \({(R_s)}_{\beta _3,\alpha _1}\) is the corrected tail-index orientation of the display at lines 1422–1438, recorded in docs/paper-gaps/cpgsv17_mpdo_sal_zcl_eta_local_structure.tex.
The preceding theorem identifies the left side with the transformed outer inverse contraction. The collapse theorem identifies the contraction with zero when \(s\ne t\) and with \({(R_s)}_{\beta _3,\alpha _1} \kappa ^{(s)}_{\beta _1,\alpha _3}\) when \(s=t\). Conjugating this matrix by \(U_B\) gives the right side.
Fix a normal sector \(s\) and virtual indices \(\alpha _1,\beta _3\) for which \({(R_s)}_{\beta _3,\alpha _1}\ne 0\). Then, for every \(\beta _1,\alpha _3\) and every pair of distinct Markov sectors \(k\ne k'\), one has
Thus each physical slice of the \(s\)-th normal tensor is block diagonal in the Markov decomposition. This is equation Qks in [ CPGSV16 , Appendix C.2, lines 1682–1688 and 1730–1735 ] .
The nonzero closing entry uses the corrected orientation \({(R_s)}_{\beta _3,\alpha _1}\) recorded in docs/paper-gaps/cpgsv17_mpdo_sal_zcl_eta_local_structure.tex.
In the entrywise BNT–Markov identity, take equal normal-sector labels and distinct Markov-sector labels. The left side vanishes, while the right side is the selected closing entry times the displayed block. Cancel the nonzero closing entry.
For a normal sector \(s\) and a Markov sector \(k\), let
The assertion \(O_s^{(k)}\ne 0\) means that this block is nonzero for at least one pair of virtual indices \(\beta ,\alpha \).
If \(p_k\ne 0\), then there is a normal sector \(s\) for which \(O_s^{(k)}\ne 0\).
This is the existence part of equation QkKjs in [ CPGSV16 , Appendix C.2, lines 1733–1737 ] . Restricting to \(p_k\ne 0\) is the literal version of the source’s preceding removal of summands for which \(Q_k\sigma _3^{(N)}Q_k=0\).
Choose nonzero diagonal entries of the trace-one left and right states in sector \(k\). The corresponding diagonal entry of the Hayashi decomposition is nonzero because \(p_k\ne 0\). Expanding the same entry by the three-site normal-sector family therefore leaves at least one nonzero summand, whose middle tensor has \(O_s^{(k)}\ne 0\).
Suppose, in addition, that every closing matrix \(R_s\) is nonzero. If \(p_k\ne 0\), there is a unique normal sector \(j_k\) for which \(O_{j_k}^{(k)}\ne 0\).
This is the cancellation argument in equation QkKjs. The stated nonzero-closing-matrix condition isolates the step obtained in the source from simplicity and the common-weight choice at [ CPGSV16 , Appendix C.2, lines 1714–1718 ] . Its derivation from those preceding hypotheses is Theorem 26.10.1.11.
Assume that sectors \(s\ne t\) both have nonzero \(k\)-blocks. A diagonal instance of the BNT–Markov identity for \(s\) gives a nonzero left factor. A mixed instance with normal-sector labels \(s,t\) forces the right factor selected from \(t\) to vanish. A diagonal instance for \(t\) then has zero left-hand side and a nonzero right-hand side, a contradiction.
Under the hypotheses of Theorem 26.10.2.10, define
This is equation Pis, restricted to the positive-weight Markov support retained here.
Every \(P_s\) is an orthogonal projection, distinct \(P_s\) are mutually orthogonal, and \(\sum _s P_s=P_{\mathrm{act}}\).
Each \(P_s\) is a sum of mutually orthogonal Markov-sector projections. Distinct labels have disjoint sums. Finally, summing over \(s\) counts each positive-weight Markov sector exactly once.
Let \(1\leq m\leq \lfloor N/2\rfloor \), and divide the chain into four consecutive regions \(A,B,C,D\) of lengths \(m-1,1,N-m-1,1\). If the tensor saturates the area law, then its marginal on \(ABC\) satisfies \(S(ABC)+S(B)=S(AB)+S(BC)\). Indeed, this is the entropy form of \(I_1=I_m\). Taking \(m=2\) gives the regions of lengths \(1,1,N-3\), and then taking \(N=4\) shows that the three-site reduced state of the four-site chain satisfies equality in strong subadditivity.
Conversely, suppose the tensor generates positive semidefinite periodic operators and their traces are nonzero at every positive chain length. If the displayed equality holds for every \(N\) and every \(1\leq m\leq \lfloor N/2\rfloor \), then the tensor saturates the area law. This is the choice of regions used in [ CPGSV16 , Appendix C.2, lines 1760–1780 ] . The endpoint corrects the strict inequality printed at source line 1771: Definition 4.6 and the concluding comparison both require \(m=\lfloor N/2\rfloor \). This local correction is recorded in docs/paper-gaps/cpgsv17_mpdo_sal_zcl_eta_local_structure.tex.
By saturation, \(I_1=I_m\). Expanding the two mutual informations and cancelling the entropy of the full chain gives \(S_{N-1}+S_1=S_m+S_{N-m}\), which is the stated equality for the regions \(A,B,C\). Conversely, this equality gives \(I_1=I_m\). If it holds for every admissible \(m\), comparing the cases \(m=L\) and \(m=L+1\) gives \(I_L=I_{L+1}\) and hence saturation of the area law.
Suppose that the periodic operators are positive semidefinite and have nonzero trace at every positive chain length. For every \(N\) and every \(1\leq m\leq \lfloor N/2\rfloor \), divide the chain into four consecutive regions of lengths \(m-1,1,N-m-1,1\). If the marginal on the first three regions admits a quantum Markov decomposition on its one-site middle system, then the tensor saturates the area law.
The quantum Markov decomposition gives equality in strong subadditivity at every admissible cut: \(S(\rho _{ABC})+S(\rho _B)=S(\rho _{AB})+S(\rho _{BC})\). These equalities at every admissible cut imply saturation of the area law by Theorem 26.10.2.13.
If a normalized three-site reduced state satisfies the equality case of strong subadditivity — the local entropy form of the simple-MPDO strong area law — then it admits the local \(\eta \)-structure of Definition 26.10.2.1.
This is the forward implication of the Hayashi equality characterization. The product-reference raw Petz formula, including its singular-support compression, is given by Theorem 21.2.49. The singular-support equality-to-recovery theorem is still separate. Beyond it, the remaining structural ingredient is the family-level Koashi–Imoto decomposition and the induced action of the middle-system channel on its common direct-sum factors.
Suppose that \(C\simeq C'\otimes E\). The partial trace over \(E\) is given entrywise by
It preserves positivity and trace.
Reindex \(C\) as \(C'\otimes E\) and apply the ordinary right partial trace. Its entry formula gives the displayed sum. Positivity follows by tracing a positive matrix, and invariance of the full trace under reindexing gives trace preservation.
Suppose that \(C\simeq C'\otimes E\) and
is a quantum Markov decomposition on \(B\). Then the marginal \(\rho _{ABC'}=\operatorname{tr}_E(\rho _{ABC})\) has the decomposition
The middle-system decomposition, its unitary change of basis, and the probabilities \(p_k\) are unchanged. Partial trace preserves positivity and trace, so each \(\operatorname{tr}_E(\rho _{b_2 C}^{(k)})\) is a density operator. Applying \(\operatorname{tr}_E\) to the block-diagonal identity gives the displayed formula.
Let \(a,b,c,N\in \mathbb N\) satisfy \(a+b+c\leq N\). For configurations \(u,v\) on the first \(a+b\) sites,
where the sum ranges over configurations on the last \(c\) sites. The same identity holds after replacing an arithmetically equal block length by its canonical identification.
Expand both reduced states as partial traces of the normalized periodic state. The left-hand side first traces the complement of the \((a+b+c)\)-site block and then its last \(c\) sites; combining the two sums gives the partial trace onto the first \(a+b\) sites.
Let \(K\) satisfy the strong area law and let \(N\geq 4\). If \(\sigma _3^{(N)}(K)\) is obtained from the normalized \(N\)-site periodic state by tracing out all but its first three sites, then \(\sigma _3^{(N)}(K)\) admits a quantum Markov decomposition on its middle site. This is the three-site marginal result proved in [ CPGSV16 , Appendix C.2, lines 1351–1371 ] . The decomposition may depend on \(N\).
Divide the first \(N-1\) sites into consecutive regions of lengths \(1\), \(1\), and \(N-3\). Let \(\rho ^{\mathrm{flat}}_{ABC}\) denote this marginal in the corresponding index set \(\{ 0,\ldots ,d^1{-}1\} \times \{ 0,\ldots ,d^1{-}1\} \times \{ 0,\ldots ,d^{N-3}{-}1\} \). By Theorem 26.10.2.13,
The first two singleton blocks have the canonical identification \(e_{\mathrm{site}}:\{ 0,\ldots ,d{-}1\} \simeq \{ 0,\ldots ,d^1{-}1\} \), and reindexing these factors preserves the displayed equality. Positivity and unit trace follow from the normalized reduced state, so the Hayashi characterization gives a quantum Markov decomposition of the \((N-1)\)-site marginal. Finally split the third region as \(C\simeq c\otimes E\), where \(c\) is its first site and \(E\) contains the remaining \(N-4\) sites. Theorem 26.10.2.17 gives the same middle-site direct sum after tracing out \(E\). The contiguous-block reduction gives
which identifies the resulting operator with \(\sigma _3^{(N)}(K)\).
Let \(K\) satisfy the strong area law. If \(\sigma _3^{(4)}(K)\) is obtained from its normalized four-site periodic state by tracing out the fourth site, then \(\sigma _3^{(4)}(K)\) admits a quantum Markov decomposition on its middle site.
This is the case \(N=4\) of Theorem 26.10.2.19.
Suppose that the common blocking has been performed, so that the BNT representatives have a simultaneous one-letter span. For the normalized three-site marginal \(\sigma _3^{(4)}\), there are a simultaneous inverse \(C\) and a Hayashi decomposition such that every Markov sector \(k\) with \(p_k\ne 0\) has a unique BNT label \(j_k\). Consequently,
Each \(P_s\) is an orthogonal projection, \(P_sP_t=0\) for \(s\ne t\), and \(\sum _sP_s=P_{\mathrm{act}}\). No independent nonzero-closing-matrix hypothesis is required. This is the four-site specialization of equations QkKjs and Pis in [ CPGSV16 , Appendix C.2, lines 1714–1737 ] . The Hayashi decomposition uses the forward equality characterization of strong subadditivity recorded in Theorem 21.4.40; no additional analytic assumption is introduced here.
The canonical ordered-triple reindexing is the submatrix convention of the SAL-to-Hayashi theorem. Choose the simultaneous inverse from the one-letter span and the Hayashi decomposition from SAL. By Theorem 26.10.1.11, the same marginal is a BNT family closure whose closing matrices are all nonzero. The generic uniqueness and projector theorems now give the unique labels, orthogonality, mutual disjointness, and resolution of the active support.
For a one-site matrix \(V\), write \(V^{\otimes N}\) in the configuration basis as
If the copies of the \(j\)-th normal tensor have common coefficient \(\mu _j\), let \(\mathcal K_j=\mu _j A_j\) denote the representative with this coefficient absorbed into its local tensor.
Under the hypotheses of Theorem 26.10.2.21, the resulting projectors satisfy \(P_sA_iP_t=0\) whenever \(s\ne t\) or \(i\ne s\). For the matching label, they satisfy \(P_iA_iP_i=A_i\). Consequently, \(\sum _sP_sA_iP_s=A_i\), which is the local diagonal decomposition required for the sector entropy argument. Equivalently, compressing the physical legs of \(A_i\) by \(P_s\) retains \(A_i\) for \(s=i\) and gives the zero local tensor otherwise. This is the local projection relation in [ CPGSV16 , Appendix C.2, lines 1680–1755 ] .
The Markov summands of zero Hayashi weight are retained in the ambient decomposition. They annihilate every normal-sector physical slice, so the displayed local identities do not require the stronger assertion \(\sum _sP_s=\mathbb {1}\) outside the active support.
Distinct Hayashi sectors give zero off-diagonal corners. On an active diagonal sector, uniqueness of the BNT label kills every normal tensor except the matching one. A zero-weight diagonal sector also kills every physical slice: otherwise the nonzero closing matrix would make the corresponding three-site Markov block nonzero. Summing these statements over the Hayashi sectors gives the asserted \(P_sA_iP_t\) identities.
Fix a BNT label \(s_0\). Each positive-weight Markov sector retains its unique BNT label, while every zero-weight Markov sector is assigned the label \(s_0\). If \(\widehat S_s\) is the set of Markov sectors with completed label \(s\), define
Let \(K\) be simple, in biCF, and satisfy SAL. Assume in addition that the common blocking has been performed, the copy weights over each BNT representative agree, and the representatives have a simultaneous one-site span. There is a distinguished BNT label \(s_0\) such that the completed projectors are orthogonal and, for distinct labels \(s\) and \(t\),
For every physical slice of the \(i\)-th BNT representative, whenever \(s\ne t\) or \(i\ne s\),
This is the copy-independent-weight specialization of the separating-projector lemma in [ CPGSV16 , Appendix C.2, lines 1680–1737 ] .
A unit-modulus BNT weight supplies the label \(s_0\). The Markov-sector projections \(Q_k\) are mutually orthogonal and sum to the identity, so partitioning all labels by \(\widehat S_s\) proves orthogonality and the identity resolution. If \(p_k=0\), then \(Q_kK_i^{\beta \alpha }Q_k=0\) for every \(i,\beta ,\alpha \). If \(p_k\ne 0\), then \(Q_kK_i^{\beta \alpha }Q_k=0\) whenever the BNT label of sector \(k\) is not \(i\). For distinct Markov sectors, \(Q_kK_i^{\beta \alpha }Q_l=0\) whenever \(k\ne l\). Hence, whenever \(s\ne t\) or \(i\ne s\),
since every summand vanishes by one of the three preceding cases.
Let \(K\) be a simple tensor in block-injective canonical form which satisfies ZCL and SAL. For the representatives \(\mathcal K_i\) of its BNT, there are orthogonal projectors \(P_s\) such that
This is the separating-projector lemma in [ CPGSV16 , Appendix C.2, lines 1626–1691 ] .
Write the horizontal canonical form as \(K=G(\bigoplus _{j,q}\mu _{j,q}\mathcal K_j)G^{-1}\). Restricting the ZCL equation to the \((j,q)\)-th block and using the non-nilpotence supplied by simplicity gives \(\mu _{j,q}=\mu _{j,q'}\) for all \(j,q,q'\). The block-diagonal gauge preserves every positive-length periodic state, so the canonical-form tensor and \(K\) generate the same such states. The preceding copy-independent-weight theorem now supplies the projectors.
For every nonempty periodic chain and every normal-sector label \(s\),
This is equation sigmaNKj in [ CPGSV16 , Appendix C.2, lines 1756–1759 ] .
Here and throughout the MPDO discussion, a periodic chain has positive length.
Sitewise multiplication by \(P_s\) is the periodic MPO obtained by changing every local physical matrix by \(P_s\). The preceding theorem replaces each normal representative by itself when its label is \(s\) and by the zero tensor otherwise. For \(N\geq 1\), the latter has zero periodic MPO. The surviving coefficient is \(n_s\mu _s^N\), which is precisely \(n_s\) times the MPO of the representative with \(\mu _s\) absorbed locally.
Under the hypotheses of Theorem 26.10.2.27, every absorbed BNT element \(\mathcal K_s\) generates positive semidefinite operators on all nonempty periodic chains. Indeed, \(P_s^{\otimes N}\sigma ^{(N)}(K)P_s^{\otimes N}\geq 0\) and \(n_s{\gt}0\), so (??) gives \(\sigma ^{(N)}(\mathcal K_s)\geq 0\). This is the first positivity conclusion in [ CPGSV16 , Appendix C.2, lines 1753–1759 ] .
Congruence by the Hermitian projection \(P_s^{\otimes N}\) preserves positive semidefiniteness. Multiplication by the inverse of the positive integer \(n_s\) then gives positivity of the sector operator.
Let \(K\) have zero correlation length and let \(\mathcal K_s=\mu _sA_s\) be an absorbed BNT element of its horizontal canonical form. If the physical-trace transfer \(\mathcal B_s\) is not nilpotent, then \(\mathcal K_s\) has zero correlation length. More precisely,
This is the final zero-correlation-length conclusion in [ CPGSV16 , Appendix C.2, line 1781 ] .
Lemma 26.10.1.1 gives the displayed quadratic identity. The common weight is nonzero, and nonnilpotence of \(\mathcal B_s\) implies \(\mathcal B_s\ne 0\), so the absorbed transfer is nonzero.
Let \(M\) generate matrix product density operators and have source zero correlation length. Then, for every \(N\geq 1\), \(\operatorname{tr}(\sigma ^{(N)}(M)){\gt}0\). In particular, after a common blocking has made the simultaneous BNT word span full at one letter, the absorbed BNT elements selected in Theorem 26.10.2.29 have a well-defined normalized state at every physical chain length. This is the biCF hypothesis imposed at the start of Case II in the source; its relation with finite physical blocking is recorded in docs/paper-gaps/cpgsv17_bicf_block_separation.tex. This supplies the strict form of the sector normalization used in [ CPGSV16 , Appendix C.2, lines 1760–1780 ] .
Write the physical-trace transfer as \(\mathcal T_M=\lambda E\), where \(\lambda {\gt}0\) and \(E\) is a nonzero idempotent. A nonzero idempotent on a finite-dimensional complex vector space has nonzero trace, while \(E^N=E\) for \(N\geq 1\). Hence
Positivity of the density operator then makes this trace strictly positive.
Let \(K\) saturate the area law and have zero correlation length, and let its horizontal BNT canonical form, up to block-diagonal gauge, have normal representatives \(A_s\), common copy weights \(\mu _s\), and copy numbers \(n_s\). Assume that the physical-trace transfer of every \(A_s\) is nonnilpotent. After a common blocking, assume explicitly that the simultaneous one-letter word tuples of the \(A_s\) span the direct sum of their full matrix algebras. Put \(\mathcal K_s=\mu _sA_s\).
For every nonempty chain, define
Then \(p_s^{(N)}{\gt}0\), and every nonempty prefix marginal has the same sector decomposition with these probabilities. Translation invariance identifies the other contiguous blocks of the same length. Consequently, every absorbed representative \(\mathcal K_s\) saturates the area law.
The one-letter simultaneous span is the block-injective canonical-form hypothesis imposed at the start of Case II in [ CPGSV16 , line 1628 ] . Its relation with finite physical blocking is recorded in docs/paper-gaps/cpgsv17_bicf_block_separation.tex. The entropy conclusion is the argument of [ CPGSV16 , Appendix C.2, lines 1748–1781 ] .
The BNT projections carry every normalized marginal on mutually annihilating supports, so the preceding support form of entropy additivity gives
The four marginals entering strong subadditivity have the same probabilities. Substitution in the equality for \(K\) cancels all four Shannon terms. Strong subadditivity gives the corresponding inequality in each sector, and strict positivity of every \(p_s^{(N)}\) forces every such inequality to be an equality. Comparing the equalities for block lengths \(L\) and \(L+1\) proves saturation for \(\mathcal K_s\).
A simple MPO tensor \(K\) is called injective if its doubled-index MPS tensor is injective. For such a tensor, the chosen decomposition map of the doubled-index MPS furnishes a concrete inverse tensor \(K^{-1}\) together with right- and left-physical realization maps that turn virtual bond insertions back into local physical operators.
For an injective simple MPO tensor, contracting \(K^{-1}\) with \(K\) recovers the matrix units on the virtual bond space. Likewise, every right virtual insertion admits a physical realization, this realization is multiplicative, and the analogous left-insertion statement also holds.
For an injective simple tensor, contracting \(K^{-1}\) with \(K\) through the physical index identifies the corresponding virtual legs. In components, this is the matrix-unit identity
Its two Kronecker factors are the two bare cups:
![\begin{tenkz}[rows={ket, bra}, tensor style=box]
\tn{K^{-1}} \\
\tn{K}
\end{tenkz}](tenkz_svg/tenkz-5130cdba16f0585c.svg)
\( = \) ![\begin{tenkz}[rows={wire, wire}, east=cup]
\tnghost{} \\
\tnghost{}
\end{tenkz}](tenkz_svg/tenkz-ceb9a525af81d9be.svg)
\( \otimes \) ![\begin{tenkz}[rows={wire, wire}, west=cup]
\tnghost{} \\
\tnghost{}
\end{tenkz}](tenkz_svg/tenkz-01e1cf890db155d4.svg)
This is the identity in [ CPGSV16 , Appendix C.2, lines 1376–1380 ] .
Applying this inverse map to the doubled physical index and closing its virtual legs against a further copy of \(K\) recovers \(K\):
![\begin{tenkz}[
rows={ket:nopair, ket, bra},
west=cup,
east=cup,
tensor style=box]
\tn[up=p]{K} \\
\tn{K^{-1}} \\
\tn{K}
\end{tenkz}](tenkz_svg/tenkz-b08a6b7f3f7da2f7.svg)
\( = \) ![\begin{tenkz}[physical=up, tensor style=box]
\tn[up=p]{K}
\end{tenkz}](tenkz_svg/tenkz-cc12e7f4eaf193ac.svg)
This is the single-block identity in [ CPGSV16 , Appendix C.2, lines 1671–1676 ] .
Let \(\mathcal K\) be an MPO tensor and set \(Z_4=\operatorname{tr}[\rho ^{(4)}(\mathcal K)]\). Its normalized fourth-site virtual tail is
Thus the contraction of the fourth ket and bra indices is incorporated into the virtual closing matrix. This is the \(N=4\) specialization of the normalization convention and three-site marginal in [ CPGSV16 , lines 792–793 and 1343–1348 ] .
Let \(u=(i_1,i_2,i_3)\) and \(v=(j_1,j_2,j_3)\) be physical words of length three. The corresponding entry of the first-three-site marginal of the normalized four-site periodic operator is
![\begin{tenkz}[physical=updown, periodic]
\tn[up=$i_1$, down=$j_1$]{\mathcal K} &
\tn[up=$i_2$, down=$j_2$]{\mathcal K} &
\tn[up=$i_3$, down=$j_3$]{\mathcal K} &
\tnX{R_4}
\end{tenkz}](tenkz_svg/tenkz-228db6d5e3496d09.svg)
Expanding the partial trace over the fourth site and factoring each length-four word at its last letter gives
Linearity of the trace and (??) now yield
If \(Z_4=\operatorname{tr}[\rho ^{(4)}(\mathcal K)]\ne 0\), then \(R_4\ne 0\).
If \(R_4=0\), then the tail formula makes every entry of \(\sigma _3^{(4)}(\mathcal K)\) vanish, so \(\operatorname{tr}[\sigma _3^{(4)}(\mathcal K)]=0\). On the other hand, \(Z_4\ne 0\) implies \(\operatorname{tr}[\sigma _3^{(4)}(\mathcal K)]=1\), a contradiction.
If \(Z_4\ne 0\), there exist virtual indices \(\beta \) and \(\alpha \) such that \((R_4)_{\beta ,\alpha }\ne 0\). These are the indices selected before the sector factorization in [ CPGSV16 , Appendix C.2, lines 1431–1437 ] .
If every matrix entry of \(R_4\) vanished, then \(R_4\) would be the zero matrix, contrary to the preceding theorem.
Let \(\mathcal K\) be an injective simple MPO tensor, and let \(R\in M_{D}(\mathbb {C})\) be the virtual contraction of the sites following a three-site word. For virtual indices \(\alpha _1,\beta _1,\alpha _3,\beta _3\), define
This is the contraction used in the proof of [ CPGSV16 , Appendix C.2, lines 1415–1438 ] .
For every middle physical index \(p_2\),
![\begin{tenkz}[compact, physical=down,
west label=$\alpha_1$, east label=$\beta_1$]
\tn[down=$p_1$]{\mathcal K^{-1}}
\end{tenkz}](tenkz_svg/tenkz-e86ecd5bca9ba159.svg)
\(\; \otimes \; \mathbb {1}\; \otimes \; \)![\begin{tenkz}[compact, physical=down,
west label=$\alpha_3$, east label=$\beta_3$]
\tn[down=$p_3$]{\mathcal K^{-1}}
\end{tenkz}](tenkz_svg/tenkz-88a58e1f1df09442.svg)
\(\Bigr)\) ![\begin{tenkz}[compact, physical=up, periodic]
\tn[up=$p_1$]{\mathcal K} &
\tn[up=$p_2$]{\mathcal K} &
\tn[up=$p_3$]{\mathcal K} &
\tnX{R}
\end{tenkz}](tenkz_svg/tenkz-38727821f3cc1c0e.svg)
\( = \) ![\begin{tenkz}[compact, physical=up,
west label=$\beta_1$, east label=$\alpha_3$]
\tn[up=$p_2$]{\mathcal K}
\end{tenkz}](tenkz_svg/tenkz-5cf7643cdfc90463.svg)
\( \otimes \) ![\begin{tenkz}[compact,
west label=$\beta_3$, east label=$\alpha_1$]
\tnX{R}
\end{tenkz}](tenkz_svg/tenkz-ace09ce05fda8d03.svg)
Move the two finite sums through the trace. By the inverse-map identity
which is the matrix form of (??), the double sum collapses to
Let \(\mathcal K\) be an MPO tensor and let \(R\in M_{D}(\mathbb {C})\). A tripartite operator \(\rho \) is the three-site closure of \(\mathcal K\) against \(R\) if
This is the algebraic form of the three-site reduced operator in [ CPGSV16 , Appendix C.2, lines 1422–1433 ] .
Fix virtual indices \(\beta ,\alpha \) and set \(\kappa _{\beta ,\alpha }(i,j)=\mathcal K^{ij}_{\beta ,\alpha }\). For a Hayashi sector \(k\), define the two inverse factors
These are the two factors in the sectorwise inverse-map calculation of [ CPGSV16 , Appendix C.2, lines 1407–1428 ] .
Let \(\mathcal K\) be injective and let \(\rho \) be its three-site closure against \(R\). Applying the inverse tensor at the two outer sites collapses the closure to one entry of the middle physical slice and one entry of the tail:
Consequently, for any matrix \(U\) on the middle site,
Substituting the three-site closure relation (??) converts the left-hand side of (??) to \(C_{\alpha _1,\beta _1,\alpha _3,\beta _3}((i_2,j_2);R)\). Theorem 26.10.2.39 then gives
which is (??). The second identity follows by expanding the matrix product entrywise,
multiplying by \(R_{\beta _3,\alpha _1}\), substituting
and reordering the finite sums.
Suppose that \(\mathcal K\) is injective, \(\rho \) is its three-site closure against \(R\), and a Hayashi decomposition of \(\rho \) has been chosen. Write
Then every diagonal sector satisfies
The probability \(p_k\) remains explicit because both sector density operators are normalized. No zero-correlation-length or non-vanishing hypothesis is used in this comparison.
For fixed \(k,\alpha _1,\beta _3\), the identity (??) reads coefficientwise as follows.
![\tnpic[
physical=updown,
west label=$\beta_1$,
east label=$\alpha_3$
]{
\tn[
mpo,
up={$(l,r)$},
down={$(l',r')$}
]{\widetilde\kappa^{(k)}}
}](tenkz_svg/tenkz-36002f462bb05d85.svg)
\( = \) \(p_k\, A^{(k)}_{\alpha _1,\beta _1}(l,l') B^{(k)}_{\alpha _3,\beta _3}(r,r')\). At fixed indices \((k,l,r)\) and \((k,l',r')\), the Hayashi block equality is
For each \(i_2,j_2\), the preceding theorem gives
Apply
to the Hayashi block equality. Substituting the preceding display and interchanging the finite sums gives
The right-hand side is
The contraction of the right-hand side of the Hayashi block equality is \(p_k A^{(k)}_{\alpha _1,\beta _1}(l,l') B^{(k)}_{\alpha _3,\beta _3}(r,r')\). Since the Hayashi block equality holds, equating the two contractions gives (??).
Carried onto the tripartite site index, the three-site marginal of the normalized four-site chain is the three-site closure of \(\mathcal K\) against the normalized fourth-site tail \(R_4\).
The entry formula (??) for \(\sigma _3^{(4)}(\mathcal K)\), carried onto the tripartite site index \((i_1,i_2,i_3)\), gives
which is the three-site closure condition (??) against \(R_4\).
In the setting of Theorem 26.10.2.43, for all \(k\ne k'\), every off-diagonal sector satisfies
Together with the diagonal comparison this gives the direct-sum structure of the factorized tensor: the direct sum over sectors is inherited from the splitting of the middle site.
For \(b\) in sector \(k\) and \(b'\) in sector \(k'\) with \(k\ne k'\), the Hayashi block equality at off-diagonal indices gives
Substituting this vanishing into the expansion of Theorem 26.10.2.42 gives (??).
Fix outer virtual indices \(\alpha _1,\beta _3\) and a Hayashi sector \(k\). The sector tensors are
These are the tensors of the factorization displayed in [ CPGSV16 , Appendix C.2, lines 1435–1448 ] .
If \(p_k=0\), then \((l_k)_\beta =0\) for every virtual index \(\beta \). Consequently, \(\eta _{h,k}=0\) for every sector \(h\).
The scalar weight \(p_k\) is a factor of every matrix entry of \((l_k)_\beta \). Each neighboring operator entering sector \(k\) contains \((l_k)_\beta \) as its target factor.
Suppose that \(\mathcal K\) is injective, \(\rho \) is its three-site closure against \(R\), a Hayashi decomposition of \(\rho \) has been chosen, and the outer indices satisfy \(R_{\beta _3,\alpha _1}\ne 0\). Then for all virtual indices \(\beta _1,\alpha _3\),
The direct sum refers to the physical indices and is inherited from the splitting of the middle site. This is the factorization displayed at [ CPGSV16 , Appendix C.2, lines 1435–1448 ] .
In the middle-site basis selected by \(U_B\), the same factorization (??) is displayed as
![\begin{tenkz}[rows={op:none, op, op:none}, tensor style=box]
\tn[pill, wide=2, legs at={1,2}]{U_B} & \\
\tn[wide=2]{\kappa_{\beta_1,\alpha_3}} & \\
\tn[pill, wide=2, legs at={1,2}]{U_B^\dagger} &
\end{tenkz}](tenkz_svg/tenkz-be45e8aed410a95a.svg)
\( = \bigoplus _k\) ![\begin{tenkz}[physical=updown, east=none, tensor style=box]
\tn{l_k}
\end{tenkz}](tenkz_svg/tenkz-6b15bc154e54af8d.svg)
\( \otimes \) ![\begin{tenkz}[physical=updown, west=none, tensor style=box]
\tn{r_k}
\end{tenkz}](tenkz_svg/tenkz-4f5a5532dd467944.svg)
On a diagonal sector, the sectorwise comparison gives \(R_{\beta _3,\alpha _1}\widetilde\kappa ^{(k)} =p_kA^{(k)}B^{(k)}\), and dividing by the nonzero entry \(R_{\beta _3,\alpha _1}\) produces \((l_k)_{\beta _1}\otimes (r_k)_{\alpha _3}\). On an off-diagonal sector, the product with \(R_{\beta _3,\alpha _1}\) vanishes, and since \(R_{\beta _3,\alpha _1}\ne 0\) the sector itself vanishes.
Let \(\mathcal K\) be injective with nonzero four-site trace, let \(\rho \) be a three-site closure of \(\mathcal K\) against the normalized fourth-site tail \(R_4\), and choose a Hayashi decomposition of \(\rho \). Then there exist sector tensors \(l_k\) and \(r_k\) such that, for all virtual indices \(\beta _1,\alpha _3\),
The neighboring operators \(\eta _{k,h}\) are assembled from these tensors in Definition 26.10.3.1; the remaining steps of [ CPGSV16 , Appendix C.2, Lemma C.4 ] prove the all-length identity and primitivity.
A nonzero four-site trace supplies outer indices with \((R_4)_{\beta _3,\alpha _1}\ne 0\), and the sector factorization applies.
26.10.3 Neighboring operators and commuting bond products
Fix outer virtual indices \(\alpha _1,\beta _3\) with the chosen Hayashi decomposition, and let \(l_k,r_k\) be the sector tensors of Definition 26.10.2.46. For each pair of sectors \((k,h)\), the neighboring operator contracts the right sector tensor of one site with the left sector tensor of the following site over the virtual bond they share:
It acts on the neighboring bond space \(B_k^{R}\otimes B_h^{L}\). This is the operator displayed in [ CPGSV16 , Appendix C.2, lines 1441–1445 ] .
In the setting of Theorem 26.10.2.48, for all sectors \(k,h\) and virtual indices \(\beta _1,\alpha _3\), contracting the shared virtual bond of two neighboring factorized slices leaves the neighboring operator between the outer sector tensors:
This is the single-bond step of the identity assembling \(\bigotimes _{n}\eta _{k_n,k_{n+1}}\) from the factorized chain in [ CPGSV16 , Appendix C.2, lines 1446–1449 ] .
Substituting the sector factorization (Theorem 26.10.2.48) into each slice, the outer sector tensors decouple from the bond sum, and the inner sum over the shared bond \(\gamma \) reads
by definition of the neighboring operator.
A physical-sector factorization of an MPO tensor \(\mathcal{K}\) consists of a finite decomposition
whose left and right factors are nonzero, an isometry \(U\) on the physical space, and matrix families \((l_k)_\beta \) and \((r_k)_\alpha \) such that every physical slice satisfies
No positivity, trace factorization, or quantum-Markov decomposition is included in this datum. This is the factorization displayed in [ CPGSV16 , Appendix C.2, lines 1381–1388 ] .
For sectors \(k,h\), contract the common virtual index of the right and left sector tensors:
This is an operator on \(B_k^R\otimes B_h^L\). Positivity is not part of this definition; it is a separate hypothesis in Proposition C.7.
For a sector \(k\) and physical matrix indices \(x,y\) in its summand, let \(A_{k;x,y}\in M_{D}(\mathbb {C})\) be the corresponding entry of the sector-coordinate tensor. For a set \(S\) of sectors, define
These are the matrix spaces associated with the physical projections in [ CPGSV16 , Appendix C.2, lines 1463–1467 ] .
Let \(S\) and \(C\) be sets of sectors. If \(\eta _{k,h}=0\) for every \(k\in S\) and \(h\in C\), then \(XY=0\) for every \(X\in \mathcal{A}_S\) and \(Y\in \mathcal{A}_C\). No conclusion in the reverse product order is asserted. This is the directed local-orthogonality implication used in [ CPGSV16 , Appendix C.2, lines 1463–1470 ] .
Expand two generating matrices in sector coordinates. Their product has entries
The middle contraction is a matrix entry of \(\eta _{k,h}\) and therefore vanishes. Bilinearity extends the conclusion from the generators to the two linear spans.
Let \(z_k\in \mathbb {C}\) satisfy \(|z_k|=1\) for every sector. Replacing
leaves every physical-slice factorization unchanged.
Vertex rephasing transforms the neighboring operators by \(\eta _{k,h}\longmapsto z_k^{-1}z_h\, \eta _{k,h}\). It leaves every sector virtual matrix unchanged, preserves the nonzero neighboring support, and leaves every complete cyclic product of neighboring operators unchanged. These are the rephasing identities used in [ CPGSV16 , Appendix C.2, lines 1434–1450 ] .
Let \(S\) be a set of active sectors in a physical-sector factorization, and suppose that \((l_k)_\beta =0\) for every \(k\notin S\) and every \(\beta \). Retain every physical summand and define
Then \(\widehat l_k\otimes \widehat r_k=l_k\otimes r_k\) for every \(k\), so the physical-slice factorization is unchanged. The neighboring operators become
This is the reparameterization freedom in the factorization and neighboring contraction of [ CPGSV16 , Appendix C.2, lines 1434–1445 ] .
Let \(\mathcal{K}\) be injective, let \(\rho \) be its three-site closure against a virtual tail \(R\), and choose a Hayashi decomposition of \(\rho \). Fix outer indices such that \(R_{\beta _3,\alpha _1}\ne 0\). The inverse-map tensors \(l_k,r_k\), together with the Hayashi physical decomposition and unitary, determine a physical-sector factorization of \(\mathcal{K}\).
The left and right factor spaces are nonzero because the corresponding Hayashi density matrices have trace one; their nonzero dimensions are conclusions rather than hypotheses. No positivity of the neighboring operators is asserted. This gives the factorization in [ CPGSV16 , Appendix C.2, equations AppUkU=rl and formK, lines 1383–1387 and 1434–1439 ] .
For all sectors \(k,h\), the neighboring contraction of the resulting physical-sector factorization is the inverse-map operator
This is equation etarl in [ CPGSV16 , Appendix C.2, lines 1441–1445 ] .
In the inverse-map factorization, retain every physical sector and its factor spaces, and set
The physical-slice factorization is unchanged. This treats the zero-weight freedom left by the Hayashi decomposition used in [ CPGSV16 , Appendix C.2, lines 1415–1445 ] .
The neighboring operators of the reparameterized factorization are
Thus a zero-weight sector has no outgoing support edge. Since \(p_h=0\) also gives \(l_h=0\), it has no incoming support edge either.
The neighboring contraction uses \(r_k\) at its source and \(l_h\) at its target. The first assertion follows from the definition of \(\widehat r_k\), and the last assertion follows from Lemma 26.10.2.47.
Every injective MPO tensor satisfying SAL admits a physical-sector factorization. This is only the raw algebraic factorization from Lemma C.4 and equations formK and etarl; coherent positivity of the neighboring operators, recurrence, and primitivity are not part of this conclusion.
Let \(N\geq 1\). For a cyclic sector configuration \(k=(k_0,\ldots ,k_{N-1})\), write
Applying the one-site sector decomposition at every site gives a bijection
On \(I_k\), define the cyclic neighboring product by
where site labels are read modulo \(N\).
Suppose that the physical-sector factorization is given. For every \(N\geq 1\), the physically transformed MPO is block diagonal in the cyclic sector configurations, and
This is the all-length algebraic decomposition in [ CPGSV16 , Appendix C.2, lines 1435–1450 ] , conditional here on the displayed physical-sector factorization. It does not assert positivity of the individual neighboring operators.
Fix a sector configuration \(k\) and entries \(x,y\in I_k\). Expanding the closed horizontal contraction gives
Cyclically shifting the summation variables and collecting the factors with the same horizontal index turns this expression into \(\Omega _k(x,y)\). If the row and column sector configurations differ, one transformed physical slice lies between distinct direct summands and vanishes. Thus all off-diagonal sector blocks vanish.
Every injective MPO tensor satisfying SAL admits a physical-sector factorization such that, for every \(N\geq 1\) and every cyclic sector configuration \(k=(k_0,\ldots ,k_{N-1})\) with \(k_N=k_0\),
This is the projected-chain inequality in [ CPGSV16 , Appendix C.2, Lemma C.4, lines 1446–1450 ] . It does not assert positivity of the individual neighboring operators.
Choose the raw physical-sector factorization supplied by SAL. The physical basis change sends the positive \(N\)-site MPO to a positive congruence. Each cyclic neighboring product is a diagonal sector compression of this congruence and is therefore positive semidefinite.
For an arbitrary virtual matrix \(X\), define the outer boundary operator on \(B_k^L\otimes B_h^R\) by
The map \(X\mapsto B_{k,h}(X)\) is complex-linear. No positivity of \(X\) or \(B_{k,h}(X)\) is assumed.
First pass to the physical basis selected by \(U\), so that each slice is written in the sector coordinates of Definition 26.10.3.3. Form the two-site closure of this sector-coordinate tensor, restrict it to the physical sectors \((k,h)\), and reindex its physical factors by
Denote the resulting matrix by \(\mathcal{K}_{2;k,h}(X)\).
For every virtual matrix \(X\) and every sector pair \((k,h)\),
At the regrouped matrix entry indexed by \(((\lambda _k,\rho _h),(\rho _k,\lambda _h))\) and \(((\lambda '_k,\rho '_h),(\rho '_k,\lambda '_h))\), the virtual trace expands as the scalar
Commuting scalar factors and collecting the sums over \((a,b)\) and \(c\) gives respectively \(B_{k,h}(X)\) and \(\eta _{k,h}\).
First pass to the physical basis selected by \(U\) and form the three-site closure of the resulting sector-coordinate tensor. Restrict this closure to the sectors \((k,l,h)\) and reindex its physical factors by
Denote the resulting matrix by \(\mathcal{K}_{3;k,l,h}(X)\).
For every virtual matrix \(X\) and fixed sectors \((k,l,h)\),
![\begin{tenkz}[physical=updown, boundary=none, compact]
\tn[pill, wide=6, legs at={1,2,3,4,5,6},
up={$L_k$,$R_h$,$R_k$,$L_l$,$R_l$,$L_h$},
down={$L_k$,$R_h$,$R_k$,$L_l$,$R_l$,$L_h$}]
{{\cal K}_{3;k,l,h}(X)}
\end{tenkz}](tenkz_svg/tenkz-ccb86d3aac349f41.svg)
\(\; =\; \) ![\begin{tenkz}[physical=updown, boundary=none, compact]
\tn[up=$L_k$, down=$L_k$]{l_k} & \tnX{X} &
\tn[up=$R_h$, down=$R_h$]{r_h}
\end{tenkz}](tenkz_svg/tenkz-a0fe5a695000010e.svg)
\(\; \otimes \; \) ![\begin{tenkz}[physical=updown, boundary=none, compact]
\tn[up=$R_k$, down=$R_k$]{r_k} &
\tn[up=$L_l$, down=$L_l$]{l_l}
\end{tenkz}](tenkz_svg/tenkz-6d20930b54f8b6b6.svg)
\(\; \otimes \; \) ![\begin{tenkz}[physical=updown, boundary=none, compact]
\tn[up=$R_l$, down=$R_l$]{r_l} &
\tn[up=$L_h$, down=$L_h$]{l_h}
\end{tenkz}](tenkz_svg/tenkz-3e6a5479e9cfa9d5.svg)
At the regrouped matrix entry indexed by \(((\lambda _k,\rho _h),((\rho _k,\lambda _l),(\rho _l,\lambda _h)))\) and its primed counterpart, the left-hand side is the scalar
Reordering the four finite sums separates the \((a,b)\) boundary contraction from the neighboring contractions indexed by \(c\) and \(e\), which gives the three factors on the right-hand side.
For every outer-sector pair \((k,h)\), define
This is the direct sum of the two neighboring contractions appearing in the three-site closure factorization [ CPGSV16 , Appendix C.2, lines 1510–1516 ] .
![\begin{tenkz}[physical=updown, boundary=none, compact]
\tn[pill, wide=2, legs at={1,2},
up={{$L_k$},{$R_h$}}, down={{$L_k$},{$R_h$}}]
{\mathfrak R_2({\cal K}_2(X))_{k,h}}
\end{tenkz}](tenkz_svg/tenkz-35f49691c1c227ec.svg)
\(\; =\; a_kb_h\; \) ![\begin{tenkz}[physical=updown, boundary=none, compact]
\tn[up=$L_k$, down=$L_k$]{l_k} & \tnX{X} &
\tn[up=$R_h$, down=$R_h$]{r_h}
\end{tenkz}](tenkz_svg/tenkz-0e1ef0c8ec2a4d92.svg)
Suppose that every neighboring operator is positive semidefinite and that there are real numbers \(a_k,b_h\) such that
These are precisely the remaining hypotheses used in the construction of Proposition C.7 after the physical-sector factorization has been fixed [ CPGSV16 , Appendix C.2, lines 1389–1403 ] .
Assume the neighboring-operator trace factorization of Definition 26.10.3.24. Then, for every outer-sector pair \((k,h)\),
This is the normalization used in the coarse-graining channel of [ CPGSV16 , Appendix C.2 ] , lines 1547–1555.
The trace of a direct sum is the sum of the traces of its blocks, and the trace of a tensor product is the product of the traces. Hence
For every virtual matrix \(X\),
Similarly,
and summing the scalar coefficient over \(l\) gives \(a_kb_h\).
The partial trace of \(A\otimes B\) is \(\operatorname{tr}(B)A\). The two-site identity follows immediately. For three sites, the coefficient is
The two- and three-site physical spaces are reindexed by their canonical direct sums over sector pairs and triples, with each summand further regrouped into its boundary and neighboring factors. Write \(\mathfrak G_2\) and \(\mathfrak G_3\) for these reindexings of matrices.
After the canonical regrouping of all physical-sector indices, the full closures are
Every matrix entry between distinct sector tuples vanishes.
On a diagonal sector tuple these are the fixed-sector factorizations. For distinct sector labels \(k\ne p\), the block structure of the sector-coordinate tensor gives \([U\kappa _{\beta ,\alpha }U^\dagger ]_{k,p}=0\). Consequently, if \((k,h)\ne (p,q)\) or \((k,l,h)\ne (p,m,q)\), respectively, then
In the two-site sector coordinates, define
Thus the outer factors carry the identity, while the neighboring factors carry the operator \(\eta _{k,h}\). This is the local bond in [ CPGSV16 , Appendix C.2, Proposition C.8, lines 1581–1593 ] .
Suppose that \(\eta _{k,h}\) is positive semidefinite for every pair of sectors. Then \(B_{\mathrm{sector}}\) is positive semidefinite. The positivity premise is the conclusion at [ CPGSV16 , Appendix C.2, lines 1446–1450 ] .
Each summand \(\mathbb {1}_{B_k^L\otimes B_h^R}\otimes \eta _{k,h}\) is positive semidefinite. Positivity is preserved by finite block diagonals and by the reindexing from the regrouped direct sum to the two-site sector coordinates.
Let \(V_2\) be the two-site tensor product of the project’s coordinate map, so that \(V_2XV_2^\dagger \) expresses a physical operator \(X\) in sector coordinates. Thus \(V_2=U_{\mathrm{paper}}^\dagger \) in the convention of [ CPGSV16 , Appendix C.2, lines 1581–1583 ] . Define
and identify pairs of physical indices with functions \(\{ 0,1\} \to \{ 0,\ldots ,d-1\} \). This is the physical two-site bond in [ CPGSV16 , Appendix C.2, Proposition C.8, lines 1581–1593 ] .
Suppose that \(\eta _{k,h}\) is positive semidefinite for every pair of sectors. Then \(B_{\mathrm{physical}}\) is positive semidefinite, both with pair indices and with two-site configuration indices.
Apply positivity of \(B_{\mathrm{sector}}\) to the congruence \(V_2^\dagger B_{\mathrm{sector}}V_2\). Reindexing the resulting matrix by two-site configurations preserves positivity.
Let \(B\) be an operator on two copies of a finite-dimensional space \(H\). On the right-associated space \(H\otimes (H\otimes H)\), define
where \(\alpha :(H\otimes H)\otimes H\to H\otimes (H\otimes H)\) is the canonical associator.
Under the canonical identification of three-site configurations with \(H\otimes (H\otimes H)\), the translates of the physical bond beginning at sites \(0\) and \(1\) are respectively \((B_{\mathrm{physical}})_{01}=B_{01}\) and \((B_{\mathrm{physical}})_{12}=B_{12}\).
An entry of the first translate vanishes unless the third indices agree; its remaining entry is that of \(B_{\mathrm{physical}}\). Similarly, an entry of the second translate vanishes unless the first indices agree. These are precisely the matrix entries of \(B_{01}\) and \(B_{12}\).
Fix sectors \((k,l,h)\). On the regrouped sector space, set
Then \(C_{01}^{k,l,h}C_{12}^{k,l,h} =C_{12}^{k,l,h}C_{01}^{k,l,h}\). This is the fixed-sector calculation in [ CPGSV16 , Appendix C.2, Proposition C.8, lines 1589–1593 ] .
On the sector triple \((k,l,h)\), the two operators act on distinct neighboring factors, while the outer factors \(B_k^L\) and \(B_h^R\) remain unchanged. Their commutativity is therefore the local identity \(\eta _{k,l}^{(12)}\eta _{l,h}^{(23)} =\eta _{l,h}^{(23)}\eta _{k,l}^{(12)}\).
The operators \(\eta _{k,l}\) and \(\eta _{l,h}\) act on the distinct factors \(B_k^R\otimes B_l^L\) and \(B_l^R\otimes B_h^L\), respectively. The claim follows from the mixed-product identity for tensor products.
The two translates of the physical bond on three sites commute:
This is the adjacent-bond calculation in [ CPGSV16 , Appendix C.2, Proposition C.8, lines 1589–1593 ] .
Regroup the three-site sector coordinates into the direct sum over triples \((k,l,h)\). The two bonds become block diagonal, with blocks \(C_{01}^{k,l,h}\) and \(C_{12}^{k,l,h}\), which commute by Theorem 26.10.3.35. Conjugating by the three-fold physical coordinate unitary preserves the equality. The three-site translation identities then give the displayed physical commutator.
Let \(W_i^L=\{ i,i+1,\ldots ,i+L-1\} \) be a cyclic window of length \(L\leq N\). Two configurations agree outside \(W_i^L\) precisely when their values agree at every site not contained in \(W_i^L\).
Replacing the entries in \(W_i^L\) does not affect any complementary site. Conversely, if the two configurations agree on the complement, replacing the window of the first by the window of the second reconstructs the second configuration.
Let \(E_i^L\) embed an operator on \(L\) consecutive sites into an \(N\)-site periodic chain at the cyclic window beginning at \(i\). Then \(E_i^L(BC)=E_i^L(B)E_i^L(C)\).
Identify a periodic configuration with its restriction to \(W_i^L\) and its restriction to the cyclic complement. In these coordinates, \(E_i^L(B)=B\otimes \mathbb {1}\), so the assertion is the mixed-product identity.
Let \(L\leq N\), and let \(W_i^L\) and \(W_j^L\) be disjoint cyclic windows in an \(N\)-site periodic chain. For arbitrary operators \(B\) and \(C\) on \(L\) sites, \(E_i^L(B)E_j^L(C)=E_j^L(C)E_i^L(B)\).
Apply both sides to a chain amplitude. The first order of application is a double sum over replacements on \(W_i^L\) and \(W_j^L\). Since the windows are disjoint, each replacement leaves the other restricted configuration unchanged, and the two replacements commute. Interchanging the two finite sums gives the reverse order.
Let \(N\geq 3\). For every site \(i\in \mathbb {Z}/N\mathbb {Z}\), including \(i=N-1\), the adjacent translates of the physical bond commute:
This is the periodic-chain form of the adjacent-bond calculation in [ CPGSV16 , Appendix C.2, Proposition C.8, lines 1571–1593 ] . The case \(N=2\) is treated separately below, because its two translated bonds have the same support with opposite cyclic order.
Enclose the two bonds in the cyclic three-site window beginning at \(i\). In these coordinates they are the bonds beginning at sites \(0\) and \(1\), respectively. Embedding an operator into a fixed cyclic window preserves products, so the assertion follows from Theorem 26.10.3.36.
On the periodic chain of length two, the translates beginning at sites zero and one read the physical sites in the opposite cyclic orders \((0,1)\) and \((1,0)\). Nevertheless they commute: \(B_{01}B_{10}=B_{10}B_{01}\). This is the length-two instance of the translated-bond commutativity in [ CPGSV16 , Appendix C.2, Proposition C.8 ] . The source does not discuss the crossed finite-size ordering separately.
Fix sectors \((k,h)\) and regroup the two physical sites as \((L_k\otimes R_h)\otimes (R_k\otimes L_h)\). The translate in the order \((0,1)\) is \(\mathbb {1}_{L_k\otimes R_h}\otimes \eta _{k,h}\), while the translate in the order \((1,0)\) is \(\eta _{h,k}^{\mathrm{op}}\otimes \mathbb {1}_{R_k\otimes L_h}\), where \(\eta _{h,k}^{\mathrm{op}}\) denotes the same matrix after exchanging the factors \(R_h\) and \(L_k\). These operators act on complementary factors and commute. Taking the direct sum over \((k,h)\) and conjugating by the two-fold physical coordinate unitary proves the claim.
Let \(N\geq 2\), and suppose that the cyclic bonds beginning at \(i\) and \(j\) have disjoint two-site supports. Then their physical bond operators commute: \((B_{\mathrm{physical}})_{i,i+1} (B_{\mathrm{physical}})_{j,j+1} =(B_{\mathrm{physical}})_{j,j+1} (B_{\mathrm{physical}})_{i,i+1}\). This is the locality part of the pairwise commutation assertion in [ CPGSV16 , Appendix C.2, Proposition C.8, lines 1571–1593 ] .
This is Theorem 26.10.3.39 applied to the same physical bond operator in both windows. No positivity or additional property of the sector factorization is needed for this locality step.
Suppose that the physical-sector factorization is given. For every periodic chain of length \(N\geq 2\) and every pair of sites \(i,j\in \mathbb {Z}/N\mathbb {Z}\), the corresponding translates of the physical bond commute:
This is the pairwise commutation assertion in [ CPGSV16 , Appendix C.2, Proposition C.8, lines 1571–1593 ] , conditional here on the physical-sector factorization constructed in the preceding results.
For \(N=2\), equal translates commute trivially, while the two distinct translates commute by Theorem 26.10.3.41. Let \(N\geq 3\). If the two cyclic supports are disjoint, apply Theorem 26.10.3.42. Otherwise, Lemma 14.1.11 shows that the two starting sites either coincide or are cyclic neighbors. The coincident case is immediate, and the two neighboring orientations follow from Theorem 26.10.3.40 and its symmetric equality.
Suppose that \(K\) is equipped with a physical-sector factorization \(F\). Let \(B_i\) denote the cyclic translate beginning at site \(i\) of the physical two-site bond determined by \(F\). For every \(N\geq 2\), the \(N\)-site periodic operator is
No positivity, zero-correlation-length, injectivity, or normalization hypothesis is required. This is the product identity in [ CPGSV16 , Appendix C.2, Proposition C.8, lines 1581–1593 ] , conditional here on the given physical-sector factorization.
For a fixed cyclic sector sequence, contraction of the virtual index between the right tensor at site \(i\) and the left tensor at site \(i+1\) produces \(\eta _{k_i,k_{i+1}}\). Taking the virtual trace therefore gives the cyclic product of these neighboring operators. The translated sector bonds give the same ordered product. Conjugating by the tensor power of the one-site coordinate unitary gives the stated physical identity. For \(N=2\), the second translate has the opposite cyclic order; the two factors are the ordinary and crossed bonds.
Under the physical-sector factorization \(F\) of the preceding theorem, for every \(N\geq 2\) there is a real number \(c{\gt}0\) such that \(\rho _N=c\, B_0B_1\cdots B_{N-1}\). Here the bonds are the canonical bonds determined by \(F\), and one may take \(c=1\). No positivity hypothesis on the neighboring operators \(\eta _{k,h}\) is required.
Take \(c=1\) in Theorem 26.10.3.44.
Suppose that every neighboring operator \(\eta _{k,h}\) is positive semidefinite. Then the physical bond is positive semidefinite and all of its cyclic translates commute on every periodic chain of length at least two. Thus it determines a single translation-invariant positive bond datum. This assertion does not include the product formula for the finite-chain density operator.
Positivity is Theorem 26.10.3.32; pairwise commutativity is Theorem 26.10.3.43.
Suppose that a physical-sector factorization of \(K\) is given and that every neighboring operator \(\eta _{k,h}\) is positive semidefinite. Then the physical two-site bond determines an \(\eta \)-local structure for \(K\). Its translated copies commute pairwise, and for every \(N\geq 2\), \(\rho _N=B_0B_1\cdots B_{N-1}\). If the virtual dimension of \(K\) is positive, then \(K\) itself is a fixed tensor representation of this product family, with the same virtual dimension. Thus the positive normalization scalar may be chosen to be \(1\). This is Proposition C.8 conditional on the coherently positive physical-sector factorization that the source derives from SAL.
Theorem 26.10.3.46 gives the positive translation-invariant bond and its pairwise commuting translates. Theorem 26.10.3.44 realizes every finite-chain operator with normalization scalar \(1\).
Every injective MPO tensor satisfying SAL admits an \(\eta \)-local structure. Thus there is a single positive two-site bond whose cyclic translates commute pairwise and whose ordered product is the finite-chain density operator at every length \(N\geq 2\), with normalization scalar one. This is the conclusion of Proposition C.8.
Theorem 26.10.4.41 supplies the positive physical-sector factorization. Apply Theorem 26.10.3.47.
Let \(K\) be injective and satisfy SAL, and suppose that its doubled-index matrix-product tensor is normal. There is a positive physical-sector factorization whose selected two-site bond \(B\) has \(K\) itself as a normal fixed tensor representation. Its virtual dimension is the virtual dimension of \(K\), and for every \(N\geq 2\), \(\rho _N(K)=B_0B_1\cdots B_{N-1}\). In particular, the scalar is one. This is an existential statement about the bond selected in the proof of Proposition C.8; it does not assert uniqueness or normal rescalability for an arbitrary proportional commuting-bond presentation.
SAL implies that the virtual dimension is positive. The positive physical-sector factorization is supplied by Theorem 26.10.4.41. For its selected bond, use \(K\) as the fixed tensor. The exact product identity is Theorem 26.10.3.44, and normality is the given hypothesis on the doubled-index tensor.
26.10.4 Closed-sector contractions and coherent rephasing
Tracing the two physical legs of a sector tensor leaves a vector over the virtual bond:
These are the closed tensors of [ CPGSV16 , Appendix C.2, lines 1473–1477 ] .
For sector tensors \((l_k)_\beta \) and \((r_k)_\alpha \), close their physical legs and set
This is the pairing operator displayed in [ CPGSV16 , Appendix C.2, lines 1473–1493 ] .
For all sectors \(k,h\), the trace of the neighboring operator is the pairing of the closed sector tensors: \(T_{k,h}=\operatorname{tr}(\eta _{k,h})=(r_k|l_h)\). This is [ CPGSV16 , Appendix C.2, lines 1452–1455 and 1478–1481 ] .
Expanding the definition of the neighboring operator and applying \(\operatorname{tr}(A\otimes B)=\operatorname{tr}(A) \operatorname{tr}(B)\) gives
For the inverse-map sector tensors, define the complex matrix \(T_{k,h}:=(r_k|l_h)\). Equivalently, \(T_{k,h}=\operatorname{tr}(\eta _{k,h})\). This is the trace matrix in [ CPGSV16 , Appendix C.2, lines 1478–1481 ] .
More generally, any sector factorization \(U_B\kappa _{\beta ,\alpha }U_B^\dagger =\bigoplus _k(l_k)_\beta \otimes (r_k)_\alpha \) gives the closed-sector pairing operator
This identifies the transfer matrix of Definition 26.4.1 with the operator on the left-hand side of the zero-correlation-length identity in [ CPGSV16 , Appendix C.2, lines 1489–1493 ] .
Trace the sector factorization over the physical index. Unitary invariance gives \(\operatorname{tr}(U_B\kappa _{\beta ,\alpha }U_B^\dagger ) =\operatorname{tr}(\kappa _{\beta ,\alpha })\). The trace of the block diagonal is the sum of the block traces. In each sector,
Summing over \(k\) gives (??).
Let \({\cal K}\) be injective. Let \(R\) and \(\rho \) give a three-site closure of \({\cal K}\), choose a Hayashi decomposition of \(\rho \), and fix virtual indices \(\alpha _1,\beta _3\) such that \(R_{\beta _3,\alpha _1}\ne 0\). For the corresponding closed sector tensors, \(\mathcal T_{\cal K}=\sum _k |l_k)(r_k|\).
The inverse-map sector tensors satisfy the sector factorization of Theorem 26.10.2.48. Apply Theorem 26.10.4.5.
Let \({\cal K}\) be injective. Let \(R\) and \(\rho \) give a three-site closure of \({\cal K}\), choose a Hayashi decomposition of \(\rho \), and fix virtual indices \(\alpha _1,\beta _3\) such that \(R_{\beta _3,\alpha _1}\ne 0\). Form the corresponding closed sector tensors \(|l_k)\) and \((r_k|\). If \({\cal K}\) has source zero correlation length, set \(S:=\sum _k |l_k)(r_k|\). Then there is a real number \(\lambda {\gt}0\) such that
If \({\cal K}\) is the canonically normalized representative satisfying \(\mathcal T_{\cal K}^2=\mathcal T_{\cal K}\), then the raw pairing operator itself satisfies \(S^2=S\). This is the display preceding [ CPGSV16 , Appendix C.2, Lemma C.5 ] ; the first identity records the rescaling-invariant form of the same zero-correlation-length condition.
Under the hypotheses of Theorem 26.10.4.7, suppose that \({\cal K}\) has source zero correlation length. There is a real number \(\lambda {\gt}0\) such that, for \(\widehat T:=\lambda ^{-1}T\), one has
For every positive integer \(N\), one also has
This is the valid trace-power display in [ CPGSV16 , Appendix C.2, lines 1490–1497 ] . No idempotence or rank-one conclusion for \(\widehat T\) is asserted.
Let \(L\) be the matrix whose \(h\)th column is \(|l_h)\), and let \(Q\) be the matrix whose \(k\)th row is \((r_k|\). Then \(S=LQ\) and \(T=QL\). Place the normalization scalar on \(L\). Since \(\lambda ^{-1}LQ\) is idempotent, associativity gives
Cyclicity of trace and the idempotence of \(\lambda ^{-1}LQ\) give
By (??), \(\widehat T^N=\widehat T^2\) for every \(N\geq 2\), which proves (??).
Fix a Hayashi decomposition witness \(h_\eta \). An explicit neighboring \(\eta \)-family is a dependent family \((\eta _{k,h})\) indexed by sector pairs, where \(\eta _{k,h}\) acts on the neighboring bond space \(B_k^{R} \otimes B_h^{L}\). Requiring positivity for each pair gives the corresponding positive \(\eta \)-data.
Let \((l_k)_a\) and \((r_k)_a\) be sector tensors indexed by the horizontal virtual index \(a\). For neighboring sectors \(k,h\), define an operator on \(B_k^R\otimes B_h^L\) by
Thus \(\eta _{k,h}=r_k l_h\) is the contraction over the shared horizontal index in [ CPGSV16 , Appendix C.2, equation (etarl) ] . Positivity is not part of this definition.
For every pair of sectors and every pair of matrix indices, the entry of \(\eta _{k,h}\) is the contraction displayed in (??).
This is immediate from the definition of \(\eta _{k,h}\).
Neighboring right and left sector tensors meet in the bond operator \(\eta _{k,h}=r_k l_h\):
![\begin{tenkz}[physical=updown, boundary=none]
\tn[pill, wide=2, legs at={1,2},
up={{$x_R$},{$x_L$}}, down={{$y_R$},{$y_L$}}]{\eta_{k,h}}
\end{tenkz}](tenkz_svg/tenkz-f00245dccec8d2ff.svg)
\( = \) ![\begin{tenkz}[physical=updown, boundary=none,
bond label={$a$ at 1-2}]
\tn[up=$x_R$, down=$y_R$]{r_k} & \tn[up=$x_L$, down=$y_L$]{l_h}
\end{tenkz}](tenkz_svg/tenkz-285e38da3e9ae6ef.svg)
This is the contraction in [ CPGSV16 , Appendix C.2, lines 1441–1445 ] .
Let \(N\geq 1\) and let \(k_0,\ldots ,k_{N-1}\) be a cyclic sequence of sectors. Fix matrix indices in the left and right factor of every sector, and abbreviate
With all site labels read modulo \(N\),
This is the entrywise cyclic contraction at [ CPGSV16 , Appendix C.2, lines 1446–1450 ] .
Expand the product of the sums defining the neighboring operators and set \(g'_n=g_{n+1}\). Cyclic invariance of the finite product and commutativity of scalar multiplication give
The last expression is the product of the corresponding entries of \(\eta _{k_n,k_{n+1}}\), as in (??).
Under the hypotheses of the preceding theorem, define \(M_n(a,b)=L_n(a)R_n(b)\). Then
Expanding the trace of the matrix word gives
Substituting \(M_n(a,b)=L_n(a)R_n(b)\) gives the cyclic sum in (??); applying that identity yields (??).
For a sector configuration \(k=(k_0,\ldots ,k_{N-1})\), let
Applying the one-site Hayashi decomposition at every site gives a bijection
We write \(\operatorname{reindex}_{\Phi _N}(M)\) for the matrix obtained by transporting both indices of \(M\) along this bijection.
Let \(U\) be a matrix on the physical space. The physical conjugation of \({\cal K}\) by \(U\) is the tensor \(\widetilde{\cal K}\) defined by
Its horizontal bond indices are unchanged. When \(U\) is unitary, this is the corresponding physical basis change.
Let \(N\geq 1\) and let \(k\) be a sector configuration. Define the matrix \(\Omega _k\) on \(I_k\) by
Thus \(\Omega _k=\eta _{k_0,k_1}\otimes \cdots \otimes \eta _{k_{N-1},k_0}\) after the neighboring tensor factors are ordered cyclically.
Suppose that the physical slices in the Hayashi basis satisfy
Let \(\eta _{q,h}=r_q l_h\). For every \(N\geq 1\) and every sector configuration \(k\), the corresponding diagonal block of \(\operatorname{reindex}_{\Phi _N}(M_N(\widetilde{\cal K}))\) is \(\Omega _k\).
Fix \(x,y\in I_k\) and expand the closed horizontal trace of the MPO word. The sector factorization writes each local entry as a product of entries of \(l_{k_n}\) and \(r_{k_n}\). Set
Then
The second equality is Theorem 26.10.4.13, and the last is (??).
Under the same sector factorization hypothesis, for every \(N\geq 1\),
This is the algebraic sector decomposition in [ CPGSV16 , Appendix C.2, lines 1446–1450 ] . It does not assert that the individual neighboring operators are positive.
If the row and column sector configurations coincide, the corresponding block is Theorem 26.10.4.17. If they differ, choose a site \(n\) at which their labels \(k_n\) and \(h_n\) are unequal. Block diagonality gives \([U_B\kappa _{\beta ,\alpha }U_B^\dagger ] _{(k_n, \cdot ),(h_n, \cdot )}=0\) for every pair of horizontal indices. This local factor occurs in every cyclic horizontal-index product, so every summand in the closed trace is zero, proving (??).
Let \(A\) and \(B\) be nonzero complex square matrices whose Kronecker product \(A\otimes B\) is positive semidefinite. Then there is a scalar \(c\ne 0\) with \(cA\ge 0\) and \(c^{-1}B\ge 0\). Since \((cA)\otimes (c^{-1}B)=A\otimes B\), the rescaled factors represent the same product.
On a product vector \(x\otimes y\) the quadratic form of \(A\otimes B\) factors, so \((x^\dagger A\, x)(y^\dagger B\, y)\ge 0\) for all vectors \(x\) and \(y\). By polarization a complex matrix is determined by its quadratic form; since \(A\) and \(B\) are nonzero there are vectors with \(a=v_0^\dagger A\, v_0\ne 0\) and \(c=w_0^\dagger B\, w_0\ne 0\). Every value \(x^\dagger (cA)\, x=(x^\dagger A\, x)\, c\) is non-negative, and over the complex numbers a non-negative quadratic form already forces the matrix to be Hermitian, hence positive semidefinite. For the second factor, \(ac{\gt}0\) and \(a\, (y^\dagger B\, y)\ge 0\) give \(c^{-1}(y^\dagger B\, y)=(a\, (y^\dagger B\, y))(ac)^{-1}\ge 0\).
If \(X\) and \(-X\) are positive semidefinite complex matrices, then \(X=0\).
The quadratic form of \(X\) is both non-negative and nonpositive, hence vanishes identically. Polarization gives \(X=0\).
For square matrices \(A_n\) indexed by \(n=0,\ldots ,N-1\), possibly with different index types, their finite Kronecker product is the matrix with entries
A finite Kronecker product of nonzero matrices is nonzero. Replacing one factor \(A_i\) by \(cA_i\) multiplies the whole finite product by \(c\).
Choose a nonzero entry in each factor and evaluate the product at the resulting row and column. The rescaling identity follows immediately from the entry formula in Definition 26.10.4.21.
Let \(M\ne 0\). If \(c_1,c_2\ne 0\) and both \(c_1M\) and \(c_2M\) are positive semidefinite, then \(c_1=t c_2\) for some positive real number \(t\). Every complex scalar of unit modulus is nonzero. In particular, every nonzero scalar \(c\) has a unit-modulus part \(u=c/|c|\), positivity of \(cM\) implies positivity of \(uM\), and there is at most one such unit scalar when \(M\ne 0\).
Hermiticity of \(c_1M\) and \(c_2M\) gives \(c_2\overline{c_1}=c_1\overline{c_2}\), so \(c_1/c_2\) is real. It cannot be negative: otherwise \(c_1M\) and its negative would both be positive semidefinite, forcing \(c_1M=0\). Thus the quotient is positive. Multiplication by \(|c|^{-1}{\gt}0\) replaces \(c\) by its unit-modulus part without changing positivity. Two admissible unit scalars differ by a positive real number of modulus one, and are therefore equal.
Let \(N\geq 1\), and let \(A_0,\ldots ,A_{N-1}\) be nonzero complex square matrices, with possibly different dimensions. If their finite Kronecker product is positive semidefinite, then there are nonzero scalars \(c_0,\ldots ,c_{N-1}\) such that
The assertion is proved by induction on \(N\). Separate the first factor from the remaining Kronecker product and apply Theorem 26.10.4.19. Absorb the inverse of the first rescaling coefficient into one factor of the tail, then apply the induction hypothesis. The resulting coefficients are nonzero, their product is one, and every rescaled factor is positive semidefinite.
Let \(\mathcal A_1,\mathcal A_2\subseteq M_{D}(\mathbb {C})\) be nonzero linear subspaces. If \(XY=0\) for every \(X\in \mathcal A_1\) and \(Y\in \mathcal A_2\), then \(\mathcal A_1+\mathcal A_2\ne M_{D}(\mathbb {C})\). No vanishing condition on products in the reverse order is required. This is the matrix-algebra obstruction used in [ CPGSV16 , Appendix C.2, lines 1465–1470 ] .
Choose nonzero matrices \(X\in \mathcal A_1\) and \(Y\in \mathcal A_2\), together with nonzero entries \(X_{ip}\) and \(Y_{qj}\). If the two subspaces spanned the full matrix algebra, write the matrix unit \(E_{pq}=E_1+E_2\) with \(E_1\in \mathcal A_1\) and \(E_2\in \mathcal A_2\). The one-sided product hypothesis gives
Its \((i,j)\) entry is \(X_{ip}Y_{qj}\ne 0\), a contradiction.
The nonzero support graph of a neighboring-operator family has a directed edge \(k\to h\) precisely when \(\eta _{k,h}\ne 0\). Its support is recurrent when every such edge belongs to a directed cycle, equivalently when \(h\) can reach \(k\) whenever \(k\to h\) is an edge.
In sector coordinates, let \(A_{k;x,y}\) be the virtual matrix obtained from the \((x,y)\) entry within sector \(k\). The sector-coordinate family contains all transformed physical matrices, while the virtual-matrix family consists of all matrices \(A_{k;x,y}\).
Let \({\cal K}\) be injective and let
be a physical-sector factorization. Then the virtual matrices \(A_{k;x,y}\) obtained from all within-sector physical entries span the full virtual matrix algebra \(M_{D}(\mathbb {C})\).
Since \(U\) is unitary, every original physical matrix of \({\cal K}\) is a linear combination of the matrices in the transformed physical coordinates. The off-diagonal sector entries in those coordinates vanish, while each diagonal sector entry is one of the matrices \(A_{k;x,y}\). Injectivity says that the original physical matrices span \(M_{D}(\mathbb {C})\), and the conclusion follows.
For a physical-sector factorization, let \(A_{k;x,y}\in M_{D}(\mathbb {C})\) be the virtual matrix obtained from the physical matrix entry \((x,y)\) within sector \(k\). Suppose that all such matrices span \(M_{D}(\mathbb {C})\) and that both endpoint sectors of every nonzero neighboring operator contain a nonzero matrix of this form. Then every nonzero \(\eta _{k,h}\) admits a two-edge return: there is a sector \(j\) for which
In particular, the nonzero neighboring support is recurrent.
A nonzero entry of \(\eta _{k,h}\), together with nonzero sector matrices at \(k\) and \(h\), gives matrices \(A\) from sector \(k\) and \(B\) from sector \(h\) with \(AB\ne 0\). By nondegeneracy of the trace pairing, choose \(Y\) with \(\operatorname{tr}(ABY)\ne 0\). Expanding \(Y\) in the spanning family, some sector matrix \(C\) from a sector \(j\) satisfies \(\operatorname{tr}(ABC)\ne 0\). Cyclicity of the trace gives \(BC\ne 0\) and \(CA\ne 0\). The sector multiplication identity then forces \(\eta _{h,j}\ne 0\) and \(\eta _{j,k}\ne 0\).
A vertex \(h\) is reachable from \(k\) in the nonzero sector support if and only if there is a finite directed walk from \(k\) to \(h\). Consequently, the support is recurrent if and only if every directed edge admits a directed return walk.
A walk gives an element of the reflexive transitive closure by induction on its edges. Conversely, append the final edge at each induction step in the reflexive transitive closure.
Every nonempty closed directed walk of length \(N\) determines a cyclic family of vertices \(v_0,\ldots ,v_{N-1}\). Each \(v_n\to v_{n+1}\) is an edge, where \(v_N=v_0\), and for any edge weights in a commutative monoid, \(\operatorname {wt}(w)=\prod _{n=0}^{N-1}\kappa _{v_n,v_{n+1}}\).
Enumerate the sources of the successive edges. The last edge ends at the initial vertex because the walk is closed. The product identity follows by induction on the number of edges.
Let \(E\) be a directed relation on a set \(V\), and suppose that every edge \(a\to b\) admits a directed return walk from \(b\) to \(a\). Let \(G\) be a group and assign a weight \(\kappa _{a,b}\in G\) to each ordered pair of vertices. If the product of the edge weights along every closed directed walk is one, then there are vertex weights \(z_a\in G\) such that \(\kappa _{a,b}=z_a^{-1}z_b\) whenever \(a\to b\) is an edge. This is the graph-theoretic coboundary step needed to make the phase choices in [ CPGSV16 , Appendix C.2, lines 1446–1450 ] coherent across a recurrent support component.
Reachability is an equivalence relation because every edge has a return walk. Choose one vertex in each reachability class and, for every vertex \(a\), choose a walk from the class representative to \(a\). Define \(z_a\) as the product of the edge weights along this walk. If two such walks have the same endpoints, append a common return walk; the two resulting closed walks have weight one, so the original walk weights agree. Comparing the chosen walk to \(b\) with the chosen walk to \(a\) followed by the edge \(a\to b\) gives \(z_b=z_a\kappa _{a,b}\).
For a cyclic sector assignment \((k_n)_{n\in \mathbb Z/N\mathbb Z}\), the horizontal bond fiber at \(n\) is
Shifting the left component by one site identifies \(\prod _n H_n\) with the vertex-indexed sector fiber used in the cyclic neighboring-operator product.
Reindexing a cyclic neighboring-operator block along its horizontal bond fiber gives the finite Kronecker product \(\bigotimes _{n=0}^{N-1}\eta _{k_n,k_{n+1}}\).
Expand both sides entrywise. The shifted left component of the bond fiber at site \(n+1\) is precisely the left index paired with the right component at site \(n\).
Let \(N\geq 1\), and fix a cyclic sector assignment \(k_0,\ldots ,k_{N-1}\) such that every operator \(\eta _{k_n,k_{n+1}}\) is nonzero. If the corresponding cyclic tensor product is positive semidefinite, then there are nonzero scalars \(c_n\) satisfying
This is a choice on one fixed cycle. It does not assert that choices made on two different cycles agree on a common edge.
Reindex the cyclic sector block by the shared horizontal-bond indices. Its matrix entries then become the finite Kronecker product of the consecutive neighboring operators. Apply Theorem 26.10.4.24.
Let \({\cal K}\) be an injective MPDO, let \(R\) give a three-site closure, and fix an \(\eta \)-structure and a nonzero tail entry. Let \(N\geq 1\). For a cyclic sector assignment \(k_0,\ldots ,k_{N-1}\), suppose that every concrete operator \(\eta _{k_n,k_{n+1}}\) obtained from these data is nonzero. Then there are nonzero scalars \(c_n\) such that
This is the specialization of Theorem 26.10.4.35 to the inverse-map sector operators; positivity of the cyclic product follows from the MPDO property.
The fixed-sector cyclic product is positive semidefinite by Theorem 26.10.4.38. Apply Theorem 26.10.4.35.
For every matrix \(U\) on the physical space and every \(N\ge 1\),
In particular \(\rho ^{(N)}(\widetilde{\cal K})\ge 0\) whenever \(\rho ^{(N)}({\cal K})\ge 0\).
Expand the closed trace over bond configurations \(g\) (with \(g_{N+1}=g_1\)), and each conjugated local entry over its two physical indices:
Distributing the site product over the physical sums factors the two \(U\)-products out of the bond sum, which leaves the matrix entry of \(U^{\otimes N}\rho ^{(N)}({\cal K})(U^{\otimes N})^\dagger \). Positivity is preserved by any congruence.
Let \({\cal K}\) be an MPDO whose transformed physical slices satisfy the sector factorization hypothesis. Then for every \(N\ge 1\) and every cyclic sector assignment \(k_1,\ldots ,k_N\) with \(k_{N+1}=k_1\),
This is the positivity half of the display at [ CPGSV16 , Appendix C.2, lines 1446–1450 ] .
The cyclic tensor product \(\Omega _k\) of Definition 26.10.4.16 is a diagonal sector block of the basis-conjugated chain by Theorem 26.10.4.17. The conjugated chain is positive semidefinite by Theorem 26.10.4.37 and the MPDO property, and a principal submatrix of a positive semidefinite matrix is positive semidefinite.
Let \({\cal K}\) be an injective MPDO with the chosen Hayashi decomposition and a nonzero tail entry. Then for all sectors \(k,h\):
\(\eta _{k,k}\ge 0\);
\(\eta _{k,h}\otimes \eta _{h,k}\ge 0\);
if \(\eta _{k,h}\ne 0\) and \(\eta _{h,k}\ne 0\), there is a scalar \(c\ne 0\) with \(c\, \eta _{k,h}\ge 0\), \(c^{-1}\eta _{h,k}\ge 0\), and \((c\, \eta _{k,h})\otimes (c^{-1}\eta _{h,k})=\eta _{k,h}\otimes \eta _{h,k}\).
The chain of length one projected to the sector \(k\) is \(\eta _{k,k}\) up to the exchange of its two tensor factors, and the chain of length two projected to \((k,h)\) is \(\eta _{k,h}\otimes \eta _{h,k}\), both positive by Theorem 26.10.4.38. The third item, including the Kronecker-product-preserving equality, is Theorem 26.10.4.19.
Let \((\eta _{k,h})\) be a neighboring-operator family for which every nonempty cyclic tensor product is positive semidefinite. Suppose, in addition, that every nonzero edge \(k\to h\) admits a directed return path from \(h\) to \(k\). Then there are unit complex numbers \(z_k\) such that \(z_k^{-1}z_h\, \eta _{k,h}\geq 0\) for all \(k\) and \(h\). Consequently, if the neighboring operators arise from the inverse-map physical-sector factorization of an injective MPDO and its nonzero support is recurrent, the sector tensors admit a reciprocal rephasing for which all neighboring operators are positive semidefinite. This factorization determines the positive commuting nearest-neighbor product structure of Proposition C.8.
This is a scope-restricted result. Recurrence is not a hypothesis of Lemma C.4 in [ CPGSV16 , Appendix C.2, lines 1406–1450 ] . Its derivation for the zero-weight reparameterized inverse-map factorization is treated in Theorem 26.10.4.41 and recorded in docs/paper-gaps/cpgsv17_mpdo_sal_zcl_eta_local_structure.tex.
For a nonzero edge, choose a directed return path and close it with the edge. Positive rescaling of the corresponding cyclic tensor product supplies a scalar that makes the edge operator positive; retain only its unit-modulus part. On any closed directed walk, positive rescaling of the associated cyclic tensor product gives phases whose product is one. Uniqueness of the unit phase identifies them edge by edge with the chosen phases. Hence the chosen edge phases have product one around every closed walk. The closed-walk criterion gives \(\kappa _{k,h}=z_k^{-1}z_h\) on every nonzero edge. Zero edge operators are already positive semidefinite. Finally, reciprocal rephasing of the left and right sector tensors multiplies each neighboring operator by this phase and preserves the physical-slice factorization. The conditional eta-local construction then gives the positive commuting product data.
Let \({\cal K}\) be an injective MPO tensor satisfying SAL. Then it admits a physical-sector factorization for which every neighboring operator \(\eta _{k,h}=\sum _a (r_k)_a\otimes (l_h)_a\) is positive semidefinite. Zero-weight Hayashi sectors remain in the physical direct sum, but their right tensors may be set to zero without changing the physical slices.
First set the right tensor to zero in every zero-weight sector. The left tensor already vanishes there, so this does not alter the factorization and removes every support edge incident to such a sector. Every remaining endpoint has positive weight. If all virtual matrices in a positive-weight sector vanished, then the corresponding middle-site block of the three-site closure would vanish; this contradicts its Hayashi expression as a nonzero weight times two trace-one sector states. Thus every endpoint of a nonzero edge contains a nonzero virtual matrix. Injectivity supplies spanning by all sector virtual matrices, and triangle closure gives recurrence. The projected chain identity makes every cyclic Kronecker product positive semidefinite. Apply Theorem 26.10.4.40 to obtain the coherent vertex rephasing.
In the setting of Theorem 26.10.4.39, assume in addition that the nonzero neighboring operators occur in symmetric pairs: \(\eta _{k,h}=0\) forces \(\eta _{h,k}=0\). Then there is a positive semidefinite family \(\eta '_{k,h}\) with \(\eta '_{k,k}=\eta _{k,k}\) and, for every pair of sectors, scalars \(c_{k,h}\ne 0\) with \(\eta '_{k,h}=c_{k,h} \eta _{k,h}\) and \(\eta '_{h,k}=c_{k,h}^{-1} \eta _{h,k}\). In particular every two-site block \(\eta '_{k,h}\otimes \eta '_{h,k}=\eta _{k,h}\otimes \eta _{h,k}\) of the sector-adapted decomposition is unchanged.
The source asserts at [ CPGSV16 , Appendix C.2, lines 1446–1450 ] that the \(\eta \)’s can be chosen positive semidefinite, without the symmetric-support hypothesis. That hypothesis excludes a scalar obstruction: a nonzero neighboring operator whose reverse pair vanishes is constrained only by cycles through at least three sectors, and a coherent choice for the whole family can then fail.
The diagonal operators are positive by Theorem 26.10.4.39, and their scalar is taken to be one. For a pair of distinct sectors, either both neighboring operators vanish and any nonzero scalar works, or both are nonzero by symmetric support and Theorem 26.10.4.39 supplies the reciprocal pair of scalars. Choosing the scalar once per unordered pair makes the two assignments reciprocal, and the block-preserving equality of Theorem 26.10.4.19 applied to that scalar gives \(\eta '_{k,h}\otimes \eta '_{h,k}=\eta _{k,h}\otimes \eta _{h,k}\).
26.10.5 Normalized preparations and controlled partial traces
Every explicit neighboring \(\eta \)-family determines a sector-by-sector trace matrix. We give both its complex-valued form and the real-part version, which is the direct data for the real Perron–Frobenius matrix \(T\) used in the rank-one step.
Let \((\eta _{k,h})\) be a positive neighboring-operator family. A rank-one trace factorization consists of real families \((a_k)\) and \((b_h)\) such that
These identities occur in [ CPGSV16 , Appendix C.2, lines 1395–1402 ] .
For sectors \(k,h\), define the positive-operator candidate
The summand indexed by \(l\) acts on \((B_k^R\otimes B_l^L)\otimes (B_l^R\otimes B_h^L)\). This is the operator adjoined by \(\mathcal T_{k,h}\) in [ CPGSV16 , Appendix C.2, lines 1527–1535 ] .
Let \((A_i)_{i\in I}\) be a finite family of positive semidefinite matrices, where the size of \(A_i\) may depend on \(i\). Then the block-diagonal matrix \(\bigoplus _{i\in I}A_i\) is positive semidefinite.
Write each \(A_i=B_i^*B_i\) using its positive square root. The dependent block diagonal then satisfies \(\bigoplus _i A_i=(\bigoplus _i B_i)^*(\bigoplus _i B_i)\).
Every operator \(\Omega _{k,h}\) is positive semidefinite.
For every \(l\), positivity is preserved by the Kronecker product: \(\eta _{k,l}\succeq 0\) and \(\eta _{l,h}\succeq 0\) imply \(\eta _{k,l}\otimes \eta _{l,h}\succeq 0\). Apply Theorem 26.10.5.4 to the family \((\eta _{k,l}\otimes \eta _{l,h})_l\).
The trace is
Under the rank-one trace factorization, this becomes
This is the trace calculation in [ CPGSV16 , Appendix C.2, lines 1531–1535 ] .
The direct-sum and Kronecker trace identities give
Substitute \(\operatorname{tr}(\eta _{k,h})=a_kb_h\) and \(\sum _l a_lb_l=1\).
Every product \(a_kb_h\) is non-negative. If \(a_kb_h=0\), then \(\eta _{k,h}=0\) and \(\Omega _{k,h}=0\).
Since \(\eta _{k,h}\succeq 0\), its trace is non-negative, and hence \(a_kb_h=\operatorname{tr}(\eta _{k,h})\geq 0\). If \(a_kb_h=0\), then \(\eta _{k,h}\succeq 0\) and \(\operatorname{tr}(\eta _{k,h})=a_kb_h=0\), which imply \(\eta _{k,h}=0\). The same implication gives \(\Omega _{k,h}=0\) because \(\Omega _{k,h}\succeq 0\) and \(\operatorname{tr}(\Omega _{k,h})=a_kb_h=0\).
With the convention \(0^{-1}=0\), define for every pair \((k,h)\)
These operators are normalized when the pair is active, namely when \(a_kb_h\ne 0\).
For every pair \((k,h)\), the operators \(\widehat\eta _{k,h}\) and \(\widehat\Omega _{k,h}\) are positive semidefinite.
By Theorem 26.10.5.7, \(a_kb_h\geq 0\), and hence \((a_kb_h)^{-1}\geq 0\). Since \(\eta _{k,h}\succeq 0\) and \(\Omega _{k,h}\succeq 0\), multiplication by \((a_kb_h)^{-1}\) gives \(\widehat\eta _{k,h}\succeq 0\) and \(\widehat\Omega _{k,h}\succeq 0\).
If \(a_kb_h\ne 0\), then \(\widehat\eta _{k,h}\) and \(\widehat\Omega _{k,h}\) have trace one. Consequently the maps
are trace-preserving and completely positive. These are the individual preparations \(\mathcal S_{k,h}\) and \(\mathcal T_{k,h}\) in [ CPGSV16 , Appendix C.2, lines 1527–1535 and 1551–1555 ] .
On an active pair, \(\operatorname{tr}(\widehat\eta _{k,h}) =(a_kb_h)^{-1}\operatorname{tr}(\eta _{k,h}) =(a_kb_h)^{-1}a_kb_h=1\). Likewise, \(\operatorname{tr}(\widehat\Omega _{k,h})=1\) follows from \(\operatorname{tr}(\Omega _{k,h})=a_kb_h\). Apply Lemma 26.1.37 to each operator.
If a matrix indexed by a finite type has trace one, then its index type is nonempty.
A matrix on an empty index space has trace zero, contrary to the hypothesis.
Every left and right factor in a Hayashi sector is nonzero-dimensional.
A matrix on an empty index space has trace zero. Hence \(\operatorname{tr}(\rho _k^R)=1\) implies \(\dim (B_k^R\otimes C){\gt}0\), which implies \(\dim B_k^R{\gt}0\), and the same argument applied to \(\rho _k^L\) gives \(d_k^L{\gt}0\).
The set of Hayashi sectors is nonempty.
The sector-weight identity \(\sum _k p_k=1\) implies \(m{\gt}0\).
For every pair of sectors \((k,h)\), the spaces carrying \(\eta _{k,h}\) and \(\Omega _{k,h}\) are nonzero-dimensional.
The tensor product carrying \(\eta _{k,h}\) is nonzero-dimensional by Lemma 26.10.5.12. Choosing an intermediate sector \(l\) by Lemma 26.10.5.13 then supplies an element of the direct-sum space carrying \(\Omega _{k,h}\).
On a nonzero finite-dimensional space \(H\), define
The matrix \(\tau _H\) is positive definite and has trace one.
Since \(\dim H{\gt}0\), the scalar \((\dim H)^{-1}\) is positive. Thus \(\tau _H\) is positive definite, and \(\operatorname{tr}(\tau _H)=(\dim H)^{-1}\dim H=1\).
For every sector pair define
On an active pair this is precisely the preparation \(\mathcal S_{k,h}\) of [ CPGSV16 , Appendix C.2, lines 1551–1555 ] ; the second branch gives a normalized extension on a pair where the displayed source quotient is undefined.
For every pair \((k,h)\), the operator \(\overline\eta _{k,h}\) is positive semidefinite with trace one. Hence \(\overline{\mathcal S}_{k,h}\) is trace-preserving and completely positive. If \(a_kb_h\ne 0\), then both the density matrix and the preparation equal their normalized active forms.
The two branches give
The state-preparation result applies in either case.
For every sector pair define
Thus \(\mathcal H^{\Omega }_{k,h}\) is the carrier space, whereas \(\Omega _{k,h}\), \(\widehat\Omega _{k,h}\), and \(\overline\Omega _{k,h}\) are operators on it: respectively the unnormalized neighboring operator, its normalization on an active pair, and its completed density on an arbitrary pair. On an active pair this is precisely the preparation \(\mathcal T_{k,h}\) of [ CPGSV16 , Appendix C.2, lines 1527–1535 ] ; the inactive branch is a normalized extension.
For every pair \((k,h)\), the operator \(\overline\Omega _{k,h}\) is positive semidefinite with trace one. Hence \(\overline{\mathcal T}_{k,h}\) is trace-preserving and completely positive. On every active pair it equals the normalized active preparation.
The two branches give
The state-preparation result therefore applies to \(\overline{\mathcal T}_{k,h}\).
Let \(H=\bigoplus _{k,h}H_{k,h}\). Orthogonally controlling the completed preparations over the sector pairs gives a map
where \((A_{k,h,a})_a\) is a Kraus family for \(\overline{\mathcal S}_{k,h}\) and the tilde denotes its extension by zero outside \(H_{k,h}\). Thus the map discards inter-sector coherences and applies the completed preparation in each diagonal sector. On active sectors this is the sector-control formula for \(\mathcal S_1\) in [ CPGSV16 , Appendix C.2, lines 1548–1555 ] .
If \(X_{k,h}\) is the \((k,h)\) diagonal block of \(X\), then
The diagonal-block formula and invariance under Kraus-index relabeling give
The map \(\overline{\mathcal S}_1\) is trace-preserving and completely positive.
If \((A_{k,h,a})_a\) is the sectorwise Kraus family, then
Thus the controlled Kraus family is trace-preserving.
Orthogonally controlling the completed direct-sum preparations gives
where \((B_{k,h,a})_a\) is a Kraus family for \(\overline{\mathcal T}_{k,h}\). This map discards inter-sector coherences and, on active sectors, is the sector-control formula for \(\mathcal T_1\) in [ CPGSV16 , Appendix C.2, lines 1523–1535 ] .
If \(X_{k,h}\) is the \((k,h)\) diagonal block of \(X\), then
The diagonal-block formula and invariance under Kraus-index relabeling give
The map \(\overline{\mathcal T}_1\) is trace-preserving and completely positive.
For the sectorwise Kraus family \((B_{k,h,a})_a\),
The orthogonal-control theorem now gives the claim.
The conclusions above begin with the rank-one trace factorization \(\operatorname{tr}(\eta _{k,h})=a_kb_h\) and \(\sum _l a_lb_l=1\). Deriving this factorization for the inverse-map neighboring operators from the strong area law and zero correlation length is the Perron–Frobenius step of [ CPGSV16 , Appendix C.2, lines 1404–1410 and 1484–1498 ] . The completed maps therefore give the global controlled preparation once that factorization is supplied; they do not by themselves prove the full two-map statement [ CPGSV16 , Appendix C.2, lines 1510–1517 ] . The regrouping maps, shift maps, and the required closure identities are separate parts of that statement.
Let
The controlled dependent partial trace discards the off-diagonal blocks between distinct \(i\) and applies \(\operatorname{tr}_{B_i}\) to the \(i\)th diagonal block.
If \(X_{ii}\) denotes the \(i\)th diagonal block of \(X\), then \([\mathcal C_{\operatorname{tr}}(X)]_{ii}=\operatorname{tr}_{B_i}(X_{ii})\).
If \((K_{i,j})_j\) is the Kraus family chosen for \(\operatorname{tr}_{B_i}\), then the diagonal-block formula gives
The controlled dependent partial trace is trace-preserving and completely positive.
Choose Kraus operators \((V_{i,a})_a\) for \(\operatorname{tr}_{B_i}\) and extend each \(V_{i,a}\) by zero outside the \(i\)th summand. Their controlled family satisfies
Its Kraus form gives complete positivity, and the displayed resolution of the identity gives trace preservation.
In sector-adapted coordinates, regroup two sites by
Globally this gives the dependent-sum equivalence
The forward equivalence prepares the neighboring factors \(R_k\otimes L_h\) for the partial trace in \(\mathcal T_0\); its inverse is the shift \(\mathcal S_2\). These are the reorderings in [ CPGSV16 , Appendix C.2, lines 1521–1522 and 1555–1559 ] .
Define \(\mathcal T_0\) by the two-site regrouping followed, in each pair \((k,h)\), by the partial trace over \(R_k\otimes L_h\). Its output is indexed by the retained outer factors \(L_k\otimes R_h\). Thus, writing \(R_{\Phi _2}\) for the regrouping reindexing,
The map \(\mathcal T_0\) is trace-preserving and completely positive.
By (??), \(\mathcal T_0\) is the composition of the controlled dependent partial trace with reindexing along the two-site equivalence. The first map is trace-preserving and completely positive by Theorem 26.10.5.30, and the second by Theorem 26.1.17. Their composition is therefore trace-preserving and completely positive.
The map \(\mathcal S_2\) is matrix reindexing along the inverse two-site equivalence. Thus
The shift \(\mathcal S_2\) is trace-preserving and completely positive.
If \(\Phi _2\) is the two-site regrouping, then \(\mathcal S_2=R_{\Phi _2^{-1}}\). It is therefore trace-preserving and completely positive by Theorem 26.1.17.
Regroup three sites by
Equivalently,
The forward equivalence prepares the four middle subspins for \(\mathcal S_0\); its inverse is the shift \(\mathcal T_2\) in [ CPGSV16 , Appendix C.2, lines 1535–1540 and 1547 ] .
Define \(\mathcal S_0\) by the three-site regrouping followed, for every outer pair \((k,h)\), by the partial trace over
Thus, writing \(R_{\Phi _3}\) for the regrouping reindexing,
The map \(\mathcal S_0\) is trace-preserving and completely positive.
By (??), \(\mathcal S_0\) is the composition of the controlled dependent partial trace with reindexing along the three-site equivalence. The controlled trace and equivalence reindexing are trace-preserving and completely positive by Theorems 26.10.5.30 and 26.1.17, respectively. Their composition has the same property.
The map \(\mathcal T_2\) is matrix reindexing along the inverse three-site equivalence. On each summand it sends
The shift \(\mathcal T_2\) is trace-preserving and completely positive.
If \(\Phi _3\) is the three-site regrouping, then \(\mathcal T_2=R_{\Phi _3^{-1}}\). It is therefore trace-preserving and completely positive by Theorem 26.1.17.
For fixed sectors \((k,h)\), define the fiberwise form of \(\mathcal T_0\) by regrouping two sites and tracing \(R_k\otimes L_h\). For fixed \((k,l,h)\), define the fiberwise form of \(\mathcal S_0\) by regrouping three sites and tracing \((R_k\otimes L_l)\otimes (R_l\otimes L_h)\).
Every fiberwise form of \(\mathcal T_0\) and \(\mathcal S_0\) is trace-preserving and completely positive.
The fiberwise forms factor as
where \(\phi _{k,h}\) is the two-site equivalence of Definition 26.10.5.31 on the \((k,h)\) sector, \(\phi _{k,l,h}\) is the three-site equivalence of Definition 26.10.5.36 on the \((k,l,h)\) sector, and \(R_\phi \) denotes equivalence reindexing. Both factors in each composition are trace-preserving and completely positive, so the composition is as well.
On fixed sector pairs, define \(\mathcal S_2\) and \(\mathcal T_2\) by the same inverse regroupings as the corresponding global shifts.
Every fiberwise shift is trace-preserving and completely positive.
The shifts \(\mathcal S_2^{(k,h)}\) and \(\mathcal T_2^{(k,h)}\) are reindexings along the inverse two-site and three-site equivalences of Definitions 26.10.5.31 and 26.10.5.36, respectively. They are therefore trace-preserving and completely positive by Theorem 26.1.17.
To record the domains and codomains, set
and introduce the matrix algebras
Thus the global maps have types
The controlled maps \(\mathcal T_0\) and \(\mathcal S_0\) discard coherences between distinct outer-sector pairs. For fixed \((k,h)\), the global map \(\mathcal S_0\) traces the entire carrier \(\mathcal H^{\Omega }_{k,h}\), including its direct sum over \(l\). The shifts \(\mathcal S_2\) and \(\mathcal T_2\) are global reindexings, with all displayed tensor factors present at their inputs [ CPGSV16 , Appendix C.2, lines 1521–1559 ] .
For the fiberwise maps, write
The map \(\mathcal S_0^{(k,l,h)}\) fixes the middle sector \(l\) before taking the partial trace; it is therefore distinct from the fixed-\((k,h)\) restriction of the global controlled map \(\mathcal S_0\). The four fiberwise types are
![\begin{tenkzcd}[maps, species={channel}, row sep=6mm]
\mathcal A_2^{k,h} & \mathcal A_{\partial}^{k,h} \\
\mathcal A_{\eta}^{k,h} & \mathcal A_2^{k,h} \\
\mathcal A_3^{k,l,h} & \mathcal A_{\partial}^{k,h} \\
\mathcal A_{\Omega}^{k,h} & \mathcal A_3^{k,\bullet,h}
\tnarrow[from={(1,1)}, to={(1,2)}, species=channel]
{\mathcal T_0^{(k,h)}}
\tnarrow[from={(2,1)}, to={(2,2)}, species=channel]
{\mathcal S_2^{(k,h)}}
\tnarrow[from={(3,1)}, to={(3,2)}, species=channel]
{\mathcal S_0^{(k,l,h)}}
\tnarrow[from={(4,1)}, to={(4,2)}, species=channel]
{\mathcal T_2^{(k,h)}}
\end{tenkzcd}](tenkz_svg/tenkz-2c83ec1bdef44a59.svg)
26.10.6 Primitivity and rank-one trace matrices
If every neighboring operator \(\eta _{k,h}\) of Definition 26.10.3.1 is positive semidefinite, the family constitutes explicit neighboring \(\eta \)-data. The source obtains the positivity from the projected chain identity: writing \(\tilde\sigma = U^{\dagger \otimes N} \sigma ^{(N)}({\cal K})\, U^{\otimes N}\) for the \(N\)-site operator in the sector basis and \(Q_k\) for the projector onto the \(k\)-th sector of the physical splitting, every cyclic sector assignment \(k_1,\ldots ,k_N\) (with \(k_{N+1}=k_1\)) satisfies
and the sector tensors can be chosen so that each factor is positive semidefinite [ CPGSV16 , Appendix C.2, lines 1446–1450 ] . The all-length sector identity is Theorem 26.10.4.18. That algebraic identity does not by itself choose coherent positive representatives for all neighboring pairs, so positive semidefiniteness remains a hypothesis on the chosen representatives here. Theorem 26.10.4.39 provides the diagonal and paired positivity, and Theorem 26.10.4.42 a rescaled positive family under symmetric support.
Under the hypotheses of Definition 26.10.6.1, the trace matrix of the explicit \(\eta \)-data is the pairing of the closed sector tensors:
Immediate from Theorem 26.10.4.3 and the definition of the trace matrix.
Let \(T\) be a non-negative irreducible matrix on a finite index set. If \((T^2)_{k,k}{\gt}0\) and \((T^3)_{k,k}{\gt}0\) for every index \(k\), then \(T\) is primitive.
Choose a positive path between every ordered pair of indices, and let \(L\) be the sum of the chosen path lengths. Every chosen length is at most \(L\). For a chosen path of length \(m\), write the difference \(2L+2-m\) as \(2a+3b\) with \(a,b\geq 0\). Appending these returns at the terminal index gives a positive path of common length \(2L+2\) between every ordered pair.
For sector weights \(p_k\), let \(I_*=\{ k:p_k\ne 0\} \). The one-site matrices and their spaces on \(I_*\) are
The real trace matrix on this index set is
Only the trace matrix is restricted: the physical direct sum still contains every zero-weight summand.
For every sector entry, the one-site matrix used in the directed-cut argument equals the virtual matrix used in the spanning argument. If all sector virtual matrices span \(M_{D}(\mathbb {C})\) and the matrices in every zero-weight sector vanish, then \(\operatorname{span}\{ A_{k;x,y}:k\in I_*\} =M_{D}(\mathbb {C})\).
Suppose that the active one-site matrices span \(M_{D}(\mathbb {C})\) and each active sector contains a nonzero such matrix. For every \(S\subseteq I_*\), \({\cal A}_S+{\cal A}_{I_*\setminus S}=M_{D}(\mathbb {C})\), and both spaces are nonzero when the corresponding sector sets are nonempty. If \(\eta _{k,h}=0\) for \(k\in S\) and \(h\in C\), then \({\cal A}_S{\cal A}_C=0\).
Suppose that the active one-site matrices span \(M_{D}(\mathbb {C})\) and each active sector contains a nonzero such matrix. Then for every \(k,h\in I_*\) there is a directed walk
such that \(\eta _{k_j,k_{j+1}}\ne 0\). In particular, every active sector has an outgoing nonzero neighboring operator.
![\begin{tenkz}[rows={op:none, op, op:none}]
\tn[mpo]{P_j} \\
\tn[mpo]{\mathcal K_i} \\
\tn[mpo]{P_{j^\prime}}
\end{tenkz}](tenkz_svg/tenkz-87e976524d52b3d7.svg)
\( = 0\quad \text{unless }i=j=j^\prime \) If every neighboring operator is positive semidefinite, then \(T^*\) is entrywise non-negative and \(T^*_{k,h}{\gt}0\Longleftrightarrow \eta _{k,h}\ne 0\).
Suppose that the active one-site matrices span \(M_{D}(\mathbb {C})\) and each active sector contains a nonzero such matrix. For every \(k\in I_*\) there is an \(h\in I_*\) such that
If the neighboring operators are positive semidefinite, then \((T^{*2})_{k,k}{\gt}0\).
Suppose that the neighboring operators are positive semidefinite, the active one-site matrices span \(M_{D}(\mathbb {C})\), and each active sector contains a nonzero such matrix. Then the non-negative matrix \(T^*\) is irreducible.
Suppose that the neighboring operators are positive semidefinite, the active one-site matrices span \(M_{D}(\mathbb {C})\), each active sector contains a nonzero such matrix, and every nonzero edge closes through a third active sector. Then \((T^{*3})_{k,k}{\gt}0\) for every \(k\in I_*\). Together with the length-two returns and irreducibility, this implies that \(T^*\) is primitive.
Triangle closure supplies a positive closed walk of length three through each active sector. Combine these walks with the length-two returns and apply Theorem 26.10.6.3.
Let \({\cal K}\) be an injective tensor that generates MPDOs and satisfies SAL, and choose the positive semidefinite neighboring operators of Theorem 26.10.4.41. Restrict the sector index to the positive-weight summands of the Hayashi decomposition. Then positive semidefiniteness makes every neighboring trace real and non-negative, and the matrix
is primitive. Zero-weight summands remain in the physical direct sum but are not indices of \(T^*\).
First use the zero-weight reparameterization to make every zero-weight sector isolated. Injectivity makes the nonzero-weight one-site matrices span \(M_{D}(\mathbb {C})\), and every such sector contains a nonzero matrix. Hence Theorem 26.10.6.7 gives strong connectivity of the active neighboring support. Its proof is the one-sided directed-cut argument at [ CPGSV16 , Appendix C.2, lines 1463–1470 ] .
Finally, nondegeneracy and cyclicity of the trace pairing give a closed support walk of length two through every active sector. Triangle closure gives one of length three. Strong connectivity together with these two coprime return lengths yields a common length at which every entry of the active trace-matrix power is positive. Thus \(T\) is primitive.
Let \({\cal K}\) be injective and suppose that it satisfies SAL and source zero correlation length. Then there are a physical-sector factorization, weights \(p_k\geq 0\) with \(\sum _k p_k=1\), and positive semidefinite neighboring operators \(\eta _{k,h}\) such that the active trace matrix
is primitive. There is also a real number \(\lambda {\gt}0\) for which, upon setting \(\widehat T^*:=\lambda ^{-1}T^*\), one has \((\widehat T^*)^2=(\widehat T^*)^3\). For every positive integer \(N\), \(\operatorname{tr}((\widehat T^*)^N)=\operatorname{tr}(\widehat T^*)\). Thus the primitivity and the two normalized identities concern the same active, positively rephased trace matrix. No idempotence, rank-one, or semisimplicity conclusion is asserted.
Choose the coherent sector phases from Theorem 26.10.6.12. For the resulting factorization, let \(L\) have columns \((\operatorname{tr}((l_h)_\beta ))_\beta \) and let \(Q\) have rows \((\operatorname{tr}((r_k)_\alpha ))_\alpha \), with both indices restricted to \(I_*\). The left tensor vanishes when \(p_k=0\), so the rectangular product \(LQ\) is the full physical-trace transfer. Source zero correlation length therefore makes \(\lambda ^{-1}LQ\) idempotent for some \(\lambda {\gt}0\).
Positivity of the neighboring operators makes their traces real, and hence \(QL\) is the complexification of \(T^*\). Associativity and cyclicity of trace give, for every \(N\geq 1\),
Injectivity of the real-to-complex inclusion gives the two real matrix identities.
There is a tensor of bond dimension two with four one-dimensional physical sectors and weights \(p_k\) such that the following statements hold simultaneously: the tensor is injective and has source zero correlation length; its physical-trace transfer is idempotent; for every \(k\in \{ 0,1,2,3\} \), the weights satisfy
its sector virtual matrices span \(M_{2}(\mathbb {C})\); every neighboring operator is positive semidefinite; and its active trace matrix is primitive and has trace one. Nevertheless, if \(S=LQ\) is the physical-trace transfer and \(T=QL\) is the active trace matrix, then \(Q(1-S)L=T-T^2\ne 0\). In particular, full virtual-matrix spanning does not by itself imply that the active trace matrix is idempotent.
Take
For each \(k\), the \(k\)th sector virtual matrix is the outer product of the \(k\)th column of \(L\) with the \(k\)th row of \(Q\). Their four coordinate functions are proportional to the Walsh functions \(1,x,y,xy\) on four points. More explicitly, the matrix whose columns are their coordinates in the order \((00,01,10,11)\) has determinant \(-1/64\). Hence they form a basis of \(M_{2}(\mathbb {C})\). Placing them on the diagonal physical entries gives an injective tensor and a physical-sector factorization with scalar sectors.
Direct multiplication gives
Thus \(LQ\) is idempotent, whereas every entry of \(QL\) is strictly positive, so \(QL\) is primitive. Its trace is one. Since each sector is one-dimensional, the neighboring operators are the scalar matrices \(\eta _{k,h}=\begin{pmatrix} (QL)_{k,h} \end{pmatrix}\succeq 0\).
The \((0,1)\) entry of \((QL)^2\) is \(1/4\), while the corresponding entry of \(QL\) is \(1/8\). Hence \(QL\) is not idempotent, and associativity gives \(Q(1-LQ)L=QL-(QL)^2\ne 0\).
If \(\rho \) is positive semidefinite on a bipartite space \(A \otimes B\), then both reduced operators \(\operatorname{tr}_A \rho \) and \(\operatorname{tr}_B \rho \) are positive semidefinite.
Write each partial trace as the finite sum, over the traced index, of a principal submatrix of \(\rho \). Principal submatrices of a positive semidefinite matrix are positive semidefinite, and finite sums of positive semidefinite matrices remain positive semidefinite.
Every Hayashi decomposition witness \(h_\eta \) canonically produces an explicit neighboring \(\eta \)-family by Kronecker-multiplying the sector-reduced states on the neighboring bond spaces:
Positive semidefiniteness of each \(\eta _{k,h}\) follows from the fact that partial traces preserve positivity and that positive semidefiniteness is closed under Kronecker products. This is the sector-reduced extraction; the remaining connection to [ CPGSV16 , Appendix C.2 ] is the identification of this family with the inverse-map construction in the original simple-MPDO tensor coordinates.
The trace matrix induced by the extracted family is entrywise \(T_{k,h}=1\) in both its complex-valued form and its real-valued form. If there are \(m\) sectors, then \(T=\mathbf{1}\mathbf{1}^{\mathsf T}\) and \(\operatorname{tr}(T)=m\).
Each \(\rho _{b_2 C}^{(k)}\) is a density matrix, hence has unit trace; the partial trace \(\operatorname{tr}_C\) preserves the trace. The analogous statement holds for \(\operatorname{tr}_A \rho _{A b_1}^{(h)}\). Multiplicativity of the trace under Kronecker products gives \(T_{k,h}=1\cdot 1=1\), and taking real parts gives the real-valued statement. Summing the \(m\) diagonal entries gives \(\operatorname{tr}(T)=m\).
The real trace matrix of the sector-reduced family is positive semidefinite.
This trace matrix is the all-ones matrix \(\mathbf{1}\mathbf{1}^{\mathsf T}\), hence is positive semidefinite.
Positivity of the neighboring operators gives entrywise nonnegativity of the real trace matrix. Matrix-level positive semidefiniteness of \(T\) is a separate structural condition.
The trace of a positive semidefinite neighboring operator is non-negative in the complex order. Taking real parts gives the stated entrywise nonnegativity.
The auxiliary predicates state two matrix-theoretic ingredients used in the rank-one step of [ CPGSV16 , Appendix C.2, Lemma C.5 ] : constant traces of all positive powers and existence of a rank-one factorization \(T=ab^\top \).
The proposed implication from primitivity and constant trace powers to a rank-one factorization is not a valid general theorem. A concrete \(3\times 3\) counterexample appears in the archive. The corrected criterion proved here adds positive semidefiniteness and trace normalization, which provide the diagonalizability needed to turn the trace-power condition into a rank-one factorization.
Let \(T\) be a finite Hermitian matrix over a real or complex scalar field, with Hermitian eigenvalues \(\lambda _i\). Then \(\operatorname{tr}(T^2)=\sum _i\lambda _i^2\).
The spectral theorem writes \(T\) as a unitary conjugate of the diagonal matrix of its Hermitian eigenvalues. Squaring commutes with this conjugation, and cyclicity of trace reduces the identity to the trace of the squared diagonal matrix.
Let \(A\) be a finite Hermitian matrix. Choose an orthonormal eigenbasis \((u_k)_k\) with eigenvalues \((\lambda _k)_k\), and let \(E_k=|u_k\rangle \! \langle u_k|\) be the rank-one projection onto \(u_k\). Then \(A=\sum _k\lambda _kE_k\).
The finite-dimensional spectral theorem gives \(A=U\operatorname{diag}(\lambda _k)U^\dagger \), where the columns of \(U\) are the vectors \(u_k\). Expanding the diagonal matrix as the sum of its rank-one coordinate projections gives the stated identity.
Let \(T\) be a finite real square matrix. If \(T\) is positive semidefinite, normalized by \(\operatorname{tr}(T)=1\), and has constant traces on all positive powers, then \(T\) has a rank-one factorization.
The spectral theorem diagonalizes the positive semidefinite matrix with non-negative real eigenvalues. The trace gives that their sum is \(1\), while the second trace moment gives that the sum of their squares is also \(1\). Hence exactly one eigenvalue is equal to \(1\) and all others vanish, so the diagonal form is rank one. Unitary conjugation preserves the existence of an outer-product factorization.
Let \(T\) be a finite real square matrix. If \(T^2=T\) and \(\operatorname{tr}(T)=1\), then there are real vectors \(a,b\) such that \(T=ab^\top \).
For an idempotent endomorphism \(f\), one has \(\operatorname{tr}(f)=\dim (\operatorname{range}(f))\). Hence the range of \(T\) has dimension one. Choosing a nonzero vector \(a\) spanning it expresses every column of \(T\) as \(b_j a\) for a scalar \(b_j\), and therefore \(T=ab^\top \).
Let \(T\) be a finite real square matrix with \(T^2=T\). Then, for every positive integer \(N\), one has \(\operatorname{tr}(T^N)=\operatorname{tr}(T)\). This recovers the trace identity displayed in the proof of [ CPGSV16 , Appendix C.2, Lemma C.5 ] from idempotence.
Induction on \(N\): for \(N\geq 1\), one has \(T^{N+1}=T^NT=TT=T\), so every positive power of \(T\) equals \(T\).
Let \(T\) be a primitive non-negative matrix. Then every row of \(T\) contains a strictly positive entry, and so does every column.
If row \(k\) of \(T\) vanishes, then, for every \(m\geq 1\),
contradicting the strict entrywise positivity of some power of \(T\). Likewise, if column \(h\) of \(T\) vanishes, then, for every \(m\geq 1\),
again contradicting the strict positivity of some power of \(T\).
Let \(I\) be a nonempty finite index set, and let \(P\) be a real matrix indexed by \(I\times I\). Suppose that \(P^2=P\) and \(P_{i,j}{\gt}0\) for every \(i,j\in I\). Then \(\operatorname{rank}(P)=1\).
Put \(u=P\mathbf1\), so every coordinate of \(u\) is positive and \(Pu=u\). For a fixed vector \(x\) satisfying \(Px=x\), choose an index \(k\) at which \(x_i/u_i\) is maximal and put \(c=x_k/u_k\). Then \(y=cu-x\) is non-negative, \(y_k=0\), and \(Py=y\). Hence
Since every \(P_{k,j}\) is strictly positive, all coordinates of \(y\) vanish. Thus every fixed vector is proportional to \(u\). The range of an idempotent is its fixed space, so it is one-dimensional.
Let \(T\) be a primitive real \(N\times N\) matrix, where \(N\geq 1\). Multiplication by a positive scalar preserves primitivity. If \(T^2=T^3\), then \(\operatorname{rank}(T^2)=1\). There are vectors \(a,b\in \mathbb R^N\) such that
This finite-dimensional matrix lemma supports the argument sought at [ CPGSV16 , Appendix C.2, line 1613 ] ; it is not a theorem asserted there.
Primitivity gives \(m\geq 1\) such that \((T^m)_{i,j}{\gt}0\) for every \(i,j\). Stabilization gives \(T^{2m}=T^2\). Thus, for every \(i,j\),
The same stabilization identity gives \((T^2)^2=T^2\). The preceding theorem applied to \(P=T^2\) now gives the conclusion. Since an idempotent has trace equal to its rank, \(\operatorname{tr}(T^2)=1\). Factoring its one-dimensional range gives \(T^2=ab^{\mathsf T}\), and taking the trace yields \(\sum _i a_i b_i=1\).
Let \(V\) be a module over a commutative ring, let \(l_1,\ldots ,l_n\in V\), let \(r_1,\ldots ,r_n\) be linear functionals on \(V\), and let \(T_{k,h}=(r_k|l_h)\) be the pairing matrix; for the tensors of [ CPGSV16 , Appendix C.2 ] this is the sector trace matrix. Assume the identity
whose right-hand side is the square of its left-hand side, so that the operator \(M=\sum _k|l_k)(r_k|\) satisfies \(M^2=M\). If the vectors \(l_1,\ldots ,l_n\) are linearly independent, then \(T^2=T\). The same conclusion holds if instead the functionals \(r_1,\ldots ,r_n\) are linearly independent.
Evaluating \(M^2v=Mv\) at an arbitrary \(v\in V\) gives
so linear independence of the \(l_k\) yields \((r_k|Mv)=(r_k|v)\) for all \(k\) and \(v\). Taking \(v=l_h\) and expanding \(Ml_h=\sum _jT_{j,h}\, l_j\) turns the left-hand side into \(\sum _jT_{k,j}T_{j,h}=(T^2)_{k,h}\) and the right-hand side into \(T_{k,h}\). For linearly independent functionals, apply the same argument in the dual space, with the \(r_k\) as vectors and evaluation at \(l_k\) as functionals. The dual pairing matrix therefore satisfies \((T^\top )^2=T^\top \). Taking transposes gives \(T^2=T\).
Equivalently, the zero-correlation-length identity is the algebraic equality
which is the displayed identity of [ CPGSV16 , Appendix C.2, lines 1490–1493 ] .
Let \(V\) be a real vector space, let \(l_1,\ldots ,l_n\in V\), let \(r_1,\ldots ,r_n\) be real linear functionals on \(V\), and let \(T_{k,h}=(r_k|l_h)\) be the pairing matrix, as in Theorem 26.10.6.29. Assuming only the identity \(M^2=M\) for the operator \(M=\sum _k|l_k)(r_k|\), without assuming linear independence of \(l_1,\ldots ,l_n\) or of \(r_1,\ldots ,r_n\), one has the following identities: the first unconditionally, the second for every \(N\geq 2\), and the third for every positive integer \(N\):
Multiplying the identity \(M^2=M\) on the left by the functional map and on the right by the coefficient map turns the two sides directly into \(T^2\) and \(T^3\), with no independence hypothesis; induction then gives \(T^N=T^2\) for every \(N\geq 2\). For the trace identity, restrict the operator \(M\) to the finite-dimensional span of \(l_1,\ldots ,l_n\): there it is an idempotent endomorphism of a finite free module, and cyclicity of the trace of a composition between that span and the coefficient space identifies \(\operatorname{tr}(T)\) with the trace of the restriction and \(\operatorname{tr}(T^2)\) with the trace of its square, which agree since the restriction is idempotent.
In the setting of Theorem 26.10.6.29 with real scalars and linearly independent vectors \(l_1,\ldots ,l_n\), if in addition \(\operatorname{tr}(T)=1\), then there are real vectors \(a,b\) such that \(T=ab^\top \). This is the factorization \(T_{k,h}=a_kb_h\) obtained in the proof of [ CPGSV16 , Appendix C.2, Lemma C.5 ] for the sector trace matrix, derived here from the displayed zero-correlation-length identity together with the linear independence of the \(l_k\), instead of the Perron–Frobenius trace-power argument of the source.
Theorem 26.10.6.29 gives \(T^2=T\), and the idempotent trace-one criterion of Theorem 26.10.6.24 gives the factorization \(T=ab^\top \).
Let \(T\) be a finite real square matrix. Assume that \(T\) is primitive, positive semidefinite, normalized by \(\operatorname{tr}(T)=1\), and has constant traces on all positive powers. Then \(T=ab^\top \) for some vectors \(a\) and \(b\), with \(a\cdot b=1\).
The positive-semidefinite matrix criterion diagonalizes \(T\), uses the first two trace moments to show that exactly one eigenvalue is nonzero, and hence gives the factorization. Taking traces and using \(\operatorname{tr}(ab^\top )=a\cdot b\) gives the normalization \(a\cdot b=1\).
For a real \(n\times n\) matrix \(T\) and a real vector space \(V\), this structure consists of vectors \(|l_0),\ldots ,|l_{n-1})\in V\) and linear functionals \((r_0|,\ldots ,(r_{n-1}|\) on \(V\) satisfying the pairing identity \(T_{k,h}=(r_k|l_h)\) of [ CPGSV16 , Appendix C.2 ] together with the zero-correlation-length identity
displayed in the proof of [ CPGSV16 , Lemma C.5 ] . The vectors are the closed sector tensors obtained there from the sector splitting of an injective simple tensor.
For sector tensors paired into \(T\) as in Definition 26.10.6.33, the operator \(M=\sum _k|l_k)(r_k|\) satisfies \(M^2=M\). If in addition the vectors \(|l_0),\ldots ,|l_{n-1})\) are linearly independent, then \(T^2=T\).
Substituting \(T_{k,h}=(r_k|l_h)\) into the right-hand side of the zero-correlation-length identity gives, for every \(v\in V\),
so \(M^2=M\). Linear independence of the \(|l_k)\) then gives \(T^2=T\) by Theorem 26.10.6.29.
For sector tensors paired into \(T\) as in Definition 26.10.6.33, without any independence hypothesis on the closed sector tensors or the functionals, one has \(T^2=T^3\), and, for every positive integer \(N\), \(\operatorname{tr}(T^N)=\operatorname{tr}(T)\).
The operator identity \(M^2=M\) established in the proof of Theorem 26.10.6.34 gives \(T^2=T^3\) and the constant trace powers by Theorem 26.10.6.30, with no independence hypothesis.
Suppose the sector trace matrix \(T\) of Definition 26.10.6.33 is primitive. Then every closed sector tensor satisfies \(|l_k)\neq 0\). If, in addition, each \(|l_k)\) lies in its own member of an independent family of subspaces of \(V\), then the vectors \(|l_0),\ldots ,|l_{n-1})\) are linearly independent.
By Theorem 26.10.6.26, some entry of the \(k\)-th column of \(T\) is positive, and that entry is the pairing \((r_j|l_k)\), so \(|l_k)\neq 0\). Nonzero vectors chosen from an independent family of subspaces, one from each member, are linearly independent.
Let \(T\) be a sector trace matrix with \(\operatorname{tr}(T)=1\), paired from sector tensors satisfying the zero-correlation-length identity of Definition 26.10.6.33. If the closed sector tensors \(|l_0),\ldots ,|l_{n-1})\) are linearly independent, then there are real numbers \(a_k,b_h\) such that
The same conclusion holds if, instead of the independence, \(T\) is primitive and the closed sector tensors lie in distinct members of an independent family of subspaces. In contrast with the positive-semidefinite criterion of Theorem 26.10.6.32, the constant trace powers are derived rather than assumed.
Theorem 26.10.6.34 gives \(T^2=T\), and Theorem 26.10.6.35 gives the constant trace powers unconditionally. The idempotent trace-one criterion of Theorem 26.10.6.24 yields the factorization \(T=ab^\top \), and taking traces gives \(\sum _k a_kb_k=\operatorname{tr}(T)=1\). When the independence is not assumed directly, Theorem 26.10.6.36 derives it from primitivity once each sector tensor lies in its own member of an independent family of subspaces.
26.10.7 Refinement and coarse-graining channels
The sector set and every middle-subspin space are nonempty. Each \(\Omega _{k,h}\) is positive semidefinite, and \(a_kb_h=0\) implies \(\Omega _{k,h}=0\).
On a sector pair with \(a_kb_h\ne 0\), set \(\widehat\Omega _{k,h}=(a_kb_h)^{-1}\Omega _{k,h}\). If \(a_kb_h=0\), positivity and vanishing trace imply \(\Omega _{k,h}=0\); choose any density operator on that summand and denote the resulting completed density by \(\overline\Omega _{k,h}\). The sectorwise preparation is \(X\longmapsto X\otimes \overline\Omega _{k,h}\). Taking the orthogonal direct sum over the outer sectors \((k,h)\) gives the preparation stage \(\mathcal T_1\) of [ CPGSV16 , Appendix C.2, lines 1523–1535 ] . The zero-weight choice makes the source map total without imposing a nonzero-weight hypothesis.
The normalized density is positive semidefinite and has trace one on every active pair. The density chosen on an inactive pair is positive definite and has trace one. Hence \(\overline\Omega _{k,h}\) is positive semidefinite with trace one for every pair, and agrees with \(\widehat\Omega _{k,h}\) on active pairs. The corresponding sectorwise and controlled maps prepare these densities on their diagonal summands.
Each completed preparation is trace-preserving and completely positive. The same holds for their orthogonally controlled direct sum.
On an active pair, \(\operatorname{tr}(\widehat\Omega _{k,h})=(a_kb_h)^{-1}a_kb_h=1\); the chosen density on a zero-weight pair also has trace one. Preparation by a positive operator of trace one is trace-preserving and completely positive. The orthogonal control discards coherences between distinct outer-sector pairs and applies these preparations on the diagonal summands.
Define the channel from two physical sites to three physical sites by \(\mathcal T=\mathcal T_2\mathcal T_1\mathcal T_0\). Thus \(\mathcal T_0\) traces the neighboring subspins, \(\mathcal T_1\) adjoins the completed three-site neighboring density, and \(\mathcal T_2\) restores the three complete physical sectors. This is the refinement map in [ CPGSV16 , Appendix C.2, lines 1522–1545 ] .
The refinement channel \(\mathcal T\) is trace-preserving and completely positive.
Each constituent map is trace-preserving and completely positive, and this property is preserved under composition.
Applied to the two-site closure, on the active \((k,h)\)-block, \(\mathcal T_0\) traces \(B_k^R\otimes B_h^L\) and sends \(\mathfrak R_2(\mathcal K_2(X))_{k,h}\) to \(a_kb_hB_{k,h}(X)\) on \(B_k^L\otimes B_h^R\). The preparation \(\mathcal T_1\) then adjoins
and \(\mathcal T_2\) orders the result as
The refinement uses the sitewise sector coordinates \(\mathfrak R_2\) and \(\mathfrak R_3\) [ CPGSV16 , Appendix C.2, lines 1510–1516 and 1522–1545 ] . For each outer pair, write
for the carrier Hilbert space of \(\Omega _{k,h}\). The symbols \(\widehat\Omega _{k,h}\) and \(\overline\Omega _{k,h}\) denote respectively its normalized active density and its completed density on every sector pair. With the matrix algebras introduced above, the refinement stages have type
![\begin{tenkzcd}[maps, species={channel}]
\mathcal A_2 &
\mathcal A_{\partial} &
\mathcal A_{\Omega} &
\mathcal A_3
\tnarrow[from={(1,1)}, to={(1,2)}, species=channel]{\mathcal T_0}
\tnarrow[from={(1,2)}, to={(1,3)}, species=channel]{\mathcal T_1}
\tnarrow[from={(1,3)}, to={(1,4)}, species=channel]{\mathcal T_2}
\end{tenkzcd}](tenkz_svg/tenkz-066b6d27c7117089.svg)
Here \(\mathcal T_1\) adjoins \(\overline\Omega _{k,h}\) on each diagonal outer-sector summand. The composite channel has type
![\begin{tenkzcd}[maps, species={channel}]
\mathcal A_2 & \mathcal A_3
\tnarrow[from={(1,1)}, to={(1,2)}, species=channel]{\mathcal T}
\end{tenkzcd}](tenkz_svg/tenkz-09a87f607a789af3.svg)
On a sector pair with \(a_kb_h\ne 0\), let \(\mathcal S_{k,h}(X)=X\otimes \frac{\eta _{k,h}}{a_kb_h}\). If \(a_kb_h=0\), positivity and vanishing trace imply \(\eta _{k,h}=0\); choose any density operator on that unused summand. Taking the orthogonal direct sum over \((k,h)\) defines \(\mathcal S_1\).
The map \(\mathcal S_1\) is trace-preserving and completely positive.
Define the channel from three physical sites to two physical sites by \(\mathcal S=\mathcal S_2\mathcal S_1\mathcal S_0\). This is the coarse-graining map in Proposition C.7 [ CPGSV16 , Appendix C.2, lines 1547–1563 ] .
The channel \(\mathcal S\) is trace-preserving and completely positive.
Each of the three factors is trace-preserving and completely positive, and this property is preserved under composition.
The coordinate equivalences used by the coarse-graining channel expose the direct sums over site-sector pairs and triples. Write \(\mathfrak R_2\) and \(\mathfrak R_3\) for the corresponding matrix reindexings. Their inverses send every sector block to the corresponding physical matrix entries. These reindexings preserve the left-right order within every physical site. They differ from the regrouped coordinates \(\mathfrak G_2\) and \(\mathfrak G_3\) of Definition 26.10.3.27, which place the two outer subspins before the neighboring subspins. If \(R_{\Phi _2}\) and \(R_{\Phi _3}\) denote the two regrouping maps, then
For every virtual matrix \(X\) and outer-sector pair \((k,h)\),
On the summand labelled by the middle sector \(l\), this is the fixed-sector factorization \(B_{k,h}(X)\otimes (\eta _{k,l}\otimes \eta _{l,h})\). Taking the direct sum over \(l\) gives the stated block.
Every matrix entry of \(R_{\Phi _3}\mathfrak R_3(\mathcal K_3(X))\) between distinct outer-sector pairs \((k,h)\ne (p,q)\) vanishes.
Distinct outer-sector pairs differ either in their first sector or in their third sector. In the corresponding physical slice, the sector-coordinate tensor is block diagonal, so that factor vanishes.
For every virtual matrix \(X\) and outer-sector pair \((k,h)\),
This is the fixed-sector two-site factorization after identifying the sitewise sector coordinates with the regrouped boundary and neighboring coordinates.
For every two-site matrix \(Y\), outer-sector pair \((k,h)\), and boundary indices \(a,b\),
The controlled partial trace acts on a diagonal outer-sector block by the ordinary partial trace over its neighboring factor.
If the \((k,h)\) diagonal block of \(R_{\Phi _2}Y\) is \(B\otimes \eta _{k,h}\), then \([\mathcal T_0(Y)]_{(k,h),(k,h)}=a_kb_hB\).
The partial trace of \(B\otimes \eta _{k,h}\) is \(\operatorname{tr}(\eta _{k,h})B=a_kb_hB\).
For every outer-sector pair \((k,h)\) and boundary matrix \(B\), the completed sectorwise preparation satisfies \(\overline{\mathcal T}_{k,h}(a_kb_hB) =B\otimes \Omega _{k,h}\).
If \(a_kb_h\ne 0\), the scalar cancels the normalization in \(\widehat\Omega _{k,h}=(a_kb_h)^{-1}\Omega _{k,h}\). If \(a_kb_h=0\), both sides vanish because \(\Omega _{k,h}=0\).
If the \((k,h)\) diagonal block of a boundary matrix \(Y\) is \(a_kb_hB\), then \([\mathcal T_1(Y)]_{(k,h),(k,h)} =B\otimes \Omega _{k,h}\).
On a diagonal outer-sector block, the controlled map is the completed sectorwise preparation, to which the preceding theorem applies.
The map \(\mathcal T_1\) annihilates every matrix entry between distinct outer-sector pairs.
This is the orthogonal control in the definition of \(\mathcal T_1\).
Let \(\Phi _3\) be the three-site regrouping. For every prepared matrix \(Y\) and three-site indices \(p,q\), \([\mathcal T_2(Y)]_{p,q}=Y_{\Phi _3(p),\Phi _3(q)}\).
This is matrix reindexing along the inverse regrouping \(\Phi _3^{-1}\).
Assume the neighboring-operator trace factorization of Definition 26.10.3.24. For every virtual matrix \(X\), in the canonical physical-sector coordinates selected by the isometry \(U\), the refinement channel satisfies
This is the refinement half of Proposition C.7 [ CPGSV16 , Appendix C.2, lines 1510–1516 and 1522–1545 ] .
The regrouping contained in \(\mathcal T_0\) changes \(\mathfrak R_2\) into \(\mathfrak G_2\). On the \((k,h)\) diagonal block, the partial trace then gives
On an active pair, \(\mathcal T_1\) adjoins \((a_kb_h)^{-1}\Omega _{k,h}\), and hence
If \(a_kb_h=0\), both sides vanish because \(\Omega _{k,h}=0\). After the canonical reassociation of the intermediate sector sum, these are precisely the diagonal blocks of \(\mathfrak G_3(\mathcal K_3(X))\); all off-diagonal sector blocks vanish. Finally, the inverse regrouping \(\mathcal T_2\) changes \(\mathfrak G_3\) into \(\mathfrak R_3\), which gives (??).
Assume the neighboring-operator trace factorization of Definition 26.10.3.24. For every virtual matrix \(X\), in the canonical physical-sector coordinates selected by the isometry \(U\), the coarse-graining channel satisfies
On the \((k,h)\) diagonal block, \(\mathcal S_0\) traces \(\Omega _{k,h}\) and gives \(a_kb_hB_{k,h}(X)\). On an active sector pair, \(\mathcal S_1((a_kb_h)B_{k,h}(X)) =B_{k,h}(X)\otimes \eta _{k,h}\). If \(a_kb_h=0\), positivity and zero trace give \(\eta _{k,h}=0\), so both sides of this identity vanish. Finally, \(\mathcal S_2\) regroups the four factors into two physical sites. For distinct sector pairs \((k,h)\ne (p,q)\),
These diagonal and off-diagonal identities give (??).
For the following construction, through Theorem 26.10.7.42, assume the neighboring-operator trace factorization of Definition 26.10.3.24. Write \(q=\dim \! \left(\bigoplus _k L_k\otimes R_k\right)\) for the physical dimension of the sector-coordinate tensor \(\widehat{\mathcal K}\).
Transporting \(\mathcal T\) through the canonical two-site and three-site coordinate identifications gives a channel \(\widehat{\mathcal T}:M_{q^2}(\mathbb {C})\longrightarrow M_{q^3}(\mathbb {C})\), where all multiple physical indices are associated to the right.
The map \(\widehat{\mathcal T}\) is trace-preserving and completely positive.
The coordinate identifications are unitary permutation channels. Composing them with a trace-preserving completely positive map preserves both properties.
For every virtual matrix \(X\), \(\widehat{\mathcal T}(\widehat{\mathcal K}_2(X)) =\widehat{\mathcal K}_3(X)\).
Transport the sector-coordinate identity through the inverse three-site coordinate identification.
Transporting \(\mathcal S\) through the canonical three-site and two-site coordinate identifications gives a channel \(\widehat{\mathcal S}:M_{q^3}(\mathbb {C})\longrightarrow M_{q^2}(\mathbb {C})\).
The map \(\widehat{\mathcal S}\) is trace-preserving and completely positive.
This follows by composition with the two coordinate-permutation channels.
For every virtual matrix \(X\), \(\widehat{\mathcal S}(\widehat{\mathcal K}_3(X)) =\widehat{\mathcal K}_2(X)\).
Transport the sector-coordinate identity through the inverse two-site coordinate identification.
Reassociating three right-associated sites as \((12)3\), applying \(\widehat{\mathcal T}\otimes \operatorname{id}\) to the first two, and reassociating the result back to \(1(2(34))\) defines the localized refinement channel \(\widehat{\mathcal T}_{(12)3}:M_{q^3}(\mathbb {C})\longrightarrow M_{q^4}(\mathbb {C})\) on right-associated physical coordinates.
The localized refinement map is trace-preserving and completely positive.
Tensoring a channel with the identity preserves complete positivity and trace, as do the two associativity permutations.
For every virtual matrix \(X\), \((\widehat{\mathcal T}\otimes \operatorname{id}) (\widehat{\mathcal K}_3(X)) =\widehat{\mathcal K}_4(X)\), with the map understood in right-associated coordinates.
Fixing the last ket and bra indices turns the three-site closure into a two-site closure with the last tensor matrix absorbed into \(X\). Apply the two-to-three-site refinement identity to this virtual matrix.
Reassociating four right-associated sites as \((123)4\), applying \(\widehat{\mathcal S}\otimes \operatorname{id}\) to the first three, and reassociating the result back to \(1(23)\) defines the localized coarse-graining channel \(\widehat{\mathcal S}_{(123)4}:M_{q^4}(\mathbb {C})\longrightarrow M_{q^3}(\mathbb {C})\) on right-associated physical coordinates.
The localized coarse-graining map is trace-preserving and completely positive.
Tensor the physical coarse-graining channel with the identity and compose with the two associativity permutations.
For every virtual matrix \(X\), \((\widehat{\mathcal S}\otimes \operatorname{id}) (\widehat{\mathcal K}_4(X)) =\widehat{\mathcal K}_3(X)\) in right-associated coordinates.
Fixing the last ket and bra indices identifies the four-site closure with a three-site closure having the last tensor matrix absorbed into \(X\). Apply the three-to-two-site coarse-graining identity.
Let \(\widehat{\mathcal K}\) be the sector-coordinate tensor selected by the physical isometry of Proposition C.7. Define
Here the second application acts on the first two sites and leaves the third site unchanged. Thus the superscript denotes two successive overlapping applications, as in the proof of [ CPGSV16 , Theorem 4.9, (iv)\(\Rightarrow \)(v) ] .
The map \(\widehat{\mathcal T}^2\) is trace-preserving and completely positive.
It is the composition of the physical refinement channel and its localization on the first two sites.
For every virtual matrix \(X\), the sector-coordinate tensor satisfies \(\widehat{\mathcal T}^2(\widehat{\mathcal K}_2(X)) =\widehat{\mathcal K}_4(X)\).
The first refinement gives \(\widehat{\mathcal K}_3(X)\); applying the localized refinement to its first two sites gives \(\widehat{\mathcal K}_4(X)\).
Define
Here the first application acts on the first three sites and leaves the fourth site unchanged. The superscript again denotes successive overlapping applications.
The map \(\widehat{\mathcal S}^2\) is trace-preserving and completely positive.
It is the composition of the localized coarse-graining channel and the physical coarse-graining channel.
For every virtual matrix \(X\), the sector-coordinate tensor satisfies \(\widehat{\mathcal S}^2(\widehat{\mathcal K}_4(X)) =\widehat{\mathcal K}_2(X)\).
The localized coarse-graining first gives \(\widehat{\mathcal K}_3(X)\); the second coarse-graining gives \(\widehat{\mathcal K}_2(X)\).
![\tnpic[compact, physical=updown, tensor style=box]{%
\tn[mpo]{\widehat{\cal K}}
\tnspan[box, label pos=south]{2}{\widehat{\cal K}^{[2]}} &
\tn[mpo]{\widehat{\cal K}}}](tenkz_svg/tenkz-00ada7cb0b19ac2f.svg)
\(\quad \xrightarrow {\widehat{\mathcal T}}\quad \) ![\tnpic[compact, physical=updown, tensor style=box]{%
\tn[mpo]{\widehat{\cal K}} &
\tn[mpo]{\widehat{\cal K}} &
\tn[mpo]{\widehat{\cal K}}}](tenkz_svg/tenkz-0881981cc0f042ec.svg)
\(\quad \xrightarrow {\widehat{\mathcal T}_{(12)3}}\quad \) ![\tnpic[compact, physical=updown, tensor style=box]{%
\tn[mpo]{\widehat{\cal K}}
\tnspan[box, label pos=south]{2}{\widehat{\cal K}^{[2]}} &
\tn[mpo]{\widehat{\cal K}} &
\tn[mpo]{\widehat{\cal K}}
\tnspan[box, label pos=south]{2}{\widehat{\cal K}^{[2]}} &
\tn[mpo]{\widehat{\cal K}}}](tenkz_svg/tenkz-0b158dc0ef6a95d5.svg)
\(\quad \xrightarrow {\widehat{\mathcal S}_{(123)4}}\quad \) \(\quad \xrightarrow {\widehat{\mathcal S}}\quad \) Let \(\widehat{\mathcal K}^{[2]}\) be the tensor obtained by blocking two adjacent sites of the sector-coordinate tensor \(\widehat{\mathcal K}\). Then \(\widehat{\mathcal K}^{[2]}\) is a renormalization fixed point in the sense of Definition 26.2.1. More precisely, after the canonical identifications of one blocked site with two original sites and two blocked sites with four original sites, the required channels are \(\widehat{\mathcal S}^2\) and \(\widehat{\mathcal T}^2\).
Decode one blocked index as a pair of physical indices and two blocked indices as four physical indices. The one-site and two-site closures of \(\widehat{\mathcal K}^{[2]}\) thereby become respectively \(\widehat{\mathcal K}_2(X)\) and \(\widehat{\mathcal K}_4(X)\). Transporting the two channels through these permutation identifications preserves complete positivity and trace, and the two closure identities give the required equations in Definition 26.2.1.
Assume that the neighboring operators are positive semidefinite and that there are real families \((a_k)_k,(b_k)_k\) satisfying \(\operatorname{tr}(\eta _{k,h})=a_kb_h\) and \(\sum _k a_kb_k=1\). Let \(\mathcal K^{[2]}\) be the tensor obtained by blocking two adjacent sites of \(\mathcal K\). Then \(\mathcal K^{[2]}\) is a renormalization fixed point in the sense of Definition 26.2.1. This is [ CPGSV16 , Theorem 4.9, (iv)\(\Rightarrow \)(v) ] .
Transport the refinement and coarse-graining channels for \(\widehat{\mathcal K}^{[2]}\) through the two-site and four-site tensor powers of the physical isometry, as in (??). These unitary congruences preserve complete positivity and trace, while the corresponding closure identities carry the two fixed-point equations back to \(\mathcal K^{[2]}\).
Let \(K\) be a simple tensor in block-injective canonical form (biCF), equipped with a basis-of-normal-tensors decomposition and satisfying the strong area law and zero correlation length. Let \(M=K^{[2]}\) be its two-site blocking. Then there are trace-preserving completely positive maps in both directions between the one-site and two-site closures of \(M\). Equivalently, \(M\) is a renormalization fixed point via these two maps. This is implication (ii)\(\Rightarrow \)(v) of [ CPGSV16 , Theorem 4.9 and Appendix C.2, lines 1810–1825 ] . The separate standing biCF and BNT hypotheses are imposed at lines 849–850 and restated at line 1628 of the same source. The printed proof of implication (ii)\(\Rightarrow \)(v) refers back to the construction of Proposition C.7 after projection to each local sector. That construction requires the rank-one neighboring-trace factorization supplied in the printed argument by the refuted Lemma C.5. No proof under SAL and ZCL alone is presently known.
Conjugation by \(U^{\otimes n}\) and \((U^\dagger )^{\otimes n}\) carries the sector-coordinate channels to the ordinary-coordinate channels. This combines Proposition C.7 with [ CPGSV16 , Appendix C.2, lines 1510–1563 and 1821–1825 ] .
Tracing the ket and bra indices of a sector tensor gives the bond vector \(|l_k)\) or the bond functional \((r_k|\). Hence a neighboring pair gives \(T_{k,h}=(r_k|l_h)=\operatorname{tr}(\eta _{k,h})\), as in [ CPGSV16 , Appendix C.2, lines 1474–1481 ] .
26.10.8 Local simple-MPDO structure
This structure records the local ingredients of [ CPGSV16 , Appendix C.2 ] : a normalized three-site reduced state with equality in strong subadditivity, the resulting Markov decomposition, and a primitive matrix \(T\) with trace one and, for every positive integer \(m\), equal positive-power traces:
together with a rank-one factorization
Lemma C.5 of [ CPGSV16 , Appendix C.2, lines 1484–1499 ] claims that an injective matrix product density operator tensor satisfying the strong area law and zero correlation length has real families \((a_k)_k,(b_k)_k\) with
This claim is false under the printed hypotheses. Theorem 26.10.8.3 gives a four-sector counterexample with the source inverse-map factorization and with no such families \(a,b\).
There is an injective matrix product density operator tensor satisfying the strong area law and source zero correlation length whose source-selected inverse-map factorization has positive semidefinite neighboring operators and a primitive trace-one active trace matrix \(T\), but for which there are no families \(a,b\) satisfying \(T_{k,h}=a_kb_h\) for all \(k,h\). Equivalently, for the corresponding closed sector tensors \(L,Q\), one has \(Q(1-LQ)L=T-T^2\ne 0\). Thus the rank-one conclusion in Lemma C.5 is false under its printed hypotheses.
The four physical sectors are one-dimensional. Their cyclic transition law is doubly stochastic. After tracing one site, conditioning on the middle physical label gives, at every admissible cut,
Thus the reduced state has a four-sector quantum-Markov decomposition and the tensor satisfies the strong area law. Its physical-trace transfer is the idempotent matrix \(LQ\), so it has source zero correlation length. The explicit Hayashi decomposition and inverse map recover the displayed tensors \(L,Q\). The active trace matrix is primitive and has trace one, but direct multiplication gives \(T^2\ne T\). If \(T=ab^\top \), then
where the last equality uses \(\operatorname{tr}(T)=1\). Hence \(T\) has no rank-one factorization.
There is a four-sector, bond-dimension-two tensor with a refined Hayashi decomposition whose four sectors are one-dimensional and have weight \(1/4\). Its three-site state is diagonal, with
The normalized four-site tail is \(LQ\). For this tail, the inverse-map construction recovers the closed sector tensors \(L\) and \(Q\) exactly. Its neighboring operators therefore coincide with those obtained directly from \(L\) and \(Q\), while \(Q(1-LQ)L\ne 0\). The inverse-map construction is the one in [ CPGSV16 , Appendix C.2, lines 1413–1455 ] .
Write \(S_i=L_{\mathord \bullet i}Q_{i\mathord \bullet }\) for the four physical slices, and let \(f_i(a,b)\) be their dual coefficients, so that
With \(T=QL\), \(p_k=1/4\), and \((LQ)_{00}=1\), the two inverse-map contractions are
Hence the neighboring operators are those obtained directly from \(L\) and \(Q\), and direct multiplication gives \(Q(1-LQ)L\ne 0\).
Let \(\mathcal K\) be an injective matrix product density operator tensor satisfying the strong area law and zero correlation length. Then there are an isometry, closed sector tensors, positive neighboring operators \(\eta _{k,h}\), and real families \((a_k)_k,(b_k)_k\) such that the inverse-map sector decomposition holds and
This is the structural corollary at [ CPGSV16 , Appendix C.2, lines 1501–1505 ] . Its printed derivation combines Lemma C.4 with the refuted Lemma C.5, so the conclusion remains open under the printed hypotheses.
These are conditional constructions under hypotheses stronger than SAL and ZCL; neither is the printed Lemma C.5 or its corollary. In the first, \(T\) is positive semidefinite, \(\operatorname{tr}(T)=1\), and \(\operatorname{tr}(T^m)=1\) for every positive integer \(m\); the spectral theorem then gives the rank-one factorization \(T=ab^\top \) with \(a\cdot b=1\). In the second, \(T\) is the sector trace matrix of linearly independent sector tensors satisfying the zero-correlation-length pairing identity. This identity implies \(T^2=T\), so trace one gives the same rank-one factorization. In this second form the constancy of the positive-power traces is a conclusion rather than an assumption.
For \(x\ne y\), the physical supports are locally orthogonal:
![\begin{tenkz}[rows={op,op}]
\tn[mpo]{\mathcal K_y} \\
\tn[mpo]{\mathcal K_x}
\end{tenkz}](tenkz_svg/tenkz-e0f13c85787d5da5.svg)
\( = 0\) In contraction notation, local orthogonality is \(\mathcal K_y^*\mathcal K_x=0\) whenever \(x\ne y\). This is [ CPGSV16 , Theorem 4.9(iv), lines 863–868 ] .
Expanding \(\sum _jP_j=\mathbb {1}\) on both physical indices and using the local orthogonality relations leaves only the diagonal term:
![\begin{tenkz}[physical=updown]
\tn[mpo]{K_i}
\end{tenkz}](tenkz_svg/tenkz-6a4a3adad136b3c2.svg)
\( = \sum _{j,j'}\) ![\begin{tenkz}[rows={op:none, op, op:none}]
\tn[mpo]{P_j} \\
\tn[mpo]{K_i} \\
\tn[mpo]{P_{j'}}
\end{tenkz}](tenkz_svg/tenkz-38aaf1743d6d6ff3.svg)
\( = \) ![\begin{tenkz}[rows={op:none, op}]
\tn[mpo]{P_i} \\
\tn[mpo]{K_i}
\end{tenkz}](tenkz_svg/tenkz-90c4975a1aea9f7f.svg)
\( = \) ![\begin{tenkz}[rows={op, op:none}]
\tn[mpo]{K_i} \\
\tn[mpo]{P_i}
\end{tenkz}](tenkz_svg/tenkz-5316c5170994cff6.svg)
Equivalently, each local tensor is supported on its own sector: \(\mathcal K_i=P_i\mathcal K_i=\mathcal K_iP_i\). This is the calculation in [ CPGSV16 , Appendix C.2, lines 1745–1751 ] .
Let \(K\) be a simple tensor in block-injective canonical form (biCF), equipped with a basis-of-normal-tensors decomposition. If \(K\) satisfies the strong area law and zero correlation length, then its basis tensors have mutually orthogonal physical supports. Moreover, each basis tensor has an inverse-map sector decomposition with positive neighboring operators \(\eta _{k,h}\) and real families \((a_k)_k,(b_k)_k\) satisfying
This is implication (ii)\(\Rightarrow \)(iv) of [ CPGSV16 , Theorem 4.9 and Appendix C.2, lines 1740–1788 ] . The separate standing biCF and BNT hypotheses are imposed at lines 849–850 and restated at line 1628 of the same source. The proof at lines 1745–1785 applies the structural corollary after establishing SAL and ZCL for each basis tensor. Since that corollary uses the refuted Lemma C.5, the printed proof does not establish this proposition.
26.11 Single-bond commuting form from the local structure
This records one positive semidefinite two-site bond \(B\ge 0\) whose translated copies commute pairwise on the periodic \(N\)-site chain at every chain length \(N\ge 2\): \([B_{i,i+1},B_{j,j+1}]=0\).
This defines the one-bond eta-local form constructed in [ CPGSV16 , Appendix C.2 ] : a two-site bond \(B\) as in Definition 26.11.1 together with, for every chain length \(N\ge 2\), a constant \(c{\gt}0\) such that
The bond acts on the full two-site space.
A translation-invariant positive two-site bond determines finite-chain commuting-form data at every length \(N\ge 2\).
Write the two-site bond on the ordered pair of physical indices, and identify a three-site configuration \((x_0,x_1,x_2)\) with the left-associated triple \(((x_0,x_1),x_2)\). These are common coordinates for the two bonds on sites \((0,1)\) and \((1,2)\).
Let \(B\) be a translation-invariant positive two-site bond, written as an operator \(\widehat B\) on an ordered pair of physical spaces. Under the same three-site identification for both translates,
Thus these two operators are respectively the left and right overlapping lifts of \(\widehat B\). Moreover, \(\widehat B\) is Hermitian and the two lifts commute. These are exactly the local hypotheses needed for the Bravyi–Vyalyi decomposition invoked at [ CPGSV16 , Appendix C.2, lines 1599–1605 ] .
Let \(\Phi \) be the matrix reindexing induced by the common three-site identification, and write \(B_{0,1}=\Phi (B_0)\) and \(B_{1,2}=\Phi (B_1)\) for the two reindexed translates. For three-site configurations \(x\) and \(y\), the two outside-window conditions are
Hence evaluation at a pair of left-associated triples gives
Positivity of \(B\) gives \(\widehat B^*=\widehat B\). Finally, matrix reindexing is multiplicative, so \(\Phi (XY)=\Phi (X)\Phi (Y)\), and
Substituting the two entrywise identities proves that the left and right overlapping lifts commute.
The \(\eta \)-local structure itself determines a single-bond commuting-form witness for every chain length \(N\ge 2\): writing \(B\) for its bond, the translated copies commute, \([B_{i,i+1},B_{j,j+1}]=0\), and there exists \(c{\gt}0\) with
This is the one-bond product conclusion used in [ CPGSV16 , Proposition C.8 ] .
At length \(N\), apply the commuting two-site bond and realization identity provided by the \(\eta \)-local structure.
The same \(\eta \)-local structure gives a single-bond product witness on every finite periodic chain.
Evaluate the local structure at each chain length \(N\ge 2\).
An \(\eta \)-local structure together with the doubled-index transfer condition gives the corresponding single-bond branch.
Use the commuting-form witness carried by the \(\eta \)-local structure and combine it with the doubled-index transfer hypothesis.
Let \(\widehat B\) be a pair-indexed positive two-site bond whose periodic translates commute. There are positive dimensions \(d_{l,q}=\dim H_{q,l}\) and \(d_{r,q}=\dim H_{q,r}\), a unitary decomposition \(H\cong \bigoplus _{q=0}^{K-1}H_{q,l}\otimes H_{q,r}\), and Hermitian operators \(R_q\) on \(H\otimes H_{q,l}\) and \(S_q\) on \(H_{q,r}\otimes H\) such that
The basis bijection for each sector enumerates the numerical indices in right–left order,
Writing \(e(q,(r,s))\) for this basis bijection, the left and right spatial coordinates are respectively
Hence the tensor factors in the displayed block actions occur in left–right order.
The pair-indexed bond is Hermitian by positivity, and its two overlapping lifts commute by the common three-site identification. Apply Theorem 25.6.11.
Let \(B\) be a pair-indexed positive bond whose periodic translates commute. There are a unitary coordinate decomposition \(H\cong \bigoplus _q H_{q,r}\otimes H_{q,l}\) and positive semidefinite operators \(\eta _{q,h}\) on \(H_{q,r}\otimes H_{h,l}\). After regrouping the two-site coordinates, the \((q,h)\)-sector is \(H_{q,l}\otimes (H_{q,r}\otimes H_{h,l})\otimes H_{h,r}\), and in these regrouped coordinates
The unitary and the neighboring operators depend only on the fixed bond, and hence are independent of a chain length or bond position.
The same unitary block decomposition at both ends of \(B\) gives identity action on \(H_{q,l}\) through one block identity and on \(H_{h,r}\) through the other. For fixed basis indices \(\ell _q^0\in H_{q,l}\) and \(r_h^0\in H_{h,r}\), let \(B'=(U^*\otimes U^*)B(U\otimes U)\). The neighboring operator has entries
Thus \(\eta _{q,h}\) is a principal compression of the positive operator \(B'\), and is therefore positive semidefinite.
Suppose that, in one fixed coordinate decomposition \(H\cong \bigoplus _q H_{q,r}\otimes H_{q,l}\), after regrouping the two-site coordinates according to this decomposition, a two-site operator has the form
For every \(N\ge 2\), regroup the cyclic chain as
Then, in these coordinates and with \(k_N=k_0\),
At \(N=2\), the two translated windows have the opposite orders \((0,1)\) and \((1,0)\) and give the two oriented factors \(\eta _{k_0,k_1}\) and \(\eta _{k_1,k_0}\).
In a fixed sector configuration, the cyclic regrouping sends the factors \((r_n,l_n)\) to the edge factors \((r_n,l_{n+1})\). The translate beginning at \(n\) acts by \(\eta _{k_n,k_{n+1}}\) on the \(n\)th edge factor and by the identity on every other edge factor. Different sector configurations do not mix. Multiplying the translated operators therefore gives the stated direct sum of tensor products.
Let \(\sigma ^{(N)}(\mathcal K)\) be realized, for every \(N\ge 2\), by translates of one positive commuting bond. There are one unitary \(U\), positive sector dimensions, and positive semidefinite operators \(\eta _{q,h}\), all independent of \(N\), such that for every \(N\ge 2\) there is a constant \(c_N{\gt}0\) satisfying, with \(k_N=k_0\),
Here \(E_N\) is the cyclic regrouping from the edge factors \(H_{k_n,r}\otimes H_{k_{n+1},l}\) to the site factors \(H_{k_n,r}\otimes H_{k_n,l}\). The factor \(c_N\) is retained from the proportionality relation in [ CPGSV16 , Appendix C.2, lines 1571–1576 ] . The passage at [ CPGSV16 , Appendix C.2, lines 1603–1605 ] suppresses this factor when invoking the coefficient-free equation sigmaNK2; the coefficient-free assertion is not part of the present theorem.
Choose the unitary and the neighboring operators from the decomposition of the fixed bond. Conjugation by \((U^*)^{\otimes N}\) carries the product of its translates to the product of the conjugated two-site bonds. The cyclic regrouping theorem changes this product into the displayed direct sum. Applying both transformations to \(\sigma ^{(N)}(\mathcal K)=c_N\prod _n B_{n,n+1}\) preserves the same positive scalar \(c_N\).
The proportional commuting-bond form alone does not permit one common local normalization. Indeed, take physical dimension \(d=1\) and bond dimension \(D=2\), with sole matrix-product letter \(M^{00}=I_2\), and take the two-site bond to be the scalar \(B=1\). Then
Any positive neighboring decomposition of the one-dimensional physical space has one sector and one scalar \(a\ge 0\). A coefficient-free identity at lengths two and four would give \(a^2=2\) and \(a^4=2\), a contradiction. This example is not a normal injective block. More generally, the source does not claim that every proportional commuting-bond presentation can be made coefficient-free by one local rescaling. In the SAL-to-commuting-form direction, the proof chooses the neighboring operators from the source tensor and obtains scalar one directly. This distinguishes the proportionality at [ CPGSV16 , Appendix C.2, lines 1571–1576 ] from the coefficient-free invocation at [ CPGSV16 , Appendix C.2, lines 1603–1605 ] . The source-selected normal representative is Corollary 26.10.3.49.
A scalar weight \(w(i,j;i',j')\) on two consecutive ket–bra pairs determines a translation-invariant matrix-product tensor with virtual dimension \(d^2\). Its incoming virtual pair is constrained to equal the current physical ket–bra pair, while its outgoing pair records the following physical pair.
For every positive length \(N\), and with \(\sigma _N=\sigma _0\) and \(\tau _N=\tau _0\), the closed matrix-product operator of the cyclic edge-weight tensor is
In the virtual trace there is precisely one potentially nonzero cyclic virtual configuration.
A fixed tensor for a bond-product family consists of a positive virtual dimension and one matrix-product tensor \(C\), independent of \(N\), such that, for every \(N\ge 2\),
If \(d{\gt}0\), a cyclic scalar edge-weight expression for the entries of the product gives such a tensor with positive virtual dimension \(d^2\).
Let \(B\ge 0\) be a two-site bond whose periodic translates commute. There is a positive integer \(R\) and a matrix-product tensor \(C\) of virtual dimension \(R\), both independent of the chain length, such that for every \(N\ge 2\),
The construction gives a positive \(R\) independent of \(N\). No normality, zero-correlation-length, or scalar normalization hypothesis is required.
Choose Beigi’s chain-independent one-site unitary and the positive neighboring operators \(\eta _{q,h}\). For a common matrix-unit index set \(\mathcal A\), write
Thus the contraction of the right factor at one site with the left factor at the next satisfies \(\sum _{a\in \mathcal A}R_a(q)\otimes L_a(h)=\eta _{q,h}\). If \(V\) is the resulting unitary change to the two-site sector coordinates, this contraction gives \(\widetilde B=VBV^\dagger \) and \(B_{\mathrm{phys}}=V^\dagger \widetilde B V=B\). For a nonzero physical space, \(\mathcal A\) is nonempty and gives a positive virtual dimension. For the zero-dimensional physical space, take one virtual state and no physical sectors. The cyclic contraction formula then gives the product of the translates of \(B\) at every length \(N\ge 2\).
Let \(C\) be the selected fixed tensor of the preceding theorem. There are physical sectors \(\mathbb {C}^d\cong \bigoplus _{q=0}^{K-1}H_q^L\otimes H_q^R\), a unitary \(U\), and matrices \(L_\beta (q)\) and \(R_\alpha (q)\) such that, writing \(C_{\beta ,\alpha }\) for the physical matrix at fixed virtual indices,
The neighboring operators \(\eta _{q,h}=\sum _a R_a(q)\otimes L_a(h)\) are positive semidefinite. If \(U_2\) denotes the induced two-site change of coordinates, then
Thus the physical-sector factorization belongs to the same tensor \(C\) whose closed operators are the fixed products of \(B\).
Take the physical-sector factorization retained with \(C\). Its defining one-site identity is
Contracting the adjacent right and left factors gives \(\eta _{q,h}=\sum _a R_a(q)\otimes L_a(h)\succeq 0\), where positivity is the retained neighboring-operator property. Finally, the retained bond identity is
A cyclic scalar edge formula need not hold in the original physical basis. For \(d=2\), let \(X\) be the Pauli flip and set \(B=\mathbb {1}+X\otimes X\). This bond is positive semidefinite and its periodic translates commute. For every \(N\ge 2\),
At \(N=2\) the periodic product is \(B^2=2B\); for \(N\ge 3\), the empty edge set and the full cycle are the two terms with zero boundary. An original-coordinate scalar edge weight would make this entry equal to \(a^N\). The cases \(N=2\) and \(N=3\) would give \(a^2=2\) and \(a^3=2\), a contradiction. Equation sigmaNK2 in [ CPGSV16 , Appendix C.2, lines 1581–1589 ] is stated only after the one-site unitary coordinate change, as used in the theorem.
Suppose one fixed tensor \(C\) generates the products of the bond carried by an eta-local structure. Then the doubled-index tensors of \(M\) and \(C\) are eventually nonzero proportional. The proof uses only lengths \(N\ge 2\) and therefore assumes no relation at length one.
At every \(N\ge 2\), substitute the exact tensor representation into the positive realization \(\rho ^{(N)}(M)=c_N\prod _n B_{n,n+1}\), where \(c_N{\gt}0\), and read ket–bra pairs as doubled physical indices.
Let \(A\) be the normal doubled-index source tensor of positive bond dimension \(D\), and let \(C\) be a fixed tensor for its commuting-bond product. Suppose that \(sC\) is normal for some \(s{\gt}0\). Then there is a complex number \(\zeta \), with \(|\zeta |=1\), such that, for every \(N\ge 2\),
This is a conditional reduction: it does not assert that \(C\), or any rescaling of \(C\), is normal.
Scaling \(C\) by \(s\) gives \(V^{(N)}(sC)_\sigma =s^N V^{(N)}(C)_\sigma \). Thus, for all sufficiently large \(N\), the proportionality \(V^{(N)}(A)_\sigma =c_NV^{(N)}(C)_\sigma \) becomes
The normal-tensor overlap dichotomy gives one \(\zeta \in \mathbb {C}\), with \(|\zeta |=1\), such that, at every positive length,
Let \(A\) be the normal doubled-index source tensor of positive bond dimension \(D\), let \(C\) be a fixed tensor for its commuting-bond product, and suppose that \(sC\) is normal for some \(s{\gt}0\). Then the unit phase is trivial. For every \(N\ge 2\),
Thus a positive normal rescaling of the fixed product tensor already supplies the positive geometric realization law.
Normality gives \(\lim _{N\to \infty }\langle V^{(N)}(A) | V^{(N)}(A) \rangle =1\). Hence at two sufficiently large consecutive lengths, say \(N\) and \(N+1\), both the source vector and the fixed bond product are nonzero. Positivity of the realization gives positive numbers \(c_N,c_{N+1}\) satisfying \((\zeta s)^N=c_N\) and \((\zeta s)^{N+1}=c_{N+1}\). Consequently \(\zeta s=c_{N+1}/c_N{\gt}0\). Since \(|\zeta |=1\) and \(s{\gt}0\), one has \(\zeta =1\).
Let \(M\) have source zero correlation length. For every \(N\geq 0\), the normalized \(N\)-site marginal obtained by tracing two sites from \(\sigma ^{(N+2)}(M)\) equals the normalized \(N\)-site marginal obtained by tracing one site from \(\sigma ^{(N+1)}(M)\).
This is the first replacement at [ CPGSV16 , Appendix C.2, line 1613 ] . It does not imply the rank-one trace factorization asserted later on that line.
Write \(E=\mathcal T_M\) and choose \(\lambda {\gt}0\) such that \(E^2=\lambda E\). For physical configurations \(u,v\) of length \(N\), the two marginal matrix elements are \(\operatorname{tr}(M^{u,v}E^2)/\operatorname{tr}(E^{N+2})\) and \(\operatorname{tr}(M^{u,v}E)/\operatorname{tr}(E^{N+1})\). The numerator and denominator on the left are respectively \(\lambda \) times those on the right, and source zero correlation length makes the denominator on the right nonzero.
Let \(M\) have source zero correlation length. For every \(L\geq 0\), the normalized \(L\)-site marginal obtained by tracing three sites from \(\sigma ^{(L+3)}(M)\) equals the normalized \(L\)-site marginal obtained by tracing one site from \(\sigma ^{(L+1)}(M)\).
First retain \(L+1\) sites and apply Lemma 26.11.23. Trace the last of these retained sites, and then apply the same lemma while retaining \(L\) sites.
A rank-one trace factorization of neighboring matrices \(\eta _{q,h}\) consists of real numbers \(a_q,b_h\) such that
The factorization is an explicit assumption; it is not deduced from zero correlation length. It is the factorization invoked at [ CPGSV16 , Appendix C.2, line 1613 ] .
Suppose \(\eta _{q,h}\) has a rank-one trace factorization. For fixed boundary sectors \(u,v\), set
Then
This is the corrected edge contraction at [ CPGSV16 , Appendix C.2, lines 1613–1617 ] .
Substitute \(\operatorname{tr}(\eta _{q,h})=a_qb_h\) and distribute the two finite sums.
Let \(\eta _{q,h}\) be arbitrary matrices on \(H_{q,r}\otimes H_{h,l}\) with a rank-one trace factorization. For a nonempty retained sector word \(k=(k_0,\ldots ,k_{L-1})\), write
The edge contraction corresponding to tracing the two boundary sites of the longer cyclic chain is
This is the edge-coordinate interpretation of the displayed fourth-region trace in [ CPGSV16 , Appendix C.2, lines 1606 and 1613–1617 ] .
Apply Lemma 26.11.26 for each retained sector word. Tensor with the unchanged product \(P_k\) and take the direct sum over \(k\).
The theorem applies, in particular, to the positive chain-independent neighboring family supplied by Theorem 26.11.12, once the rank-one trace factorization is given. It does not derive that factorization, identify this two-boundary contraction with a one-site marginal, or prove a Markov decomposition or saturation of the area law.
There is an injective normal tensor \(\mathcal K\), of physical dimension two and bond dimension one, which generates MPDOs and has source zero correlation length, together with one positive two-site bond \(B\) whose translates commute. For every \(N\geq 2\),
while the trace matrix of a chosen two-sector Beigi factorization is
In particular, \(T\) is not primitive.
This shows that primitivity cannot be inferred for an arbitrary selected factorization in the step suggested at [ CPGSV16 , Appendix C.2, line 1613 ] . The same tensor also has a one-sector factorization with positive semidefinite neighboring operator and primitive trace matrix. Hence the example does not exclude a source-faithful minimal or visible-sector selection, and it does not disprove the stated commuting-form-to-SAL implication.
Let \(\mathcal K^{0,0}=1\) and let all other physical entries vanish. The matrices \(\{ \mathcal K^{i,j}\} _{i,j=0}^1\) span \(M_1(\mathbb C)\). For every \(X\in M_1(\mathbb C)\), the doubled-index transfer \(\mathcal E_{\mathcal K}\) and the physical-trace transfer \(P_{\mathcal K}\) satisfy
Hence \(\mathcal K\) is injective and normal and has source zero correlation length. Decompose the physical space nonminimally as \(\mathbb C\oplus \mathbb C\). With \(w_0=1\) and \(w_1=0\), take both sector factors to be the scalar \(w_k\) in sector \(k\). The neighboring operators are
In the sector coordinates the corresponding physical bond is
Since only the all-zero physical-index word contributes, the factorization identity is
Consequently,
For every \(n\geq 1\), \(T^n=\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}\). Thus no positive power has all entries strictly positive, so \(T\) is not primitive. For the alternative factorization, take one sector \(B^L=\mathbb C^2\) and \(B^R=\mathbb C\), with
Its sole neighboring operator is \(\widehat\eta _{0,0}=\operatorname{diag}(1,0)\succeq 0\), and hence its trace matrix is \(\widehat T=(\operatorname{tr}(\widehat\eta _{0,0}))=(1)\). Every positive power of \(\widehat T\) equals \((1)\), so this one-sector trace matrix is primitive.
For a physical-sector factorization with neighboring operators \(\eta _{k,h}\), retain a sector \(k\) when there is a sector \(h\) such that \(\eta _{k,h}\ne 0\) and \(h\leadsto k\), where \(\leadsto \) denotes reachability by nonzero neighboring operators. The first edge \(k\to h\) is explicit, so the resulting closed walk has positive length even when the return path is empty. The set \(C\) of retained sectors is therefore exactly the set of vertices occurring in positive-length directed cycles, in the sense of [ Bei12 , Section III ] . Define, for \(k,h\in C\),
For positive semidefinite neighboring operators the trace is real, so \(T_{k,h}=\operatorname{tr}(\eta _{k,h})\), and \(T_{k,h}{\gt}0\) is equivalent to \(\eta _{k,h}\ne 0\).
Let \(N\geq 1\) and let \(k=(k_0,\ldots ,k_{N-1})\) be a cyclic sector configuration. If \(k_n\notin C\) for some \(n\), then \(\Omega _k=0\). Thus deleting the sectors outside \(C\) leaves every finite cyclic neighboring product unchanged. For every linear contraction \(\Phi \), including a composition of reindexings and partial traces, \(\Omega _k=0\) implies \(\Phi (\Omega _k)=0\).
If \(\Omega _k\ne 0\), each factor \(\eta _{k_n,k_{n+1}}\) is nonzero; otherwise every matrix entry of the product vanishes. Starting with the edge \(k_n\to k_{n+1}\) and following the remaining cyclic edges gives a return path to \(k_n\). Hence every \(k_n\) belongs to \(C\). The last assertion is the linear identity \(\Phi (0)=0\).
Suppose that the virtual dimension is positive, the MPO tensor is injective, and every \(\eta _{k,h}\) is positive semidefinite. Then \(C\) is nonempty and the matrix \(T\) is primitive. More precisely, \(T\) is irreducible and every \(k\in C\) satisfies
The virtual matrices of all physical sectors span the full matrix algebra. If \(A=A_k^{xy}\) is nonzero, nondegeneracy of the trace pairing gives a virtual matrix \(B=A_q^{uv}\) such that \(\operatorname{tr}(AB)\ne 0\). Cyclicity gives \(\operatorname{tr}(BA)=\operatorname{tr}(AB)\ne 0\), and therefore \(\eta _{k,q}\ne 0\) and \(\eta _{q,k}\ne 0\). Thus a sector containing a nonzero virtual matrix lies on a directed two-cycle. Hence every virtual matrix outside \(C\) vanishes, and the virtual matrices belonging to \(C\) still span the full algebra.
If a proper directed cut separated two nonempty subsets \(C_1,C_2\) of \(C\), let \(V_{C_i}\) denote the span of the virtual matrices in \(C_i\). Absence of an edge across the cut gives
Both spaces contain a nonzero matrix. This contradicts the one-sided product obstruction for two nonzero subspaces whose sum is the full matrix algebra. Thus \(T\) is irreducible. The same trace pairing gives a two-edge return through each sector and closes every retained edge through a third retained sector. These yield the displayed positive diagonal entries of \(T^2\) and \(T^3\); the coprime return lengths imply primitivity.
Under the hypotheses of Theorem 26.11.32, suppose that the physical-trace transfer has source zero correlation length. There is a number \(\lambda {\gt}0\) such that, with \(\widehat T=\lambda ^{-1}T\),
and \(\operatorname{tr}(\widehat T^m)=\operatorname{tr}(\widehat T)\) for every \(m\geq 1\). By matrix multiplication, the two-step coefficient is \((\widehat T^2)_{k,h}=\sum _{q\in C}\widehat T_{k,q}\widehat T_{q,h}\). The assertion neither identifies \(\widehat T\) with \(\widehat T^2\) nor implies that the unreduced Beigi trace matrix is primitive or rank one.
Set the left sector tensor equal to zero outside \(C\). The preceding span argument shows that this does not change the physical tensor, and the neighboring operators with both indices in \(C\) are unchanged. The source zero-correlation-length relation, transported through this restricted factorization, gives \(\widehat T^2=\widehat T^3\) on the cyclic-active indices. By Theorem 26.11.32, \(\widehat T\) is primitive. Its square is therefore a strictly positive idempotent matrix and has rank one.
Let \(C\) be the set of cyclic-active sectors and suppose that every neighboring operator is positive semidefinite. For \(q,h\in C\),
More generally, if \(B_{q,h}(X)\) denotes the outer boundary operator in the three-site closure, then
Here \(\operatorname{tr}_{\mathrm{mid}}\) traces both neighboring factors. No equality between \(T_C\) and \(T_C^2\) is asserted.
For every \(\lambda \in \mathbb {R}\), set \(\widehat T_C=\lambda ^{-1}T_C\). The same calculation with each fully traced edge multiplied by \(\lambda ^{-1}\) gives
Positivity makes each \(\operatorname{tr}(\eta _{a,b})\) real. Expanding matrix multiplication gives
The partial trace of the fixed-sector three-site closure is \(\operatorname{tr}(\eta _{q,r})\operatorname{tr}(\eta _{r,h})B_{q,h}(X)\). Summing over \(r\in C\) proves the second identity. Multiplying each of the two trace factors by \(\lambda ^{-1}\) gives the normalized identity.
Let \(\mathcal K\) be injective and have source zero correlation length, and suppose its neighboring operators are positive semidefinite. There are \(\lambda {\gt}0\) and strictly positive functions \(a,b\) on the cyclic-active sectors, normalized by \(a\mathbin {\boldsymbol \cdot }b=1\), such that every fourth-region marginal is a direct sum over retained sector words of the unchanged interior neighboring product tensored with the separated left and right boundary factors determined by \(a\) and \(b\).
The additional marginal replacement leaves the retained interior product unchanged and produces the two-step coefficient on the cyclic-active sectors. Its positive rank-one factorization separates the two surviving boundary sums. All sector words leaving the cyclic-active support vanish before the two outer partial traces are evaluated.
Under the hypotheses of Theorem 26.11.35, for arbitrary \(A,C\geq 0\), the fourth-region marginal on consecutive regions of lengths \(A\), \(1\), and \(C\) admits a Hayashi Markov decomposition. The middle site decomposes as the direct sum of its left and right sector factors. Each block is the tensor product of normalized positive left and right path states, and its probability is the product of their unnormalized traces divided by the positive periodic normalization.
Separate the retained chain at the distinguished middle site. The retained neighboring factors split into the left and right path products, while the two cyclic endpoint factors become their respective boundary matrices. Normalize each positive block by its trace, using an arbitrary trace-one positive matrix when the block vanishes. The trace-one identity for the full marginal shows that the resulting non-negative block weights sum to one.
Let \(\mathcal K\) have positive bond dimension and be injective, and let \(F\) be a physical-sector factorization of \(\mathcal K\). Suppose every neighboring operator \(\eta ^F_{q,h}\) is positive semidefinite. If the physical-trace transfer of \(\mathcal K\) has source zero correlation length, then \(\mathcal K\) saturates the area law.
Positivity of the neighboring operators makes every nonempty sector-coordinate chain operator positive. Source zero correlation length makes its trace nonzero. Apply the preceding theorem with \(A=m-1\) and \(C=N-m-1\) at every admissible cut, then use the all-cut Markov criterion. Finally transport saturation through the physical isometry of \(F\).
Let \(\mathcal K\) be an injective normal tensor generating MPDOs. Suppose that there is one positive two-site bond \(B\) whose translates commute and, for every \(N\geq 2\), a constant \(c_N{\gt}0\) such that
If the physical-trace transfer of \(\mathcal K\) has zero correlation length, then \(\mathcal K\) saturates the area law.
This is the proposition at [ CPGSV16 , Appendix C.2, lines 1597–1619 ] .
The single positive bond determined by the \(\eta \)-local structure gives the one-sector instance of the explicit GSNNCH decomposition.
Apply the one-sector construction to the single-bond product at every chain length.
Every injective MPO tensor satisfying SAL generates Gibbs states of a nearest-neighbor commuting Hamiltonian: the normalized finite-chain states are density operators with the explicit sector form of Definition 26.8.1, realized through one sector. The one-sector form is the commuting product form in which [ CPGSV16 , Proposition C.8 ] states its conclusion for an injective tensor, combined here with the density normalization of [ CPGSV16 , Definition 4.8 ] .
Corollary 26.10.3.48 gives the \(\eta \)-local structure, whose single positive bond gives the one-sector form by Theorem 26.11.39. SAL includes the nonvanishing of every positive-length trace, so Theorem 26.8.28 applies.
Suppose that \(K\) satisfies \(\mathrm{SAL}(K)\) and admits a physical-sector factorization whose neighboring operators \(\eta _{k,h}\) are simultaneously positive semidefinite. The associated two-site bond is
and, for every \(N\geq 2\),
Under this hypothesis, Theorem 26.10.3.47 gives the \(\eta \)-local structure and proves (??)–(??) with \(c_N=1\). Theorem 26.11.6 then gives the commuting-form conclusion. Adjacent translates commute on every periodic chain of length at least three by Theorem 26.10.3.40, and the crossed two-site case is Theorem 26.10.3.41. Disjoint translates commute by Theorem 26.10.3.42. These cases give pairwise commutativity at every length \(N\geq 2\) by Theorem 26.10.3.43. Given the physical-sector factorization, the all-length product identity is Theorem 26.10.3.44. The doubled-index transfer condition enters only in the subsequent RFP and single-bond branch conclusions.