Dependencies

For \(t\ge 0\) and \(n\in \mathbb {N}\), one has \(T_{nt}=(T_t)^n\).

Every density matrix is nonzero.

If a corner compression is invariant under \(E\), it remains invariant under every power \(E^n\).

If \(L(X)=\mu X\), then \(e^{tL}(X)=e^{t\mu }X\) for all \(t\in \mathbb {R}\).

If \(\mu \) is in the spectrum of \(L\), then \(e^{t\mu }\) is in the spectrum of \(e^{tL}\).

If \(e^{it\theta }\) is a root of unity for every \(t{\gt}0\), then \(\theta =0\).

Peripheral spectral points of the generator have vanishing real part.

The number of peripheral eigenvalues is bounded by \(\dim (M_{D}(\mathbb {C}))\).

If all powers of a peripheral eigenvalue remain peripheral, then some positive power not exceeding \(\dim (M_{D}(\mathbb {C}))\) equals \(1\).

For an irreducible channel with faithful fixed point, every power of a peripheral eigenvalue is again peripheral.

If all eigenvalues satisfy \(|\lambda |{\lt}1\), then the spectral radius is \({\lt}1\).

A linear map is two-positive if it is \(2\)-positive.

A linear map is \(k\)-positive if and only if its blockwise ampliation is positive on \(M_{D}(\mathbb {C})\otimes M_{k}(\mathbb {C})\).

Let \(K=(K_i)_{i=0}^{d-1}\) be a unital Kraus family, that is, \(\sum _i K_i K_i^\dagger =\mathbb {1}\). Then for all \(X\in M_{D}(\mathbb {C})\), \(\mathcal{K}(X^\dagger X)\ge \mathcal{K}(X)^\dagger \mathcal{K}(X)\), where \(\mathcal{K}(Y)=\sum _i K_i Y K_i^\dagger \).

For an orthogonal projection \(P\) and linear map \(E\), the corner \(PM_{D}(\mathbb {C})P\) is preserved if \(E(PXP)=PE(PXP)P\) for every \(X\).

The corner subspace associated with \(P\) is \(\{ PXP:X\in M_{D}(\mathbb {C})\} \).

Irreducibility of the corner-restricted map.

The rank of an orthogonal projection \(P\in M_{D}(\mathbb {C})\), equivalently the dimension of its range.

If the parent-ground-space dimension is eventually constant, then for every \(N{\gt}2\) it equals the number of positive loops in the sector graph. In particular, this conclusion holds under the BNT and all-chain nearest-neighbor parent-ground-space hypotheses of Theorem 14.1.70.

If \(D\ge 1\) and \(A\) is \(0\)-block injective, then \(D=1\).

Under the same hypotheses, suppose additionally that every periodic single-block chain space is contained in its matrix-product-vector line. Then

Let \((C_j^s)\) and \((E_j^s)\) be two marked families over a BNT canonical form. If their closed marked chains agree for every marked index \(s\) and every unmarked word \(w\), then \(C_j^s=E_j^s\) for every representative \(j\) and every marked index \(s\).

Let \(A\) be a CPSV canonical-form renormalization fixed point, and let \((A_j)_j\) be a basis of normal tensors for \(A\). CPSV asserts that there are invertible matrices \(X_j\), positive diagonal matrices \(\rho _j\) with \(\operatorname{tr}(\rho _j)=1\), and tensors \(U_j\) such that

Thus the BNT-basis tensors share the physical index \(i\), and the family \((U_j)_j\) satisfies the source joint-isometry condition. This is the square-root-consistent form of [ CPGSV16 , Corollary 3.12 ] .

Local fix (square-root diagonal). CPSV prints a bare \(\Lambda _j\) in Corollary 3.12, whereas the reference tensor at line 1300 has coefficients \(\sqrt{(\rho _j)_{\alpha ,\alpha }}\), which forces the factor \(\sqrt{\rho _j}\) used above. This repair is documented in docs/paper-gaps/cpsv16_rfp_isometry_scope.tex.

Let \(M\) be an MPO tensor in normalized BNT-refined horizontal form. For two coefficient families \(f_{u,z}\) and \(g_{u,z}\), put \(C^u=\sum _z f_{u,z}M^z\) and \(E^u=\sum _z g_{u,z}M^z\). If \(\operatorname{tr}(C^uM^w)=\operatorname{tr}(E^uM^w)\) for every marked index \(u\) and every word \(w\), then \(C^u=E^u\) for every \(u\).

The marks are required to lie in the span of the physical letters. No assertion is made for arbitrary bond-space matrices, since off-diagonal matrices between repeated sectors are invisible to closed chains. The corollary is the physical-insertion consequence of [ CPGSV16 , Appendix C.3, lines 1835–1858 ] used in the contractions of [ CPGSV16 , Proposition 4.13, lines 1909–1919 ] .

For \(s\in [0,1]\), \(t\in [0,1]\), and positive-definite matrices \(A_1,A_2,B_1,B_2\), the map \((A,B)\mapsto \Re \operatorname{tr}(A^s B^{1-s})\) is jointly concave:

Obtained from Theorem 20.6.13 by taking \(K=\mathbb {1}\).

For \(s\in [0,1]\), \(t\in [0,1]\), matrix \(K\), and positive-definite matrices \(A_1,A_2,B\), the map \(A\mapsto \Re \operatorname{tr}(K^\dagger A^s K B^{1-s})\) is concave:

For \(s\in [0,1]\), \(t\in [0,1]\), matrix \(K\), and positive-definite matrices \(A,B_1,B_2\), the map \(B\mapsto \Re \operatorname{tr}(K^\dagger A^s K B^{1-s})\) is concave:

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 ] .

For every positive blocked length \(n\), the blocked-basis coefficients are traces of powers of the BNT-label \(\chi \)-matrices selected by the source and target label maps. Equivalently, they are traces of powers of the pulled-back blocked-basis matrices \(\chi _{n,i,j,k}\):

For every positive blocked length \(n\), the blocked-basis coefficients obtained from an existential BNT-label theorem witness are traces of powers of the BNT-label \(\chi \)-matrices selected by the source and target maps. Equivalently, they are traces of powers of the pulled-back blocked-basis \(\chi \)-matrices:

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.

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.

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 \(n_x\).

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.

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, \(\mathcal T_{\cal K}=\sum _k |l_k)(r_k|\).

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.

For the diagonal MPO associated to a pure MPS tensor, the fusion-isometry formulation reduces to the usual pure-state RFP condition.

For every ancillary dimension, the spin reduction obtained by tracing the ancillary index is trace-preserving and completely positive.

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.

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.

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.

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 \(\mathcal K\) 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.

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.

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.

Let \(M\) be a matrix product density operator in normalized BNT-refined horizontal form and suppose that it satisfies the renormalization fixed-point condition. One choice of the vertical canonical decomposition and fusion coisometries satisfies, for every \(N{\gt}0\),

Equivalently, \([A_N,T_N]=0\), as asserted in [ CPGSV16 , lines 1000–1002 ] . No additional compatibility hypothesis is imposed. This statement makes no length-independence, terminal spectral, projection, Hamiltonian, or Gibbs-state assertion.

Scope restriction (BNT-refined horizontal form): the normalized BNT-refined horizontal hypothesis is stronger than the literal CPSV canonical-form hypothesis.

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 \(I(X:Y)\leq 2\log D\).

Every nondegenerate purification renormalization fixed point satisfies the positive-length global purification predicate of Definition 4.3 in [ CPGSV16 , lines 744–758 ] .

Let \(m\geq 1\) and let \(\lambda _0,\ldots ,\lambda _{m-1}\in \mathbb {C}\) be nonzero. Then there is an exponent \(N\) with \(1\leq N\leq m\) such that \(\sum _{q=0}^{m-1}\lambda _q^N\neq 0\). This is the consequence of the Appendix power-sum lemma used in the proof of Lemma L [ CPGSV16 , Appendix C.3, lines 1851–1858 ] ; the finite-range comparison is [ CPGSV16 , Appendix A, lines 1155–1163 ] .

For \(t \ge 0\), setting \(s=t\) in (??) gives

where \(M = \sup _{u \in [0,t]} \| e^{uL}\| \).

Let \(A\) and \(B\) be normal tensors of positive, possibly different, bond dimensions. Suppose that, for every positive \(N\), their matrix product vectors are proportional by a nonzero scalar. Then there is \(a\in \mathbb {C}\), with \(|a|=1\), such that \(|V^{(N)}(A)\rangle =a^N|V^{(N)}(B)\rangle \) for every positive \(N\). This is the positive-length one-normal-block consequence of [ CPGSV16 , Corollary A.3 and Theorem 2.10 ] .

For \(t \ge 0\), if \(\| e^{sL}\| \le 1\) and \(\| e^{sL'}\| \le 1\) for every \(s \in [0,t]\), then \(\| e^{tL'} - e^{tL}\| \le t\, \| L' - L\| \).

Let \(1\le k\) with \(k+1{\lt}D\). Then there is a map on \(M_{D}(\mathbb {C})\) that is \(k\)-positive but not \((k+1)\)-positive; the cone of \(k\)-positive maps strictly contains the cone of \((k+1)\)-positive maps. Every inclusion between consecutive amplification dimensions below the top one is therefore strict.

Let \(P\) satisfy the auxiliary BNT sector hypotheses, with distinct basis tensors \((B_j)_j\). Then the direct sum containing one copy of each basis tensor is a renormalization fixed point if and only if every \(B_j\) has the square-root fixed-point form and \(\mathcal{E}_{j,j'}=0\) for \(j\ne j'\). This is the multiplicity-one, phase-one specialization of [ CPGSV16 , Theorem 3.11 and Corollary 3.12 ] . It concerns the direct sum of the basis representatives, not the repeated-copy tensor carrying the weights \(\mu _{j,q}\).

If the basis direct sum is a renormalization fixed point, there are also invertible matrices \(X_j\), trace-one positive diagonal matrices \(\rho _j=\operatorname{diag}(\lambda _{j,\alpha })\), and tensors \(U_j\) such that \(B_j^i=X_j\sqrt{\rho _j}\, U_j^iX_j^{-1}\), and the family \((U_j)_j\) satisfies the joint isometry condition, including the cross-block equations for \(j\ne j'\).

If \(P\) is in BNT canonical form, then every basis block \(P_j\) is normal in the algebraic sense of Definition 2.4.4.

Let \(P\) and \(Q\) be BNT canonical forms whose total tensors generate the same MPV family at every positive length. Their total bond dimensions agree; writing the common value as \(D\), there is a unitary matrix \(U\in \mathrm{U}(D)\) such that, for every physical index \(i\),

This is the equal case of [ CPGSV16 , Corollary A.6, lines 1197–1199 ] .

Under the hypotheses of Theorem 11.3.1, there are a bijection \(\beta :J(Q)\simeq J(P)\), unitary matrices \(U_k\in \mathrm{U}(\dim Q_k)\), and phases \(\zeta _k\in \mathbb {C}\) with \(|\zeta _k|=1\), such that \(\dim P_{\beta (k)}=\dim Q_k\) and, for every \(k\) and every physical index \(i\),

This is the proportional part of [ CPGSV16 , Corollary A.6, lines 1197–1199 ] .

The Dyson–Phillips series \(\sum _n \widetilde{T}^{(n)}_t\) converges in operator norm for every \(t \ge 0\).

Every nonzero fixed point of an irreducible channel has nonzero trace.

Let \(T:M_{D}(\mathbb {C})\to M_{D'}(\mathbb {C})\) be a trace-preserving positive linear map. Then for all density matrices \(\rho _1,\rho _2\in M_{D}(\mathbb {C})\), \(\lVert T(\rho _1)-T(\rho _2)\rVert _{\operatorname{tr}} \leq \lVert \rho _1-\rho _2\rVert _{\operatorname{tr}}\). See [ Wol12 , Chapter 8, Eq. (8.80) ] .

One has \(\sum _{n=0}^{\infty }\widetilde{T}^{(n)}_t = e^{tL'}\) in tsum form.

Let \((A_x)_{x\in X}\) be a finite separated normal-canonical family whose blocks have positive virtual bond dimension. Then there is a positive integer \(L\) such that every blocked tensor \(A_x^{[L]}\) is one-site injective. The same conclusion holds simultaneously for two finite separated normal-canonical families whose blocks have positive virtual bond dimension.

Here the family is a basis of normal-canonical representatives in the sense of Definition 9.9.1.2; the blocks are distinct and the weight moduli are non-increasing but need not be strictly ordered. The representative family is assumed to have been selected; this statement does not reconstruct the full CPSV sector decomposition with repeated copies inside one sector. This scope restriction is recorded in docs/paper-gaps/ft_one_copy_scope_restriction.tex.

If \(D {\gt} 0\), the factor system \(\omega \) from Theorem 12.1.16 satisfies the 2-cocycle identity for all \(g,h,k \in G\):

Let \(A\) be an MPS tensor with bond dimension \(D\ge 1\), satisfying \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\), and suppose that \(A\) is primitive in the sense of Definition 8.1.1.3. If some Kraus operator \(A^{i_0}\) is invertible, then

Under the hypotheses of Theorem 8.4.2, one has \(\iota (A)\le D^2-\operatorname{kr}(A)+1\), where \(\iota (A)=\min \{ n\ge 1:S_n(A)=M_{D}(\mathbb {C})\} \).

Let \(A\) be an MPS tensor with bond dimension \(D\ge 1\), satisfying \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\), and suppose that \(A\) is primitive in the sense of Definition 8.1.1.3. If some Kraus operator \(A^{i_0}\) is non-invertible and has an eigenvector \(\varphi \neq 0\) with nonzero eigenvalue \(\mu \), then

Under the hypotheses of Theorem 8.4.6, one has \(\iota (A)\le D^2\).

The fixed-point set of a complex-linear map \(E:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) is \(\operatorname{Fix}(E)=\{ X\in M_{D}(\mathbb {C}):E(X)=X\} \).

For a complex-linear map \(E:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\), set

These are [ Wol12 , Equations (5.11)–(5.12) ] , with the source’s convention that the subscript records the side on which \(A\) multiplies the varying matrix.

A complex-linear map \(E:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) satisfies the Schwarz inequality if, for every \(A\in M_{D}(\mathbb {C})\),

This is [ Wol12 , Equation (5.2) ] .

For a Lindblad operator \(L_j\), the adjoint dissipator on observables is

The adjoint fixed-point set is \(\operatorname{Fix}(E^*)=\{ X\in M_{D}(\mathbb {C}):E^*(X)=X\} \), where \(E^*(X)=\sum _iK_i^\dagger XK_i\).

The adjoint generator of a Lindblad form is

The adjoint kernel is the set of observables \(A\) satisfying \(L^*(A)=0\).

The length-\(2\) blocked AKLT tensor groups two physical sites into one, with blocked physical dimension \(9 = 3^2\) and the same bond dimension \(D = 2\).

The length-\(2\) blocked on-site action of \(\mathbb {Z}_2 \times \mathbb {Z}_2\) on the \(9\)-dimensional blocked physical space is the Kronecker square \(g \longmapsto P_g \otimes P_g\) of the single-site spin-\(1\) \(\pi \)-rotation representation \(P_g\) that implements the symmetry of Theorem 17.2.4. Identifying the blocked physical index with the pair of single-site indices, the entry of \(P_g \otimes P_g\) at \((i_1 i_2, j_1 j_2)\) is the product \((P_g)_{i_1 j_1}(P_g)_{i_2 j_2}\) over the two grouped sites. This is the on-site action that makes the blocked twist agree with blocking the single-site twist.

The Cartesian presentation of the AKLT tensor has \(d = 3\), \(D = 2\), and \(A^x = \sigma _x\), \(A^y = \sigma _y\), \(A^z = \sigma _z\), the three Pauli matrices. This is the rotationally natural form used in [ CPGSV21 ] (around line 1159); it is the AKLT state in the Cartesian basis of the spin-\(1\) site, the counterpart of the \(|m\rangle \)-basis form of Definition 17.2.1.

The AKLT factor system on \(\mathbb {Z}_2 \times \mathbb {Z}_2\) is \(\omega (g,h) = (-1)^{(g_1 + g_2)\, h_1}\) (the value \(-1\) when \(g_1 + g_2 = 1\) and \(h_1 = 1\), and \(1\) otherwise). It is the factor system of the explicit projective representation with virtual gauges \(i\sigma _y\) and \(\sigma _z\); the diagonal term \(g_1 h_1\) relative to the cluster case reflects \((i\sigma _y)^2 = -\mathbb {1}\).

Set \(d = 3\) (spin-1) and \(D = 2\). The AKLT tensor is the family \(\{ A^0, A^1, A^2\} \subset M_{2}(\mathbb {C})\) defined via the Pauli matrices by

Here \(\sigma _z = |0\rangle \! \langle 0| - |1\rangle \! \langle 1|\), so equivalently \(A^1 = \frac{1}{\sqrt{3}} \sigma _+\) and \(A^2 = -\frac{1}{\sqrt{3}} \sigma _-\) for \(\sigma _+ = \sqrt{2} |0\rangle \! \langle 1|\) and \(\sigma _- = \sqrt{2} |1\rangle \! \langle 0|\). Equivalently, the matrices are the Clebsch–Gordan coefficients coupling spin-\(1\) and spin-\(\tfrac 12\) to spin-\(\tfrac 12\).

For an MPS tensor \(A\), let \(\operatorname{alg}(A)\) be the unital \(\mathbb {C}\)-subalgebra of \(M_{D}(\mathbb {C})\) generated by the matrices \(\{ A^i\} _{i=0}^{d-1}\): \(\operatorname{alg}(A)=\mathbb {C}\langle A^i:i=0,\ldots ,d-1\rangle \subseteq M_{D}(\mathbb {C})\).

For an Appendix B structural datum and a periodic chain of length \(N{\gt}2\), set \(p_i=h_i^{(N)}\) for \(i\in \mathbb Z/N\mathbb Z\), where \(h_i^{(N)}\) is the translated length-two parent interaction. These maps are idempotent and commute pairwise: adjacent commutation follows from Theorem 14.1.60, while disjoint length-two interactions commute by locality.

Suppose that \(A\) is equipped with Appendix B structural data \((X,\Lambda ,U)\): the matrix \(X\) is invertible, the diagonal entries of \(\Lambda \) are strictly positive, \(A^i=X\Lambda U^iX^{-1}\), and \(U\) satisfies the pair-index isometry equation below. Let \(\mathcal H_a\) and \(\mathcal H_b\) have bases indexed by \(\alpha ,\beta \in \{ 0,\ldots ,D-1\} \). The coefficients \(U^i_{\alpha ,\beta }\) define maps

and

For every \(N\geq 0\), the coefficients of \(U\) define

with

The conjugate coefficient map is

This is the tensor power occurring in the basic-vector expression [ CPGSV16 , Equation (3.17) ] .

The \(AX\) coefficient-space representative is, before the \(AX\)-lift, the canonical two-site parent interaction of the core tensor, \(\widehat Q_{AX}=q_2(A_{\mathrm{core}})\). The source projector \(Q_{AX}\) of Definition D.2 acts on \(\mathcal H_A\otimes \mathcal H_X\). The displayed operator is the two-site coefficient representative that will be placed on the \(AX\) face; it is not by itself the construction of that source projector.

The \(XB\) coefficient-space representative is, before the \(XB\)-lift, the same coefficient-space representative, \(\widehat Q_{XB}=\widehat Q_{AX}\). The source projector \(Q_{XB}\) of Definition D.2 acts on \(\mathcal H_X\otimes \mathcal H_B\). The displayed operator is the coefficient representative later placed on the \(XB\) face.

For Appendix B structural data, the two-site basic support is \(G_2(A_{\mathrm{core}})=G_2(\Lambda U)\).

Let \(P_2\) be the orthogonal projector onto \(G_2(\Lambda U)\), as in the local parent-space construction of [ CPGSV16 , Definition 3.9 ] .

Put \(|\varphi \rangle =\sum _b\lambda _b|b,b\rangle \). For outer-boundary coefficients \(v_{a,c}\), insertion of this vector on the middle virtual bond is the map \(I_\varphi \) defined by

The two-site physical map is \(U^{\otimes 2}I_\varphi \). A boundary matrix \(Y\) determines outer-boundary coefficients by \((v_Y)_{a,c}=\lambda _aY_{c,a}\). Opening the outer virtual bond in the two-site contraction of [ CPGSV16 , Equations (3.17)–(3.18) ] gives these coefficients.

Write \(s=\langle \varphi ,\varphi \rangle =\sum _b\lambda _b^2\). The orthogonal projector onto \(\mathbb {C}\varphi \) has pair-index coefficients

On three virtual site pairs, let \(P_{01}\) place this projector on the bond joining the right index of site zero to the left index of site one, and let \(P_{12}\) place it on the next bond.

Let \(a\to a\) be a positive loop and choose a nonzero vector \(\varphi _a\in \operatorname{ran}P_{a,a}\). In sector coordinates define the matrix product tensor \((C_a^{(r,l)})_{x,y}=\delta _{x,l}\varphi _a(r,y)\) and set it to zero outside sector \(a\). The physical loop tensor is the image of \(C_a\) under the one-site unitary of the spatial decomposition. Its associated cyclic product pairs the right factor at site \(n\) with the left factor at site \(n+1\). The corresponding \(N\)-site product state is denoted by \(\Psi _a^{(N)}\in (\mathbb {C}^d)^{\otimes N}\).

Let \(\Phi _a=(\varphi _a(r,l))_{r,l}\) and write \(r_a=\operatorname{rank}\Phi _a\). Choose a rank factorization

The tensor on the minimal Schmidt support is

again extended by zero outside sector \(a\). Its physical tensor is its image under the same one-site unitary.

Let \(\nu _a=\lVert \varphi _a\rVert _2{\gt}0\), where the norm is the Hilbert-space norm of the bond vector, or equivalently the Frobenius norm of \(\Phi _a\). In either coordinate system, the normalized representative is \(\widehat C_a=\nu _a^{-1}\widetilde C_a\). The raw coordinate and physical tensors are retained separately.

Suppose that a one-site unitary identifies the physical Hilbert space with a finite direct sum

and that in these coordinates the complementary two-site interaction is \(\bigoplus _{a,b}\mathbb {1}_{a,l}\otimes P_{a,b}\otimes \mathbb {1}_{b,r}\), where each \(P_{a,b}\) is an orthogonal projector. The directed edge \(a\to b\) is present precisely when \(\operatorname{ran}P_{a,b}\) is nonzero. An ordered cycle of length \(N\) is a sequence \((a_0,\ldots ,a_{N-1})\) for which every cyclically adjacent edge \(a_n\to a_{n+1}\) is present. Cyclic rotations are not identified. This is the graph following equation (2) of [ Bei12 , Section III ] .

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.

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.

For \(r,s\), let \(P_{\mathrm{top}}=C_{r,s}^\dagger C_{r,s}\) and \(P_{\mathrm{bot}}=\mathbb {1}-P_{\mathrm{top}}\). Both are PSD (for \(P_{\mathrm{bot}}\), by Lemma 18.6.4) and \(P_{\mathrm{top}}+P_{\mathrm{bot}}=\mathbb {1}\).

A matrix on \(\bigoplus _k \mathbb {C}^{D_k}\) is block diagonal if it is a direct sum of square matrices, one on each summand.

Given \(r\) blocks with bond dimensions \((D_k)_{k=1}^r\), per-block tensors \(\{ A_k^i\} \), and scaling factors \((\mu _k)_{k=1}^r\), the associated block-diagonal tensor is the tensor of total bond dimension \(D = \sum _{k=1}^r D_k\) defined by

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 a block of \(L\) physical sites, the induced action is the Kronecker-power representation

For a finite direct sum \(\bigoplus _k \mathbb {C}^{D_k}\), the block projection \(P_j\) is the matrix that is the identity on the \(j\)-th summand and zero on all other summands.

For natural numbers \(r,s\), let \(C=C_{r,s}\in \mathbb {C}^{r\times (r+s)}\) be the rectangular \(r\times (r+s)\) matrix whose rows are the first \(r\) rows of the identity on \(\mathbb {C}^{r+s}\): \(C_{ij}=1\) if \(j=i{\lt}r\) and \(0\) otherwise.

Given a tensor \(A\) and a blocking length \(L\), the \(L\)-blocked tensor is the tensor with physical index set \(\{ 0,\ldots ,d{-}1\} ^L\) (hence physical dimension \(d^L\)) and the same bond dimension \(D\), whose matrices are indexed by words \((i_1, \ldots , i_L) \in \{ 0,\ldots ,d{-}1\} ^L\):

Blocking coarse-grains \(L\) neighbouring sites into one tensor: the inner virtual bonds are contracted, the \(L\) physical legs merge into a single composite index, and the two outer bonds remain as the bond of \(A^{[L]}\).

A finite block family has no gauge-phase-equivalent distinct blocks when, for any two distinct indices with the same bond dimension, the corresponding blocks are not gauge-phase equivalent.

For a linear map \(E : M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\), its \(k\)-fold ampliation acts on block matrices by

For a fixed blocked length \(m\), the fixed-length form of the block-injective canonical-form condition is

Equivalently,

This is the fixed-length form of the block-injective canonical-form condition of [ CPGSV21 , lines 2113–2117 ] .

A basis of normal tensors (BNT) for a total tensor \(A_{\mathrm{tot}}\) consists of \(g\) block tensors \((A_1, \ldots , A_g)\) with bond dimensions \((D_1, \ldots , D_g)\) such that each \(A_j\) is normal (Definition 2.4.4); at each system size \(N\geq 1\) the MPV of \(A_{\mathrm{tot}}\) lies in the span of the block MPVs \(\{ |V^{(N)}(A_j)\rangle \} _{j=1}^g\); and for sufficiently large \(N\) those block MPVs are linearly independent. Here normality means eventual block injectivity, as in Definition 2.4.4. Thus this is the algebraic form of [ CPGSV21 , Definition IV.2 ] , rather than its primitive-channel formulation. In the canonical-form characterization of [ CPGSV16 , Section 2.3 ] , a BNT is a minimal family of representatives for \(\widetilde A_k^i=e^{i\phi _k}X_kA_j^iX_k^{-1}\) among the normal blocks.

The coefficient form of a basis of normal tensors: each block \(A_j\) is a CPSV normal tensor (Definition 9.6.1), the MPV of \(A\) is given at every positive length by explicit coefficients \(c_j^{(N)} \in \mathbb {C}\) with \(|V^{(N)}(A)\rangle = \sum _{j=1}^{g} c_j^{(N)} |V^{(N)}(A_j)\rangle \), and the block MPVs \(\{ |V^{(N)}(A_j)\rangle \} \) are eventually linearly independent. This is the literal coefficient-form definition of [ CPGSV16 , Section 2.3 ] . It differs from Definition 10.1.1 only in the chosen normality predicate. Theorem 10.2.3 proves the implication between the two forms.

For a basis of normal tensors \(A_1,\ldots ,A_g\), the vector space generated by their length-\(N\) matrix product vectors is \(\operatorname{span}\{ V^{(N)}(A_j):j=1,\ldots ,g\} \).

For a matrix \(U\) on \(M_{D}(\mathbb {C})\), the Breuer–Hall map on \(M_{D}(\mathbb {C})\) is

Here, a block-injective canonical form for a tensor consists of:

1. a number of blocks \(r \ge 1\),

2. bond dimensions \(D_1, \ldots , D_r\),

3. injective tensors \(\{ A_k^i\} _{i=0}^{d-1}\) with \(A_k^i \in M_{D_k}(\mathbb {C})\) for each block \(k \in \{ 1,\ldots ,r\} \),

4. scaling factors \(\mu _1, \ldots , \mu _r \in \mathbb {C}\).

The associated tensor is the block-diagonal tensor

This block-injective formulation of the canonical form is used in [ PGVWC07 ] ; see also  [ CPGSV21 ] for the normal canonical form.

The Cesàro mean of a linear map \(E\) at order \(N\), applied to a starting matrix \(\rho _0\), is

For \(N{\gt}0\) and \(L\le N\), define the periodic parent-Hamiltonian ground space

Here the condition \(\psi |_{[i,i+L-1]}\in G_L(A)\) means that, for every choice of physical indices outside the window, the restriction of \(\psi \) to that window lies in \(G_L(A)\). The one-index notation \(G_L(A)\) denotes the local ground space \(\mathcal G_L\) of [ CPGSV21 , Section IV.C ] . The two-index notation \(\mathcal G_{N,L}(A)\) used here denotes the periodic intersection of those local constraints over all translated length-\(L\) windows on the \(N\)-site ring. When \(N=0\) or \(L{\gt}N\), set \(\mathcal G_{N,L}(A):=\top \) by convention.

The Hilbert-space cyclic constraint space is the transport of \(G_L^{(N)}(A)\) by the canonical \(\ell ^2\) identification: \(G_{L,\mathrm{ES}}^{(N)}(A)=e_N^{-1}G_L^{(N)}(A)\).

A linear map \(E : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) is a quantum channel if it is both completely positive and trace-preserving (CPTP), following  [ Wol12 ] .

For a linear map \(T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\), the channel determinant \(\det T\) is the determinant of the matrix of \(T\) with respect to the standard matrix-unit basis of \(M_{D}(\mathbb {C})\). This is the quantity studied in [ Wol12 , Section 6.1 ] .

The period of a channel \(E\) is the number of peripheral eigenvalues, counted without multiplicity: \(p(E):=|\mathrm{peripheral}(E)|\).

The Choi matrix of a linear map \(T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) is

Concretely, \(\tau _{(i_1,i_2),(j_1,j_2)} = \frac{1}{D}\, (T(E_{i_2 j_2}))_{i_1 j_1}\) where \(E_{i_2 j_2}\) is the matrix unit. This is the Choi–Jamiol\- kowski convention of [ Wol12 , Proposition 2.1 ] .

For \(X\in M_{D,k}(\mathbb {C})(\mathbb {C})\) and a Choi matrix \(\tau \), the right-factor compression in the blockwise ampliation index convention has entries

The associated vector has coefficients \(D^{-1/2}X_{i,p}\).

For \(X\in M_{D,k}(\mathbb {C})(\mathbb {C})\), let \(R_X\) be the matrix from \(\mathbb {C}^D\otimes \mathbb {C}^D\) to \(\mathbb {C}^D\otimes \mathbb {C}^k\) with entries

On the cyclic index set \(\mathbb {Z}/d\mathbb {Z}\), the Choi-type map is

where \(D\) keeps the diagonal of a matrix and \(U_{k0}\) is the cyclic shift by \(k\).

For a vector \(v\) on \(\mathbb {Z}/d\mathbb {Z}\), set

The cluster-state factor system on \(\mathbb {Z}_2 \times \mathbb {Z}_2\) is \(\omega (g,h) = (-1)^{g_2\, h_1}\) (the value \(-1\) when \(g_2 = 1\) and \(h_1 = 1\), and \(1\) otherwise). It is the factor system of the explicit projective representation with virtual gauges \(\sigma _z\) and \(\sigma _x\), both squaring to the identity, so there is no diagonal term.

Set \(d = D = 2\). The cluster-state tensor is the family \(\{ A^0, A^1\} \subset M_{2}(\mathbb {C})\) defined by

where \(|\pm \rangle = \tfrac {1}{\sqrt{2}}(|0\rangle \pm |1\rangle )\). The review  [ CPGSV21 ] writes the reflected convention \(A^0 = |0\rangle \! \langle +|\), \(A^1 = |1\rangle \! \langle -|\); the transpose taken here is the same state read in the opposite direction and carries the same SPT order.

Two cocycles \(\omega _1,\omega _2\) are cohomologous (written \(\omega _1 \sim \omega _2\)) if there exists \(\varphi \colon G \to \mathbb {C}^{\times }\) such that, for all \(g,h \in G\),

The commutator phase of a \(2\)-cochain \(\omega \) at a pair \(g,h\) is \(\beta _\omega (g,h):=\omega (g,h) \omega (h,g)^{-1}\). It is defined for any \(\mathbb {C}^{\times }\)-valued \(2\)-cochain; the cocycle condition is not needed.

The commutant of a Lindblad form is the set of matrices commuting with \(H\), with every Lindblad operator \(L_j\), and with every adjoint \(L_j^\dagger \).

The commutator form of the Lindblad equation writes the generator as

where \([A,B] = AB - BA\). This is [ Wol12 , Equation (7.22) ] .

The local physical intertwining relation is \(\sum _j u_{ij}\, A^j = V\, A^i\, V^\dagger \). Locally, the physical action of \(u\) can therefore be pushed to the virtual legs:

= .

Transfer-map covariance is the identity that the transfer map commutes with virtual conjugation:

In doubled-layer notation, the virtual operators slide through the local transfer cell:

= .

In the transfer-map language, the commutation of the doubled transfer matrix \(E = \sum _i A_i \otimes \bar A_i\) with \(V \otimes \bar V\) is

The connected two-point function is

For \(a\in \{ 0,1\} \) and \(N\ge 0\),

A dynamical semigroup \(T\) is norm-continuous if the map \(t \mapsto T_t\) is continuous in the operator norm topology. In finite dimension this coincides with strong continuity.

The restriction of \(E\) to an invariant corner \(PM_{D}(\mathbb {C})P\).

For an idempotent \(P\in M_{D}(\mathbb {C})\), the subspace \(\{ X\mid PXP=X\} \) and the corner \(\{ PXP\mid X\in M_{D}(\mathbb {C})\} \) are identified as \(\mathbb {C}\)-linear spaces by the identity on underlying matrices.

For every \(\lambda _2\in \mathbb {C}\), define the associated correlation-length quantity by

For a nonzero subleading eigenvalue in the physical range \(0{\lt}|\lambda _2|{\lt}1\), this quantity is positive. If all subleading spectral values vanish and correlations therefore vanish after finitely many transfer steps, the total definition gives the limiting value \(\xi =0\).

For linear maps \(S,T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\), write \(T\le S\) (in the CP order) when \(S-T\) is completely positive.

A linear map \(E : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) is completely positive (CP) if it admits a Kraus representation: there exist operators \(\{ K_i\} _{i=0}^{r-1}\) with \(K_i \in M_{D}(\mathbb {C})\) such that, for every \(X \in M_{D}(\mathbb {C})\),

The Kraus representation also gives entrywise positivity on every positive block matrix by Theorem 4.2.1, and hence the associated completely positive map between matrix \(C^*\)-algebras in Theorem 4.2.2.

Let \(A\) have bond dimension \(D\). It is in the canonical form of [ CPGSV16 , Section 2.3 ] if there are positive integers \(D_1,\ldots ,D_r\), normal tensors \(A_k\) of bond dimension \(D_k\), weights \(\mu _k\in \mathbb {C}\), and, writing \(S=\sum _kD_k\), a matrix \(U\in \mathbb {C}^{S\times D}\) such that \(S\le D\) and, for every physical index \(i\),

Thus the complement of the retained direct sum consists literally of zero bond-space coordinates. In particular, \(r=0\) is possible only when \(A=0\).

The convention in line 246 of [ CPGSV16 ] is a separate normalization of these data, not part of canonical-form membership:

No weight is required to be nonzero, and no ordering or separation condition is imposed.

The basis-of-normal-tensors refinement is Definition 10.1.1; it is not part of canonical form itself.

In canonical form II of [ CPGSV16 , Appendix A ] , there are positive retained dimensions \(D_k\), normal tensors \(A_k\), weights \(\mu _k\), and, for \(S=\sum _kD_k\le D\), a matrix \(U\in \mathbb {C}^{S\times D}\) such that, for every physical index \(i\),

Thus omitted bond-space coordinates are literally zero. Every retained block additionally satisfies

where \(\Lambda _k\) is diagonal and positive definite. The two displayed equations are the only additional blockwise conditions; no ordering, injectivity, gauges, or basis-of-normal-tensors assumptions are included. The line-246 convention (??) remains a separate condition on the underlying canonical-form data.

The cumulative span at level \(n\) is \(T_n(A)=S_0(A)+S_1(A)+\cdots +S_n(A)\).

Given an MPS tensor \(A\) and a vector \(\varphi \in \mathbb {C}^D\), the cumulative vector span is

This is  [ SPGWC10 , proof of Lemma 2(a) ] .

Let \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\). A block-permutation structure for \(T\) consists of a finite index set \(\iota \), a family of orthogonal projections \(P_k\in M_{D}(\mathbb {C})\) for \(k\in \iota \), and a permutation \(\sigma \in \mathrm{Sym}(\iota )\), not necessarily a single cycle, such that, for every \(k\in \iota \) and \(X\in M_{D}(\mathbb {C})\),

In general, \(\sigma \) may have multiple disjoint cycles.

For a length-\(L\) cyclic window on \(\{ 0,\ldots ,N{-}1\} \) starting at \(i\), its support is \(S_i=\{ i,i+1,\ldots ,i+L-1\} \pmod N\), with repeated visits counted once.

Two length-\(L\) cyclic windows overlap, written \(i\sim _L j\), when their supports meet: \(S_i\cap S_j\ne \varnothing \).

A map \(\Phi \colon M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) is completely copositive if \(\Phi \circ \theta \) is completely positive, where \(\theta \) is transposition. It is decomposable if \(\Phi =\Phi _{\mathrm{cp}}+\Phi _{\mathrm{ccp}}\) with \(\Phi _{\mathrm{cp}}\) completely positive and \(\Phi _{\mathrm{ccp}}\) completely copositive. It is indecomposable if it is positive and not decomposable.

The set of density matrices in \(M_{D}(\mathbb {C})\) is

Let

and let \(\mathcal H_A=\bigoplus _{k\in I}\mathbb {C}^{n_k}\) and \(\mathcal H_B=\bigoplus _{l\in J}\mathbb {C}^{m_l}\). Write \(\iota _A,\iota _B\) for the block-diagonal embeddings and \(\pi _A,\pi _B\) for diagonal-block compression. For a linear map \(T:\mathcal A\to \mathcal B\), its canonical full-matrix extension is

For \(X\in \mathcal A\) and \(Y\in \mathcal B\), the trace adjoint \(T^*:\mathcal B\to \mathcal A\) is defined by

This is the block-diagonal realization needed for the fixed-point and classification arguments in [ CPGSV16 , Appendix C.4, lines 1980–2003 ] . It is only an auxiliary construction: it does not assert the classification conclusion of that passage.

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 ] .

A linear map \(T : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) is doubly-stochastic if \(T(\mathbb {1})\propto \mathbb {1}\) and the reduced density matrix \(\operatorname{tr}_{1}[\tau ]\) of its Choi matrix \(\tau =(T\otimes \operatorname{id})(|\Omega \rangle \! \langle \Omega |)\) is proportional to the identity. This is the normal form in [ Wol12 , Proposition 2.9 ] .

A family of linear maps \(T : \mathbb {R}\to (M_{D}(\mathbb {C}) \to _{\ell } M_{D}(\mathbb {C}))\) is a dynamical semigroup if

1. (semigroup law) \(T_{t+s} = T_t \circ T_s\) for all \(t,s \ge 0\); and

2. (initial condition) \(T_0 = \operatorname{id}\).

This is [ Wol12 , Equation (7.1) ] .

Let \(d_A{\gt}0\) and let \(\rho _{BC}\) be positive semidefinite. Define

Under the canonical reassociation from \(A\times (B\times C)\) to \((A\times B)\times C\), the right partial trace removes \(C\).

This formulation has the same value \(I(A{:}B) = S(\rho _A) + S(\rho _B) - S(\rho _{AB})\) as in Definition 21.5.1.

Let \(\rho _A\) and \(\rho _{BC}\) be positive semidefinite, and define

We use the canonical reassociation from \(A\times (B\times C)\) to \((A\times B)\times C\), so that the right partial trace removes \(C\).

This is the entropy formulation of the Hayashi Markov decomposition from Definition 21.4.39.

If \(P_A\) is the support projection of \(\rho _A\), define

This formulation has the same value \(S(\rho ) = -\sum _i \lambda _i \log \lambda _i\) as in Definition 21.2.1.

Given a word \(w = (i_1, \ldots , i_L) \in \{ 0,\ldots ,d{-}1\} ^L\), the word evaluation is the matrix product

The empty word evaluates to the identity, \(A^\varnothing = \mathbb {1}_D\). In tensor-network notation,

in which each black node denotes the same local tensor \(A\), the virtual legs remain open, and the physical legs are labelled by the word \((i_1,\ldots ,i_L)\).

The Kraus family has eventually full Kraus rank if there is a positive integer \(i\) such that \(S_i(A)=M_{D}(\mathbb {C})\).

Two tensors \(A\) and \(B\) are eventually nonzero proportional if, for all sufficiently large \(N\), there is a nonzero scalar \(c_N \in \mathbb {C}\) such that \(|V^{(N)}(A)\rangle = c_N |V^{(N)}(B)\rangle \). This auxiliary form is used with the eventual linear-independence statement Lemma F.1.10.

Given a generator \(L \in \operatorname{End}_{\mathbb {C}}(M_{D}(\mathbb {C}))\), the exponential semigroup is

For \(\sigma \in \{ 0,\ldots ,d{-}1\} ^{N}\) and a starting site \(i\), write \(\sigma |_{[i,i+L-1]}\in \{ 0,\ldots ,d{-}1\} ^{L}\) for the word defined by \((\sigma |_{[i,i+L-1]})(r)=\sigma (i+r\bmod N)\) for \(0\le r{\lt}L\).

In \(\mathbb {C}^2 \otimes \mathbb {C}^2\), let \(v = |01\rangle - |10\rangle \). Equivalently, \(v_{(0,1)}=1\), \(v_{(1,0)}=-1\), and all other coefficients are zero.

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 \(X\) 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 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

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.

For \(r\geq 1\), let \(W_0:\mathbb {C}^D\to \mathbb {C}^D\otimes \mathbb {C}^r\) be the isometry \(W_0x=x\otimes e_0\).

Let \(Y\in M_{d}(\mathbb {C})\), and write physical indices of an \((L+1)\)-site block as \(I=(i_0,\ldots ,i_L)\) and \(J=(j_0,\ldots ,j_L)\). The induced matrix on the blocked physical space is

A Fitting decomposition of a linear endomorphism \(f:V\to V\) on a finite-dimensional vector space over an algebraically closed field consists of:

1. \(f\) is nilpotent on the generalized \(0\)-eigenspace,

2. \(f\) is invertible on each generalized \(\mu \)-eigenspace for \(\mu \neq 0\),

3. the generalized eigenspaces span \(V\),

4. the generalized eigenspaces are linearly independent.

Given a linear map \(E:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) and a fixed point \(\rho \) with \(E(\rho )=\rho \) and \(\operatorname{tr}(\rho )\neq 0\), the fixed-point projection is the rank-one map

This projects onto the span of \(\rho \) along the kernel of the trace functional. When the fixed point is unique up to scaling, this span is the full fixed-point space.

The flip operator \(F\) on \(\mathbb {C}^d \otimes \mathbb {C}^d\) is

Its matrix entries are

For a possibly rectangular matrix \(X\in M_{m\times n}(\mathbb {C})\), its Frobenius norm squared is

The matched flattened data are the functions \(\widetilde\mu \), \(\widetilde P\), and \((k,q)\mapsto X_k\) defined above on the flattened copy coordinates of \(Q\).

For a permutation \(\rho \) of a finite bond-index set, let \(\Pi _\rho \) be its permutation matrix. The corresponding element of the general linear group is denoted \(G_\rho \).

The sector bijection, copy bijections, and bond-dimension equalities induce an equivalence of the dependent triples \((j,q,a)\) and \((k,q',a')\). After flattening the dependent bond coordinate, this gives an equivalence of the standard total bond-index sets.

The multiplicative domain of \(E\) is \(\mathcal{A}(E):=\mathcal{A}_R(E)\cap \mathcal{A}_L(E)\).

Two tensors \(A\) and \(B\) of the same bond dimension \(D\) are gauge equivalent if there exists an invertible matrix \(X \in \mathrm{GL}_D(\mathbb {C})\) such that, for every \(i \in \{ 0,\ldots ,d{-}1\} \),

In tensor-network notation, (??) is represented by

where the black node denotes the tensor \(A\), the upper leg is the physical index \(i\), and the two red side nodes denote the gauge matrices acting on the virtual legs.

Two tensors \(A\) and \(B\) are gauge-phase equivalent if there exist \(X \in \mathrm{GL}_D(\mathbb {C})\) and \(\zeta \in \mathbb {C}\setminus \{ 0\} \) such that, for every physical index \(i\),

A generator decomposition consists of a completely positive map \(\phi : M_{d}(\mathbb {C}) \to M_{d}(\mathbb {C})\) and a matrix \(\kappa \in M_{d}(\mathbb {C})\), defining the linear map

This is [ Wol12 , Equation (7.14) ] .

For a projection \(P\), the generator preserves the compression \(PM_{D}(\mathbb {C})P\) if, for every \(X\in M_{D}(\mathbb {C})\),

Set \(d = D = 2\). The GHZ tensor is the family \(\{ A^0, A^1\} \subset M_{2}(\mathbb {C})\) defined by

For a system of \(N\) sites, the resulting MPV is the GHZ state

The virtual \(\mathbb {Z}_2\) representation on the bond space is the homomorphism \(\mathbb {Z}_2 \to M_{2}(\mathbb {C})\) sending the identity to \(I\) and the generator to \(\sigma _z = \operatorname{diag}(1, -1)\).

Given per-block gauges \(X_k \in \mathrm{GL}_{D_k}(\mathbb {C})\), form the block-diagonal element \(\bigoplus _k X_k\) and then reindex the dependent block-coordinate space with the standard bond index \(\{ 0,\ldots ,\sum _k D_k-1\} \) of the assembled tensor. The result is an element of \(\mathrm{GL}_{\sum _k D_k}(\mathbb {C})\).

Let \(\iota _L\colon (\{ 0,\ldots ,d{-}1\} ^{L}\to \mathbb {C})\simeq \ell ^2(\{ 0,\ldots ,d{-}1\} ^{L})\) be the canonical computational-basis identification. The local ground space under this \(\ell ^2\) identification is \(G_L^{\mathrm{ES}}(A):=\iota _L(G_L(A)) \subseteq \ell ^2(\{ 0,\ldots ,d{-}1\} ^{L})\).

The map \(\Gamma _L\) in (??) is a \(\mathbb {C}\)-linear map from \(M_{D}(\mathbb {C})\) to \((\{ 0,\ldots ,d{-}1\} ^{L}\to \mathbb {C})\). In tensor-network notation,

The chain denotes the word tensor \(A^\sigma \) with open physical indices \(\sigma _1,\ldots ,\sigma _L\), and the red capsule denotes the boundary matrix \(X\) inserted on the closing virtual bond before taking the trace. The brace records the chain length \(L\).

The local ground space is the image \(G_L(A):=\operatorname{range}(\Gamma _L)\subseteq (\{ 0,\ldots ,d{-}1\} ^{L}\to \mathbb {C})\).

The second cohomology set is the quotient

where \(\sim \) is the coboundary-equivalence relation.

Condition (4): there exists a nontrivial projector \(P\) and a Lindblad form \((H,\{ L_j\} )\) for \(L\) such that \((\mathbb {1}-P)L_jP=0\) and \((\mathbb {1}-P)\kappa P=0\) for all \(j\), where \(\kappa =iH+\frac{1}{2}\sum _jL_j^\dagger L_j\).

Let \(\mathbb K\) denote either \(\mathbb {R}\) or \(\mathbb {C}\), and let \(V\) be a normed vector space over \(\mathbb K\). A linear endomorphism \(f:V\to V\) has bounded orbits if, for every \(x\in V\), the set \(\{ f^n x:n\in \mathbb {N}\} \) is bounded.

The structure used below records only the idempotent-product consequence of the parent commuting Hamiltonian condition of [ CPGSV16 , Appendix D.2 ] . It consists of commuting idempotent endomorphisms \(P_{AX}\) and \(P_{XB}\) such that, for the idempotent \(P_K\) associated to \(K\), \(P_{AX}\circ P_{XB}=P_K\). In the source, this is the equation obtained from local projectors \(Q_{AX}\) and \(Q_{XB}\) satisfying \([Q_{AX},Q_{XB}]=0\) and \(K_{AXB}=(K_{AX}\otimes H_B)\cap (H_A\otimes K_{XB})\).

A generator \(L\) has a faithful stationary state if there exists a positive definite density matrix \(\rho _0\) with \(L(\rho _0)=0\).

An MPS tensor \(A\) has an invariant projection if there exists an orthogonal projection \(P\) with \(P\neq 0\), \(P\neq \mathbb {1}\), and \((\mathbb {1}-P)A^iP=0\) for all \(i\). A tensor is irreducible if it has no nontrivial invariant projection (Definition 9.1.1.1).

Condition (3): there exists a nontrivial orthogonal projector \(P\) such that \(e^{tL}(PM_{D}(\mathbb {C})P)\subseteq PM_{D}(\mathbb {C})P\) for all \(t\ge 0\).

The commutant of the Lindblad family \(\{ L_j\} '\) equals \(\mathbb {C}\cdot \mathbb {1}\), i.e. the Lindblad operators act irreducibly on \(M_{D}(\mathbb {C})\).

For a fixed chain length \(N\), this condition is the conjunction of the nearest-neighbor commutation equation, the zero-energy equation for the MPS vector, and the ground-space spanning equation:

The source statement in [ CPGSV16 , Theorem 3.10(iii) ] quantifies this condition over all \(N{\gt}2\).

This is the source quantification in [ CPGSV16 , Theorem 3.10(iii) ] . For every \(N{\gt}2\), the fixed-chain condition of Definition 14.1.16 holds:

This condition records the ground-space spanning equation, but does not prove it for a concrete canonical-form tensor.

A tensor has physical string order in the sense of [ PGWS\(^{+}\)08 , display SOP, lines 112–122 ] if there exist a unitary \(u\ne \mathbb {1}\) and physical endpoint operators \(x,y\) for which Definition 12.3.5 holds. The comparison with arbitrary virtual boundary matrices is a separate endpoint- realizability statement.

For fixed physical operators \(x,y,u\), the correlator has physical string order if there is \(s{\gt}0\) such that \(|S_N(x,y,u)|\to s\).

Condition (1): there exists a density matrix \(\rho _0\) with nontrivial kernel such that \(e^{tL}(\rho _0)=\rho _0\) for all \(t\ge 0\).

Condition (2): there exists a density matrix \(\rho _0\) with nontrivial kernel satisfying \(L(\rho _0)=0\).

A bipartite matrix \(\rho \) on \(M_{d}(\mathbb {C})\otimes M_{d'}(\mathbb {C})\) has Schmidt number at most \(n\) when it is a finite sum of pure-state projectors of Schmidt rank at most \(n\),

A completely positive map \(E\) on \(M_{D}(\mathbb {C})\) has the restricted CP spectral properties used here if there exist a positive real number \(r\), a positive definite right eigenvector \(\rho \), and a positive definite left eigenvector \(\sigma \) for the adjoint map such that

every positive semidefinite right eigenvector for eigenvalue \(r\) is a scalar multiple of \(\rho \), and the spectral radius of \(E\) is equal to \(r\). The uniqueness clause concerns only positive semidefinite Perron eigenvectors. Unlike the full nondegenerate-eigenspace statement in [ Wol12 , Theorem 6.4 ] , it does not assert that every complex eigenvector at \(r\) is proportional to \(\rho \).

For boundary matrices \(X,Y\in M_{D}(\mathbb {C})\), set explicitly

The virtual-boundary nondecay predicate holds for \(u\) and \(\Lambda \) if there exist \(X,Y\) and a positive constant \(c{\gt}0\) such that \(c\le \| R_L(u;X,Y)\| \) for every \(L\). Thus some virtual-boundary sequence does not decay to zero.

Virtual-boundary nondecay is an existence statement over the virtual matrices \(X,Y\), for a fixed twist \(u\); no claim is made for any particular boundary choice. Definition 12.3.6 records the physical-endpoint string-order notion from the source, which quantifies existentially over \(u,x,y\) and requires a positive limit rather than a lower bound uniform in the length. That paper also notes that the correlator for a particular endpoint choice can vanish even when the physical symmetry is present.

A Lindblad form has traceless Kraus operators if \(\operatorname{tr}(L_j) = 0\) for every \(j\).

A finite-dimensional ground space \(S\) is one-dimensional if \(\dim S=1\).

A Hayashi Markov decomposition of a tripartite state \(\rho _{ABC}\) consists of a finite direct-sum decomposition

together with a unitary change of basis on \(B\), a probability vector \((p_j)_j\), and density matrices \(\rho _{A B_j^L}\) and \(\rho _{B_j^R C}\) such that, in the adapted basis, the state becomes

The terminology follows Hayashi’s presentation of quantum Markov structure  [ Hay06 ] ; the block decomposition used by the MPDO argument is the structure theorem of [ HJPW04 ] .

Let \(A\) be Hermitian, with spectral decomposition \(A=U\operatorname{diag}(\lambda _i)U^\dagger \). Its support projection is

An MPO tensor is in normalized BNT-refined horizontal form if its doubled-index MPS tensor is a BNT sector decomposition with an independent gauge on every repeated copy:

Here the \(A_j\) are the distinct minimal representatives, \(r_j{\gt}0\), and every weight \(\mu _{j,q}\) is nonzero, \(|\mu _{j,q}|\leq 1\), and at least one weight has modulus one. The BNT hypotheses include irreducibility, left-canonicality, normalized self-overlap, eventual linear independence of the representative MPV families, and the exclusion of similarity up to phase between distinct representatives. Repeated gauge-equivalent copies are retained over the same representative rather than separated as distinct blocks. The total bond dimension of the displayed direct sum is exactly \(D\). Equivalently, with the block-diagonal matrix \(X=\bigoplus _{j,q}X_{j,q}\),

This is a strengthened representative-grouped form of the canonical decomposition in [ CPGSV16 , Section 2.3 ] . In contrast with Definition 9.6.4, the retained direct sum has the full bond dimension, every copy weight is nonzero and normalized, and gauge-equivalent normal blocks are grouped over a BNT representative with a separate gauge for each copy. It is therefore not the literal CPSV canonical-form hypothesis.

An MPO tensor has a horizontal-canonical-form MPV representation if its doubled-index MPS tensor generates the same positive-length MPV family as a BNT canonical form \(\bigoplus _{j=0}^{g-1}\bigoplus _{q=0}^{r_j-1}\mu _{j,q}A_j\). Its length-\(N\) matrix product vector is

This is the MPV-level decomposition in [ CPGSV16 , Section 2.3, lines 298–301 ] .

A state \(\psi \) on \(L+1\) sites satisfies the left ground condition if, for every \(j\), the state obtained by fixing the last index to \(j\) lies in \(G_L(A)\).

A state \(\psi \) on \(L+1\) sites satisfies the right ground condition if, for every \(i\), the state obtained by fixing the first index to \(i\) lies in \(G_L(A)\).

A state \(\psi \) on \(K+L\) sites lies in the suffix ground condition if every fixed prefix \(u\) gives a suffix state in \(G_L(A)\).

A tensor \(A\) is injective if its matrices span the full matrix algebra: \(\operatorname{span}_{\mathbb {C}}\{ A^i : i = 0,\ldots ,d{-}1\} = M_{D}(\mathbb {C})\).

A quantum instrument with \(n\) outcomes is a family \(\{ \Phi _i\} _{i=0}^{n-1}\) of completely positive maps on \(M_{D}(\mathbb {C})\) whose sum \(\sum _i\Phi _i\) is trace-preserving.

Associated to an instrument are the total channel \(\sum _i\Phi _i\), the unnormalized update \(\rho \mapsto \Phi _i(\rho )\) for each outcome \(i\), the outcome probability \(p_i(\rho )=\operatorname{tr}(\Phi _i(\rho ))\), and the normalized posterior state \(\Phi _i(\rho )/p_i(\rho )\) whenever \(p_i(\rho )\neq 0\).

An MPS tensor \(A\) has invariant-projector closure if every orthogonal projection \(P\) satisfying \(PA^i=PA^iP\) for all \(i\) also satisfies \(A^iP=PA^iP\) for all \(i\). This is the projector hypothesis in the sufficient condition for canonical form of  [ CPGSV16 , Section 2 ] .

A subspace \(W\le \mathbb {C}^D\) is \(A\)-invariant if \(A^iW\subseteq W\) for every \(i\).

The Kraus operators of \(A\) act irreducibly on \(\mathbb {C}^D\) if every \(A\)-invariant subspace is trivial, i.e. equal to \(0\) or to \(\mathbb {C}^D\).

A linear map \(E : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) is irreducible if whenever \(P\) is an orthogonal projection satisfying \(E(P M_{D}(\mathbb {C}) P) \subseteq P M_{D}(\mathbb {C}) P\), then \(P = 0\) or \(P = \mathbb {1}\). This is [ Wol12 , Theorem 6.2(1) ] . The definition applies to any linear map; complete positivity is not required.

An MPS tensor \(A\) is irreducible if it has no nontrivial invariant projection.

Let \((A_j)_j\) be a basis of normal tensors. The family is locally orthogonal when, for every pair of distinct components,

This is the mixed-sector condition of [ CPGSV16 , Definition 3.5 ] .

Let \((A_j)_j\) be a basis of normal tensors for \(A\). This auxiliary condition requires virtual-insertion distance independence and \(\mathcal{E}_{j,j'}=0\) for every pair of distinct BNT components. It is not [ CPGSV16 , Definition 3.6 ] , which quantifies over physical observables on all disjoint regions.

A collection of scaling factors \((\mu _k)_{k=1}^r\) and block tensors \((A_k)_{k=1}^r\) satisfies the auxiliary block-family hypotheses if:

1. each \(A_k\) is injective,

2. each \(A_k\) satisfies the TP normalization \(\sum _i(A_k^i)^\dagger A_k^i=\mathbb {1}\),

3. the moduli are non-increasing: \(|\mu _1|\ge |\mu _2|\ge \cdots \ge |\mu _r|\),

4. all \(\mu _k\neq 0\),

5. every block bond dimension is positive: \(D_k\ge 1\) for every \(k\),

6. the self-overlap converges: \(O_{A_kA_k}(N)\to 1\) for each \(k\).

Only this one-sided normalization is assumed; no unital condition is assumed here. Condition (6) is a primitivity hypothesis: for a primitive channel ( [ Wol12 , Theorem 6.7(3) ] ), \(T^n(\rho )\to \rho _\infty \) for every initial state, which forces the self-overlap to converge to \(1\). See also  [ Wol12 , Theorem 6.8 ] for the completely-positive characterisation and Chapter 7 for the transfer-operator gap proof.

A linear map \(L : M_{d}(\mathbb {C}) \to M_{d}(\mathbb {C})\) is conditionally completely positive (CCP) if it admits a generator decomposition, i.e. there exist a CP map \(\phi \) and a matrix \(\kappa \) such that \(L(\rho ) = \phi (\rho ) - \kappa \rho - \rho \kappa ^\dagger \). This is condition 1 of [ Wol12 , Proposition 7.2 ] .

For every positive-definite right fixed point \(\rho ^R{\gt}0\), every pair of virtual bond matrices \(X,Y\), and all separations \(n,m\geq 1\), the virtual-insertion expression is independent of distance: \(C(X,Y;n)=C(X,Y;m)\).

A scalar 2-cocycle \(\omega \) is a coboundary if there exists \(\varphi \colon G \to \mathbb {C}^{\times }\) such that, for all \(g,h \in G\),

A scalar 2-cochain \(\omega \colon G \times G \to \mathbb {C}^{\times }\) is a 2-cocycle if it satisfies, for all \(g,h,k \in G\),

The parent Hamiltonian with block length \(L\) on \(N\) sites has commuting local terms if \(h_i h_j=h_j h_i\) for all \(i,j\in \{ 0,\ldots ,N{-}1\} \). This records the translated commutation equation \([\tau _j(P_L),P_L]=0\) from [ CPGSV16 , Definition 3.9 ] ; the same source definition also requires the ground space to be spanned, for \(N{\gt}L\), by the periodic MPS vectors of the BNT components.

Given a \(*\)-subalgebra \(S\subseteq M_{D}(\mathbb {C})\), a linear map \(P:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) is a conditional expectation onto \(S\) if it is idempotent and unital, has range contained in \(S\), and satisfies \(P(X)=X\) for every \(X\in S\).

Given an endomorphism \(P_K\) and sets of observables \(\mathcal{O}_A\), \(\mathcal{O}_B\), regions \(A\) and \(B\) are decorrelated w.r.t. \(K\) when \(P_K\circ O_A\circ (\mathbb {1}-P_K)\circ O_B\circ P_K=0\) for all \(O_A\in \mathcal{O}_A\) and \(O_B\in \mathcal{O}_B\).

A vector \(\psi \) is a frustration-free ground state of the parent Hamiltonian when each local interaction annihilates it, that is, when \(h_i\psi =0\) for every \(i\in \{ 0,\ldots ,N{-}1\} \).

A linear map \(L : M_{d}(\mathbb {C}) \to M_{d}(\mathbb {C})\) is a GKSL generator if \(e^{tL}\) is a quantum channel (CPTP) for all \(t \ge 0\).

Let \(\lambda _\alpha {\gt}0\) satisfy \(\sum _\alpha \lambda _\alpha =1\), and set \(\rho =\operatorname{diag}(\lambda _\alpha )\). A normal tensor \(A\) has the square-root fixed-point form when there are an invertible matrix \(X\) and a tensor \(U\) satisfying the unit pair-index isometry

such that

Local fix (square-root diagonal). The display at line 1278 uses the bare diagonal \(\Lambda \), while lines 1281–1283 impose a unit pair-index isometry and line 1300 constructs the reference tensor with coefficients \(\sqrt{\Lambda _\alpha }\). These equations force the repaired form \(A^i=X\sqrt{\rho }\, U^iX^{-1}\) used in this definition. The correction is documented in docs/paper-gaps/cpsv16_rfp_isometry_scope.tex.

Explicitly, the reference tensor is

Thus \(\rho =\operatorname{diag}(\lambda _\alpha )\) is the trace-one diagonal fixed point. This records the \(j=j'\) part of the joint isometry condition [ CPGSV16 , Appendix B, lines 1278, 1281–1283, 1300 ] .

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.

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.

The span of the Lindblad operators is closed under Hermitian conjugation, i.e. it forms a \(*\)-subspace of \(M_{D}(\mathbb {C})\).

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.

This auxiliary one-block convention means \(\mathcal{E}_A^2=\mathcal{E}_A\). It is not local orthogonality in the sense of the source, which is the family of mixed-sector equations \(\mathcal{E}_{j,j'}=0\) for distinct BNT components.

A Naimark dilation of a POVM \(\{ E_i\} \) consists of an isometry \(V\) into a larger Hilbert space together with a projective measurement \(\{ P_i\} \) there such that \(E_i=V^\dagger P_iV\) for all \(i\).

The nearest-neighbor commuting condition is the length-\(2\) specialization of the preceding commutation equation. For every \(i,j\in \{ 0,\ldots ,N{-}1\} \), one has \(h_i h_j=h_j h_i\). The source definition [ CPGSV16 , Definition 3.9 ] also requires the corresponding parent Hamiltonian to have \(L=2\) and to have the prescribed ground-space spanning property for \(N{\gt}2\).

This condition records the commutation equation and the zero-energy equation for the nearest-neighbor local terms:

This is the zero-energy part of the condition appearing in [ CPGSV16 , Theorem 3.10(iii) ] : the MPS vector has zero energy for the translated two-site local terms, and those terms commute. The missing source assertion is that the whole parent-Hamiltonian ground space is spanned by the corresponding periodic MPS vectors from the BNT components, as recorded in Definition 14.1.15.

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.

A projection is nontrivial if it is orthogonal and neither \(0\) nor \(\mathbb {1}\).

Scaling factors \((\mu _k)_{k=1}^r\) and block tensors \((A_k)_{k=1}^r\) satisfy the normal canonical form conditions if:

1. each \(A_k\) is irreducible,

2. each \(A_k\) satisfies \(\sum _i(A_k^i)^\dagger A_k^i=\mathbb {1}\),

3. each \(\mathcal{E}_{A_k}\) is primitive (equivalently, \(\sigma _\partial (\mathcal{E}_{A_k})=\{ 1\} \)),

4. \(|\mu _1|\ge |\mu _2|\ge \cdots \ge |\mu _r|\),

5. all \(\mu _k\neq 0\),

6. all bond dimensions are positive.

Here “normal” is the spectral sense of  [ CPGSV16 ] , which by  [ SPGWC10 , Proposition 3 ] is equivalent to the eventual block-injectivity formulation of Definition 2.4.4.

A family \((A_j)_j\) of MPS tensors has positive-gap BNT zero correlation length for \(A\) when \((A_j)_j\) is a basis of normal tensors for \(A\) in the sense of Definition 10.1.2, its components are locally orthogonal, and the physical correlations of \(A\) are independent of separation whenever both complementary gaps are positive. Adjacent regions are not included.

For an observable \(O\) on a block of \(L\) physical spins, write

Place two observables \(O_1,O_2\) on nonempty finite blocks of a periodic chain, with positive complementary gap lengths \(n_1,n_2\). The tensor has positive-gap physical correlations independent of distance when the expectation is unchanged upon replacing these gaps by positive \(m_1,m_2\) satisfying \(n_1+n_2=m_1+m_2\):

This positive-gap restriction is the transfer-matrix form of [ CPGSV16 , Definition 3.3 and the correlation formula preceding Theorem 3.8 ] , restricted to positive complementary gaps. It excludes adjacent regions.

A bipartite matrix \(\rho \) has the positive-partial-transpose (PPT) property when its first-factor partial transpose is positive semidefinite, \(\rho ^{T_1}\ge 0\).

A quantum dynamical semigroup is a norm-continuous dynamical semigroup whose time slices are quantum channels for every \(t \ge 0\).

A quantum dynamical semigroup is reducible if it has a nontrivial invariant compression.

A family \((U_j)_j\) satisfies the joint isometry condition when

Each block satisfies the within-block pair-index orthonormality, and the cross-block sums between distinct blocks vanish. This is the isometry condition of [ CPGSV16 , Theorem 3.11 ] , split into its \(j=j'\) and \(j\ne j'\) cases.

An MPO tensor \(M\) has transfer-map idempotence if

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:

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 ] .

For tensors \(A,B\) and a fixed physical representation \(U:G\to \mathrm{GL}_d(\mathbb {C})\), the predicate called same SPT phase asserts the existence of factor systems \(\omega _A,\omega _B:G\times G\to \mathbb {C}^\times \) and projective bond representations \(\rho _A,\rho _B\) such that

exactly, and \(\omega _A\sim \omega _B\). Thus the predicate is equality of virtual factor-system classes. It neither quantifies over Hamiltonian paths nor states physical phase equivalence.

A bipartite matrix \(\rho \) on \(M_{d}(\mathbb {C})\otimes M_{d'}(\mathbb {C})\) is separable when it is a finite sum of Kronecker products of positive semidefinite matrices,

A linear map \(L : M_{d}(\mathbb {C}) \to M_{d}(\mathbb {C})\) is trace-annihilating if \(\operatorname{tr}(L(\rho )) = 0\) for all \(\rho \in M_{d}(\mathbb {C})\). This is the infinitesimal version of trace preservation.

A generator decomposition \((\phi , \kappa )\) satisfies the trace constraint if \(\phi ^*(\mathbb {1}) = \kappa + \kappa ^\dagger \), where \(\phi ^*\) is the Hilbert–Schmidt adjoint. This is the condition in [ Wol12 , Equation (7.20) ] .

This auxiliary one-block predicate is the conjunction of \(\mathcal{E}_A^2=\mathcal{E}_A\) and virtual-insertion distance independence. It is not the physical zero-correlation-length condition of Definition 24.2.6.

A linear map \(E : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) is \(k\)-positive if \(E \otimes \operatorname{id}_{k}\) is positive on \(M_{D}(\mathbb {C}) \otimes M_{k}(\mathbb {C})\).

A Kossakowski form consists of \(H=H^\dagger \), a finite family of matrices \(\{ F_k\} _{k=0}^{n-1}\), and a positive semidefinite matrix \(C \in M_{n}(\mathbb {C})\), defining

In Wolf’s formula one takes \(\{ F_k\} \) to be a basis of traceless matrices; here we keep only the algebraic data needed for the conversion to Lindblad form.

The minimal number of Lindblad operators across all GKSL representations of \(L\), which equals the rank of the Kossakowski matrix in the orthonormal-basis representation.

The adjoint Kraus map is

When the \(\{ K_i\} \) are the matrices of an MPS tensor \(A\), this is the transfer map of the conjugate-transposed family \(i \mapsto (A^i)^\dagger \).

The Kraus commutant is

It is a \(*\)-subalgebra of \(M_{D}(\mathbb {C})\).

The fixed-point set of a Kraus map \(E(X)=\sum _iK_iXK_i^\dagger \) is \(\operatorname{Fix}(E)=\{ X\in M_{D}(\mathbb {C}):E(X)=X\} \).

Given operators \(\{ K_i\} _{i=0}^{d-1}\) with \(K_i \in M_{D}(\mathbb {C})\), the Kraus map is

A linear map is completely positive (Definition 4.1.4) if and only if it can be written in this form, as in (??).

Let

Write \(\iota _{\mathcal A},\pi _{\mathcal A}\) and \(\iota _{\mathcal B},\pi _{\mathcal B}\) for the corresponding block-diagonal embeddings and diagonal-block compressions. The canonical full-matrix extension of \(T:\mathcal A\to \mathcal B\) is

The map \(T\) is a direct-sum Kraus map when \(\widehat T\) has a Kraus representation.

Let \(A\in M_{D}(\mathbb {C})\) be Hermitian with eigenvalues \(\lambda _1\ge \lambda _2\ge \cdots \ge \lambda _D\) listed in decreasing order. For \(0\le k\le D\), set

This is the sum of the \(k\) largest signed eigenvalues. Indices beyond \(D\) contribute zero, so the value stabilizes at \(\operatorname{Re}\operatorname{tr}(A)\) once \(k\) reaches \(D\).

The Ky-Fan \(k\)-norm \(\| A\| _{(k)}\) is by definition the sum of the \(k\) largest singular values of \(A\). It agrees with \(S_k(A)\) exactly when \(A\) is positive semidefinite, where eigenvalues and singular values coincide; for an indefinite Hermitian \(A\) the two differ. For \(A=\operatorname{diag}(1,-2)\), one has \(S_1(A)=1\) but \(\| A\| _{(1)}=2\). The results below concern \(S_k(A)\) for arbitrary Hermitian \(A\), so on the positive-semidefinite cone they are statements about the Ky-Fan norm.

A tensor \(A\) is \(L\)-block injective if

An injective tensor is \(1\)-block injective.

An MPS tensor \(A=(A^i)_i\) is left-canonical when

Equivalently, its Kraus map is trace-preserving.

A Lindblad form consists of a Hermitian matrix \(H = H^\dagger \) (the Hamiltonian) and a family of matrices \(\{ L_j\} _{j=0}^{r-1}\) (the Lindblad operators), defining the linear map

where \([A,B] = AB - BA\) and \(\{ A,B\} _+ = AB + BA\). This is [ Wol12 , Equation (7.21) ] .

The \(\mathbb {C}\)-linear span of the Lindblad operators \(\{ L_j\} \).

If a Stinespring matrix \(V:\mathbb {C}^{d_{\mathrm{in}}}\to \mathbb {C}^{d_{\mathrm{out}}}\) acts on the first factor while a finite-dimensional space \(R\) is left unchanged, the corresponding local Stinespring matrix is \(W=V\otimes \mathbb {1}_R\).

The virtual local-covariance predicate holds for \(A,u,\Lambda \) if there exist a unitary matrix \(V\) and a phase \(\mu \) with \(|\mu | = 1\) such that \(V^\dagger \Lambda V = \Lambda \) and, for every physical index \(i\),

The unitary \(V\) intertwines the physical action of \(u\) with a virtual conjugation, and the boundary state \(\Lambda \) is invariant under \(V\).

For each site index \(i\in \{ 0,\ldots ,N{-}1\} \), the translated local term is determined by

For a completely positive, trace-preserving qubit channel \(T' : M_{2}(\mathbb {C}) \to M_{2}(\mathbb {C})\), write \(\widehat{T'}\) for its matrix in the normalized Pauli basis \(\{ \sigma _{0}/\sqrt{2},\sigma _{1}/\sqrt{2}, \sigma _{2}/\sqrt{2},\sigma _{3}/\sqrt{2}\} \). The channel is in diagonal Lorentz normal form when it is unital and every off-diagonal entry of \(\widehat{T'}\) is zero.

For a completely positive, trace-preserving qubit channel \(T' : M_{2}(\mathbb {C}) \to M_{2}(\mathbb {C})\), write \(\widehat{T'}\) in the normalized Pauli basis. The channel is in non-diagonal Lorentz normal form when, for some \(x \in [0,1]\),

Trace preservation supplies the first row of the displayed matrix.

For a completely positive, trace-preserving qubit channel \(T' : M_{2}(\mathbb {C}) \to M_{2}(\mathbb {C})\), write \(\widehat{T'}\) in the normalized Pauli basis. The channel is in singular Lorentz normal form when

equivalently, every input state is mapped to \((1+\sigma _{3})/2\). Trace preservation supplies the first row of the displayed matrix.

An MPO tensor is an LPDO if there exist a Kraus dimension \(d_K\), an inner bond dimension \(D'\), a bond-space identification \(e:\{ 0,\ldots ,D-1\} \simeq \{ 0,\ldots ,D'-1\} \times \{ 0,\ldots ,D'-1\} \) (equivalently, \(D\) and \((D')^2\) have the same cardinality), and matrices \(A^{(i,k)}\in M_{D'}(\mathbb {C})\) such that, for all physical indices \(i,j\),

where \(\otimes \) denotes the Kronecker product, \(\overline{(\cdot )}\) denotes entrywise complex conjugation, and \((\cdot )_{e,e}\) denotes reindexing along \(e\) back to \(\{ 0,\ldots ,D-1\} \). See [ CPGSV16 , Section 4.3 ] . The local contraction is

\( = \)

Set \(d = 2\) (spin-\(\tfrac 12\)) and \(D = 3\). The Majumdar–Ghosh tensor is the family \(\{ A^0, A^1\} \subset M_{3}(\mathbb {C})\) defined by

The three bond levels are the two spin-\(\tfrac 12\) values of an open singlet partner together with the closed inter-dimer link; the relative sign \(-\tfrac 1{\sqrt2}\) is the antisymmetry of the singlet. The equality between the matrix product vectors of these matrices and the even-ring dimer superposition, with normalization and cyclic convention fixed, is a separate verification, not part of this definition.

A linear map \(T:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) maps the positive semidefinite cone onto itself if \(T\) is positive and every positive semidefinite matrix is the image under \(T\) of a positive semidefinite matrix. This is condition (1) of [ Wol12 , Proposition 3.8 ] .

Let

be the tensor of a sector decomposition. Let \(s\) range over a second finite alphabet, and for each representative \(A_j\) let \(C_j^s\) be a matrix on its bond space. The corresponding direct sum with one marked letter is

The marked alphabet need not coincide with the alphabet of the unmarked tensor. This is the algebraic form of the marked matrices in the two diagrams of [ CPGSV16 , Proposition 4.13, lines 1909–1919 ] .

Flattening matrix entries defines an isometric embedding \(\operatorname{vec}:M_{m\times n}(\mathbb {C})\to \mathbb {C}^{mn}\) with respect to the Frobenius norm: \(\| \operatorname{vec}(X)\| ^2=\| X\| _F^2\).

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.

For every \(d\in \mathbb {N}\), define \(|\Omega \rangle \! \langle \Omega |\) on \(\mathbb {C}^d\otimes \mathbb {C}^d\) using

When \(d\geq 1\), this is the maximally entangled state: a rank-one projector with \((|\Omega \rangle \! \langle \Omega |)_{(i_1,i_2),(j_1,j_2)} = \frac{1}{d}\delta _{i_1 i_2}\delta _{j_1 j_2}\) and \(\operatorname{tr}(|\Omega \rangle \! \langle \Omega |)=1\).

Let \(V\) be finite-dimensional and let \(f:V\to V\) have bounded orbits. The mean-ergodic projection \(P_f:V\to V\) is the projection onto \(\ker (f-\mathbb {1}_V)\) along \(\operatorname{range}(f-\mathbb {1}_V)\) in (??).

The mixed transfer operator (or cross transfer operator) for two MPS tensors \(A\) and \(B\) of the same bond dimension \(D\) is the linear map \(F_{AB}:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) defined by

For MPS tensors \(A\) of bond dimension \(D_1\) and \(B\) of bond dimension \(D_2\), possibly with \(D_1\neq D_2\), the rectangular mixed transfer operator is the map \(F_{AB}^{\mathrm{rect}}: M_{D_1\times D_2}(\mathbb {C})\to M_{D_1\times D_2}(\mathbb {C})\) defined by

An MPO tensor is an MPDO if, for every \(N\geq 1\),

This is the global positivity condition of [ CPGSV16 , Section 4 ] .

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.

A family of support \(*\)-algebras \(\{ \mathcal{A}_n\} _{n \ge 0}\), one at every blocking size, equipped with blocking maps \(m_n : \mathcal{A}_n \otimes \mathcal{A}_n \to \mathcal{A}_{2n}\) and inclusion maps \(\iota _n : \mathcal{A}_n \to \mathcal{A}_{n+1}\) realized, inside the ambient bond-space matrix algebra, by

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 ] .

Let \(M_c^{il}\in M_{D_c}(\mathbb {C})\) be a finite labelled MPO tensor family. A simultaneous block left inverse consists of coefficients \(C_{(c,x,y),(i,l)}\) such that

This is the common block inverse used in [ CPGSV16 , Appendix C.2, lines 1666–1676 ] and [ BMW\(^{+}\)17 , lines 269–277 ] .

For each blocked size \(n\), fix a basis \(\{ b_i\} _{i \in I_n}\) of \(\mathcal{A}_n\) over a finite index set \(I_n\). The blocked coefficients of \(x \in \mathcal{A}_n\) are the coordinates \((x_i)_{i \in I_n} \in \mathbb {C}^{I_n}\) with

For \(i \in I_n\), the blocked inclusion coefficients are the blocked coefficients \((d_{ij})_{j \in I_{n+1}}\) of \(\iota _n(b_i) \in \mathcal{A}_{n+1}\):

For each positive blocked size \(n\) and multiplication triple \((i,j,k)\), where \(i\) and \(j\) label basis elements of \(\mathcal{A}_n\) and \(k\) labels a basis element of \(\mathcal{A}_{2n}\), a size \(r_{n,i,j,k} \in \mathbb {N}\) and diagonal entries \(\chi _{n,i,j,k,1}, \ldots , \chi _{n,i,j,k,r_{n,i,j,k}} \in \mathbb {C}\), giving the diagonal matrix

Unlike the uniform BNT-label family in [ CPGSV16 , Theorem 4.14(ii) ] , the size and diagonal entries may depend on the blocked length and on the chosen basis indices.

For \(i,j \in I_n\), the blocked multiplication coefficients are the blocked coefficients \((c^{(n)}_{i,j,k})_{k \in I_{2n}}\) of \(m_n(b_i,b_j) \in \mathcal{A}_{2n}\):

Fix a coefficient family \(c^{(L)}_{\alpha ,\beta ,\gamma }\), BNT-label operators \(O_L(M_\alpha )\), and trace scalars \(m_\alpha =\operatorname{tr}(\mu _\alpha )\). They satisfy the BNT algebra clause when there is a length-independent family of positive diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\) such that, for every \(L{\gt}0\), \(c^{(L)}_{\alpha ,\beta ,\gamma } =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\), the same-length product law holds,

and the trace scalars satisfy

This is the algebra statement in [ CPGSV16 , Theorem 4.14(ii), labelled IV.13 in the source, lines 972–985 ] , before comparison with a chosen blocked basis.

Choose a vertical canonical decomposition

Separating the two factors of the doubled physical index of \(M_\alpha \) gives an MPO tensor \(\widehat M_\alpha \). Define

The chosen decomposition satisfies the tensor-attached BNT algebra clause when the tensors \(M_\alpha \) form a basis of normal tensors and the BNT algebra clause holds for precisely these operators and these trace scalars. This is the data in [ CPGSV16 , Theorem 4.14(ii), lines 972–993, and Appendix C.4, lines 2046–2064 ] ; it does not include a comparison with a two-site vertical canonical decomposition.

A comparison between the BNT-label coefficients and the chosen blocked-basis coefficients consists of a BNT-label assignment together with the equality

This is a blocked-basis comparison: the left hand side is expanded in the chosen basis of \(\mathcal A_{2n}\), so the statement is not itself the same-length BNT product law.

The BNT-label assignment consists, for every positive blocked length \(n\), of a source label map \(\sigma _n\) from the chosen basis of \(\mathcal A_n\) and a target label map \(\tau _n\) from the chosen basis of \(\mathcal A_{2n}\) to the fixed BNT labels. It also gives the combined label map \(\sigma _n \sqcup \tau _n\) on the disjoint union of these two blocked bases.

Suppose that every entry of the one-site multiplicity matrices, every entry of the two-site multiplicity matrices, and every entry of the matrices \(\chi _{\alpha ,\beta ,\gamma }\) is positive. Suppose also that \(\sum _i\mu _{\alpha ,i}=m_\alpha \) for every \(\alpha \). If, for every \(\gamma \) and all sufficiently large \(L\),

then the entries on the two sides agree as multisets. Together with the one-site trace identity \(\sum _i\mu _{\alpha ,i}=m_\alpha \), the one-site and two-site multiplicity entries and their dimensions define a BNT multiplicity-spectrum comparison.

Positivity makes all entries nonzero. Eventual equality of two finite sums of geometric sequences gives, for every \(L\geq 1\),

The finite power-sum identity determines both the cardinality and the multiset of entries.

Suppose the fusion-isometry identity holds with positive diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\), and suppose the trace scalars satisfy the length-one idempotent law for the corresponding trace-power coefficients. For \(L{\gt}0\), the resulting BNT algebra clause has coefficients \(c^{(L)}_{\alpha ,\beta ,\gamma } =\operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\) and records positivity of every diagonal entry. The product law and the assumed idempotent condition are, respectively,

This construction realizes implication (iii)\(\Rightarrow \)(ii) of [ CPGSV16 , Theorem 4.14 ] ; it does not assert either of the implications involving the renormalization fixed-point condition.

An active-support fusion coisometry family consists of the same labelled tensors and positive diagonal matrices, together with maps \(U_{\alpha ,\beta }\) satisfying \(U_{\alpha ,\beta }U_{\alpha ,\beta }^\dagger =1\), the forward fusion identity

and exact reconstruction

Zero-dimensional retained sums are permitted. If in addition \(U_{\alpha ,\beta }^\dagger U_{\alpha ,\beta }=1\) for every pair, this family gives the preceding full-support fusion-isometry family.

Local fix (Figure-11 fusion coisometry): The source uses the retained-row orientation of Proposition 4.13. Thus its fusion maps are coisometries onto the active sectors; exact reconstruction records the discarded common zero corner. This convention is documented in docs/paper-gaps/cpsv16_figure11_fusion_coisometry.tex.

A fusion-isometry family over a finite set of labels consists of a family of tensors \(M_\gamma \) with a common physical dimension and per-label bond dimensions, diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\) with positive entries, and isometries \(U_{\alpha ,\beta }\) such that, site by site,

This is the isometry statement of [ CPGSV16 , Theorem 4.14(iii) ] ; the tensors are the vertically read basis of normal tensors of [ CPGSV16 , Proposition 4.13 ] , so the physical dimension is the bond dimension of the original tensor. Statement (iii) of the source theorem additionally asserts the idempotent identity for the trace scalars of the vertical decomposition, recorded separately by Definition 27.2.2.5.

Scope restriction (full-support fusion family): The displayed family assumes \(U_{\alpha ,\beta }^\dagger U_{\alpha ,\beta }=1\) on the whole product bond space. The unrestricted source statement permits a common zero corner and instead gives a coisometry onto the active direct sum, together with exact reconstruction. This distinction is documented in docs/paper-gaps/cpsv16_figure11_fusion_coisometry.tex.

The tensors of a fusion-isometry family generate the concrete BNT-label operator family \(O_L(M_\gamma )\), and the positive diagonal matrices give the positive trace-power witness for the coefficient family \(c^{(L)}_{\alpha ,\beta ,\gamma } = \operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L})\).

Choose a vertical canonical decomposition

The fusion clause for this decomposition consists of positive diagonal matrices \(\chi _{\alpha ,\beta ,\gamma }\) and coisometries \(U_{\alpha ,\beta }\) onto the active product sectors, satisfying

They obey \(U_{\alpha ,\beta }U_{\alpha ,\beta }^\dagger =1\), and the active direct sum reconstructs the product tensor:

The trace scalars \(m_\alpha =\operatorname{tr}(\mu _\alpha )\) satisfy the length-one idempotent identity for the coefficients determined by these same diagonal matrices. This is statement (iii) of [ CPGSV16 , Theorem 4.14, lines 986–993 ] , attached to the chosen decomposition of [ CPGSV16 , Proposition 4.13, lines 943–951 ] .

The same-length coefficient system \(c^{(L)}_{\alpha ,\beta ,\gamma }\) from [ CPGSV16 , Theorem 4.14(ii) ] , indexed by fixed BNT labels \(\alpha ,\beta ,\gamma \). The coefficient is attached to the length-\(L\) product \(O_L(M_\alpha )O_L(M_\beta )\), not to the blocked-basis multiplication \(\mathcal A_n\times \mathcal A_n\to \mathcal A_{2n}\). A diagonal \(\chi \)-family canonically determines the coefficient family \(c^{(L)}_{\alpha ,\beta ,\gamma } = \operatorname{tr}(\chi _{\alpha ,\beta ,\gamma }^{L}) = \sum _r\chi _{\alpha ,\beta ,\gamma ,r}^{\, L}\), as in [ CPGSV16 , Appendix C.4 ] .

A BNT-label trace-scalar family and a BNT-label coefficient system have idempotent coefficient form when, for every BNT label \(\gamma \),

This is the idempotent condition in [ CPGSV16 , Theorem 4.14(ii) ] , stated separately from the trace-power formula.

A BNT-label coefficient system \(c^{(L)}_{\alpha ,\beta ,\gamma }\) is length independent when \(c^{(L)}_{\alpha ,\beta ,\gamma } = c^{(1)}_{\alpha ,\beta ,\gamma }\) for every positive chain length \(L\) and all labels; equivalently, the coefficients agree at any two positive lengths. This is the case singled out in the discussion following [ CPGSV16 , Theorem 4.14 ] .

A BNT-label operator family is linearly independent at length \(L\) when the operators \(O_L(M_\alpha )\) are linearly independent, and eventually linearly independent when this holds for every length beyond some threshold. Eventual linear independence is the operator-level form of the third defining property of a basis of normal tensors in [ CPGSV16 , Section 2.3 ] , invoked for the labelled operators in [ CPGSV16 , Appendix C.4 ] .

A BNT-label operator family records, for each chain length \(L\), the operators \(O_L(M_\alpha )\) indexed by the fixed BNT labels \(\alpha \). The ambient algebra may depend on \(L\) and is not identified with a chosen blocked support algebra. Write \(\widehat M_\alpha ^{ab}=M_\alpha ^{(a,b)}\) for the vertical reading. The vertical indices of the \(L\) copies are contracted cyclically, while the horizontal legs remain open, as in [ CPGSV16 , lines 962–967 ] :

A BNT-label coefficient system and a diagonal \(\chi \)-family are compatible when, for every positive length \(L\),

The same matrix \(\chi _{\alpha ,\beta ,\gamma }\) is used for all positive \(L\).

A BNT-label operator family and a BNT-label coefficient system have same-length product form when, for every positive length \(L\),

This is the product identity appearing in [ CPGSV16 , Theorem 4.14(ii) ] , stated independently of the blocked-basis multiplication \(\mathcal A_n\times \mathcal A_n\to \mathcal A_{2n}\).

The BNT-label theorem data consist of a BNT algebra clause together with the comparison with the chosen blocked bases. Explicitly, for every \(L{\gt}0\) the product law is

while the idempotent scalar law is

For every positive blocked length \(n\) and blocked-basis indices \(i,j,k\), the source and target label maps \(\sigma _n,\tau _n\) give the comparison

They also determine the pulled-back blocked-basis \(\chi \)-family. These data are the source-side hypotheses from which the blocked-basis consequences below are derived. In the source-side special case where the coefficient family is canonically determined by the same \(\chi \)-family, theorem data are built from that \(\chi \)-family, its positivity, and the product, idempotent, and blocked-comparison statements for the resulting canonical coefficients. The same product and idempotent predicates may also be rephrased using the canonical coefficient family determined by the \(\chi \)-matrices carried by the data, with the corresponding displayed equations.

An existential BNT-label theorem witness consists of the finite BNT-label type, the same-length operator spaces, their algebraic structure, and the corresponding BNT-label theorem data, including the pulled-back blocked-basis \(\chi \)-family. The construction of such a witness from an MPDO tensor is the outstanding step toward [ CPGSV16 , Theorem 4.14(ii) ] . The proposition-level form is the nonemptiness of this witness type. In the source-side special case where the coefficients are canonically determined by a \(\chi \)-family, the witness is obtained from that same \(\chi \)-family together with the remaining product, idempotent, and blocked-comparison statements. For any such witness, the product and idempotent predicates can be rephrased with the canonical coefficient family determined by its \(\chi \)-matrices, with the corresponding displayed equations.

A BNT-label trace-scalar family records the scalars \(m_\alpha =\operatorname{tr}(\mu _\alpha )\) indexed by the fixed BNT labels \(\alpha \), as in the idempotent condition of [ CPGSV16 , Theorem 4.14(ii) ] .

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 \(\beta ,\alpha \).

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 ] .

Fix the one-site diagonal multiplicity matrices \(\mu _\alpha \) and the two-site diagonal multiplicity matrices \(\nu _\gamma \) in the respective vertical canonical decompositions. A multiplicity-spectrum comparison records \(\operatorname{tr}(\mu _\alpha )=m_\alpha \) and, for every \(\gamma \), equality with multiplicity between the entries of \(\nu _\gamma \) and the products \(\mu _{\alpha ,i}\mu _{\beta ,j}\chi _{\alpha ,\beta ,\gamma ,k}\). This is the conclusion of the vertical BNT comparison and positive power-sum argument in [ CPGSV16 , Appendix C.4, lines 2048–2058 ] . The definition does not require this comparison to follow from a BNT algebra clause; constructing it from the MPDO tensor is a separate obligation.

Under the hypotheses of Theorem 26.10.2.10, define

This is equation Pis, restricted to the positive-weight Markov support retained here.

The invertible-gauge sub-result for a source-derived two-site multiplicity spectrum additionally records, for every one-site sector \(\gamma \), equality of bond dimensions \(d_\gamma =d_{\sigma (\gamma )}\) and an invertible matrix \(Z_\gamma \in \mathrm{GL}(d_\gamma ,\mathbb {C})\) such that, for every physical index \(i\),

Thus the matched normal tensors have no residual scalar phase. This is the exact invertible conjugacy retained from [ CPGSV16 , Appendix C.4, lines 2053–2057 ] . It does not include the further conclusion on line 2057 that \(Z_\gamma \) may be chosen unitary.

For a tensor-attached BNT algebra clause, a source-derived two-site multiplicity spectrum consists of a vertical canonical decomposition

a bijection \(\sigma \) from the one-site sectors to its sectors such that \(M_\gamma \) and \(A_{\sigma (\gamma )}\) generate the same matrix product vectors at every positive length, and, for every \(\gamma \), the multiset equality

This is the decomposition and comparison retained in [ CPGSV16 , Appendix C.4, lines 2046–2058 ] .

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 ] .

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 ] .

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 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: \([B_{i,i+1},B_{j,j+1}]=0\).

Fix a final label \(d\). Summing the diagonal matrix-unit coefficients of the simultaneous block inverse defines

Here \(C_{(d,x,y),(i,j)}\) denotes the coefficient of the simultaneous block inverse corresponding to the matrix unit \(E_{xy}\) in block \(d\).

Fix labels \(a,b,c,d,e\). The five fusion trees in equation (33) of [ BMW\(^{+}\)17 ] determine five multiplicity spaces and five synthesis maps into the fourfold bond space. These spaces use the integer multiplicities \(N_{ab}^{c}\) of the complete zipper family, rather than the weighted positive-diagonal coordinates introduced earlier.

Each edge of the pentagon carries the printed \(F\)-matrix for the affected triple tensor factor and the identity on the untouched multiplicity factor. The three upper edges and the two lower edges define composites in the forward orientation. Replacing each edge by its inverse defines the opposite-oriented composites whose entries occur literally in equation (33).

Let \(\Lambda \) be a finite set. A complete zipper fusion family consists of positive bond dimensions \(D_a\), MPO blocks \(B_a^{ij}\in M_{D_a}(\mathbb {C})\), fusion multiplicities \(N_{ab}^{c}\), and matrices

The matrices \(X^{c+}_{ab,\mu }\) are independently chosen left inverses, not adjoints. For all labels and multiplicity indices,

The opposite product \(P_{ab}=\sum _{c,\mu }X^c_{ab,\mu }X^{c+}_{ab,\mu }\) is the projector onto the product-MPO support; it need not be the identity on the ambient bond space. Every product-MPO letter is reconstructed on this support:

Writing \(B_{ab}^{ik}=\sum _j B_a^{ij}\otimes B_b^{jk}\), the two zipper identities are

Each block is injective, and the labelled block-letter matrix has a common left inverse: its contraction with \(B_d^{ij}\) is \(\delta _{cd}E_{xy}\) on the block selected by \((c,x,y)\). These are the fusion-tensor, reconstruction, zipper, and simultaneous inverse relations of [ BMW\(^{+}\)17 , lines 161, 181–200, and 269–277 ] .

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 \(H=\bigoplus _{k,h}H_{k,h}\). 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 ] .

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.

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 ] .

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 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 ] .

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

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.

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 ] .

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 \(\eta _{k,h}\ne 0\).

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.

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.

A family of diagonal complex matrices, indexed by ordered triples \((\alpha ,\beta ,\gamma )\) drawn from a common index type, consisting of a size \(r_{\alpha ,\beta ,\gamma } \in \mathbb {N}\) and diagonal entries \(\chi _{\alpha ,\beta ,\gamma ,1}, \ldots , \chi _{\alpha ,\beta ,\gamma ,r_{\alpha ,\beta ,\gamma }} \in \mathbb {C}\) for each triple, giving the diagonal matrix

This is the matrix \(\chi _{\alpha ,\beta ,\gamma }\) of [ CPGSV16 , Theorem 4.14(ii) ] .

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.

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 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.

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.

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.

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 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 \((\eta _{k,h})\) 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 ] .

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.

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

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.

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.

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.

On a nonzero finite-dimensional space \(H\), define

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 \((R_k\otimes L_l)\otimes (R_l\otimes L_h)\).

On fixed sector pairs, define \(\mathcal S_2\) and \(\mathcal T_2\) by the same inverse regroupings as the corresponding global shifts.

For some common length \(S\), suppose that for each final label \(\varepsilon \) there are coefficients \(c_\varepsilon (w)\), indexed by the words \(w\) of length \(S\) in the doubled physical alphabet, such that

This is the finite-word form of the simultaneous inverse relation \(B_d^+B_{d'}=\delta _{d,d'}1\) used in the fixed-channel argument of [ BMW\(^{+}\)17 ] .

The maps \(U^{\mathrm L}_{\alpha ,\beta ,\gamma ;\varepsilon }\) and \(U^{\mathrm R}_{\alpha ,\beta ,\gamma ;\varepsilon }\) are obtained by restricting the two iterated fusion isometries to the summands with final label \(\varepsilon \). Their adjoints, after the canonical reassociation of the triple bond space, have the orientation and bracketing pattern of the weighted analogs of the two sides in the printed-\(F\) row/column convention of [ BMW\(^{+}\)17 , Section 3.4 ] .

Fix a final label \(\varepsilon \). The multiplicity spaces of the two bracketings are

These spaces have the bracketing pattern of the \((e,\mu ,\nu )\) and \((f,\lambda ,\sigma )\) index spaces in the printed-\(F\) row/column convention of [ BMW\(^{+}\)17 , Section 3.4 ] . Identifying the dimensions of the positive diagonal matrices with the fusion multiplicities of that equation requires the length-independent integer specialization and is not asserted here. The corresponding fixed-final row spaces are \(\mathcal H^{\mathrm L}_{\varepsilon }\otimes \mathbb {C}^{D_\varepsilon }\) and \(\mathcal H^{\mathrm R}_{\varepsilon }\otimes \mathbb {C}^{D_\varepsilon }\).

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 \(d^2\).

For a virtual matrix \(X\), the four-site physical closure of \(K\) is the operator \(K_4(X)\) whose matrix coefficients are

For a full-support fusion family, let \(P_\gamma \) act on the bond space of \(M_\gamma \). Define

The terminal contraction is the same as in the source operator. The additional condition is that the active support is the whole product bond space.

Scope restriction (full-support fusion family): Full support is not an additional hypothesis of the arbitrary-chain source statement. This distinction is documented in docs/paper-gaps/cpsv16_topological_projector_recursion.tex.

A transfer-map fusion-isometry datum at blocked size \(n\) for an MPO tensor consists of a subspace \(\mathcal{A}_n \subseteq M_{D}(\mathbb {C})\) together with linear maps

where \(E_n\) denotes the blocked transfer map.

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 ] .

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 ] .

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.

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.

The blocked multiplication coefficients are in trace-power form when, for every positive blocked size \(n\) and every multiplication triple \((i,j,k)\), the coefficient \(c^{(n)}_{i,j,k}\) equals \(\sum _r \chi _{n,i,j,k,r}^{\, n}\). Equivalently, by Lemma 27.2.2.19, it is the trace of the \(n\)-th power of the corresponding diagonal matrix. The exponent \(n\) is the source blocking length of the two factors in \(\mathcal{A}_n\); the product is expanded in the basis of \(\mathcal{A}_{2n}\).

A binary compatibility predicate between an abstract structure-coefficient family \(c^{(L)}_{\alpha ,\beta ,\gamma }\) and a diagonal \(\chi \) family: the pair \((c, \chi )\) is said to be in trace-power form when

for every \(L \ge 0\) and every triple \((\alpha , \beta , \gamma )\), where \(r_{\alpha ,\beta ,\gamma }\) is the size of \(\chi _{\alpha ,\beta ,\gamma }\). This is the formal analog of the target identity of [ CPGSV16 , Theorem 4.14(ii) ] ; the existential version—“there exists \(\chi \) for which \((c,\chi )\) has trace-power form”—is obtained by quantifying over \(\chi \).

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.

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.

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 ] .

The spin reduction obtained by tracing the ancillary index is trace-preserving completely positive: the ancillary trace map \(\operatorname{tr}_a\) satisfies Definition 26.1.6. This is the structure described immediately after Definition 4.3 in [ CPGSV16 , lines 761–764 ] .

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 \(B=\bigoplus _k(b_1^{(k)}\otimes b_2^{(k)})\) 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 ] .

For a linear map \(\Psi _B\) and a bipartite matrix \(\rho _{AB}\), define

where \(\Sigma \) denotes the canonical exchange of the two tensor factors.

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.

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 ] .

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 ] .

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.

For labels \(\alpha , \beta , \gamma \), the left iterated fusion isometry \(U^{\mathrm L}_{\alpha ,\beta ,\gamma }\) is obtained by applying \(U_{\alpha ,\beta }\) (tensored with the identity on the bond space of \(M_\gamma \)) to fuse the first two tensors, then applying \(U_{\delta ,\gamma }\), for each label \(\delta \) appearing in the resulting direct sum, to fuse the outcome with the third tensor. Its codomain is indexed by pairs of labels \((\delta , \varepsilon )\) together with the multiplicity indices of \(\chi _{\alpha ,\beta ,\delta }\) and \(\chi _{\delta ,\gamma ,\varepsilon }\).

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.

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.

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.

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 ] .

Say that displaced projectors of \(M\) never commute if every orthogonal projector \(Q\) on the physical space with \(Q\widetilde M\ne Q\widetilde MQ\) satisfies \(Q_1H^{(N)}\ne H^{(N)}Q_1\) at every length. This is the canonical-form step of the periodic-sector argument: the proof of [ CPGSV16 , Proposition 4.13 ] assumes the tensor is in canonical form in the horizontal direction, and the argument of lines 1874–1887 turns all-length commutation of \(Q_1\) with \(H^{(N)}\) into the letter-level invariance \(Q\widetilde M=Q\widetilde MQ\). This argument uses the normalized BNT-refined horizontal hypothesis. The all-length condition is stronger than is needed for either Theorem 23.4.47 or Theorem 23.4.48; each uses a single noncommuting length supplied by the contrapositive of Lemma L.

With the convention \(0^{-1}=0\), define for every pair \((k,h)\)

These operators are normalized when the pair is active, namely when \(a_kb_h\ne 0\).

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 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 ] .

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 ] .

The product tensor of two MPO tensors contracts the bra index of the first factor with the ket index of the second and takes the tensor product of the bond spaces:

In the vertical reading of [ CPGSV16 , Section 4.5 ] this is the tensor obtained by joining two vertically read tensors along the original horizontal bond, the tensor written \(M_\alpha M_\beta \) in [ CPGSV16 , Theorem 4.14(iii) ] .

For three bond spaces, let

be the coordinate pullback induced by canonical reassociation, defined on elementary tensors by \(A_{D_1,D_2,D_3}(x_1\otimes (x_2\otimes x_3)) =(x_1\otimes x_2)\otimes x_3\).

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 \(\{ K_x\} _{x=0}^{g-1}\) 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.

Let \(M\) be an MPO tensor. A periodic-sector projector for \(M\) with exponent \(p\ge 1\) is an orthogonal projector \(Q\) on the physical space such that for every \(N\ge 0\), with \(Q_1=Q\otimes \mathbb {1}^{\otimes N}\) acting on the first site,

These are the commutation identities that the proof of [ CPGSV16 , Proposition 4.13 ] , lines 1889–1891, attaches to a nontrivial vertical period; the second family is the displayed identity at lines 1890–1891.

Let \(M\) generate matrix product density operators, let \(p\ge 1\), and let \(Q\) be an orthogonal projector on the physical space satisfying \(QA_w=QA_wQ\) for every product \(A_w\) of \(p\) letters of \(\widetilde M\) and \(Q_1H^{(N)}\ne H^{(N)}Q_1\) for every length. Then \(Q\) is a periodic-sector projector for \(M\) with exponent \(p\).

Say that periodic vectors of \(M\) supply cyclic projectors if for every nontrivial \(p\)-periodic vector of the vertically viewed tensor \(\widetilde M\) — presented as in Definition 9.7.3 — there are an exponent \(q\ge 1\) and an orthogonal projector \(Q\) on the physical space satisfying the one-sided invariance \(QA_w=QA_wQ\) for every product \(A_w\) of \(q\) letters of \(\widetilde M\), together with \(Q_1H^{(N)}\ne H^{(N)}Q_1\) for every length. This is the shape of the cyclic decomposition of the peripheral spectrum [ Wol12 , Theorem 6.6 ] applied to the vertical transfer map: each cyclic projection of a period-\(p\) sector returns to itself after \(p\) applications of the map, while its nontrivial cyclic displacement under one application prevents commutation with \(H^{(N)}\). Compared with Definition 23.4.37, the commutation of \(Q\) with \([H^{(N)}]^p\) is no longer assumed; it is derived through the stacked-layers identity.

Say that periodic vectors of \(M\) supply projectors if for every nontrivial \(p\)-periodic vector of the vertically viewed tensor \(\widetilde M\) — an irreducible restriction \(B\) of \(\widetilde M\) to a minimal invariant subspace whose transfer map has a peripheral eigenvalue different from its spectral radius, as in Definition 9.7.3 — there is an exponent \(q\ge 1\) for which \(M\) admits a periodic-sector projector. The proof of [ CPGSV16 , Proposition 4.13 ] , lines 1889–1891, asserts this implication from the general theory of the peripheral spectrum of completely positive maps, with \(q\) equal to the period \(p\) of the vector; only the existence of some exponent enters the exclusion argument. The projector, its word invariance, and its single-letter displacement are constructed from the cyclic decomposition of the peripheral spectrum (Theorem 23.4.45); the passage from the displacement to the all-length non-commutation family is not needed for the periodic-sector contradiction below.

Transporting \(\mathcal S\) through the canonical three-site and two-site coordinate identifications gives a channel \(\widehat{\mathcal S}:M_{q^3}(\mathbb {C})\longrightarrow M_{q^2}(\mathbb {C})\).

Define

Here the first application acts on the first three sites and leaves the fourth site unchanged. The superscript again denotes successive overlapping applications.

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 \(N\).

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

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 ] .

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 \(z_k\in \mathbb {C}\) satisfy \(|z_k|=1\) for every sector. Replacing

leaves every physical-slice factorization unchanged.

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 \(\mathcal S_1\).

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 ] .

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.

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 ] .

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 ] .

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 ] .

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.

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.

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) ] .

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 ] .

A positive blocked \(\chi \) trace-power witness consists of a blocked \(\chi \) family, positivity of all its diagonal entries, and the positive-length trace-power identity for the blocked multiplication coefficients. This is the blocked-basis analog of the positive diagonal matrices in [ CPGSV16 , Theorem 4.14(ii) ] , without requiring a single family indexed only by the BNT labels.

A positive BNT-label \(\chi \) witness consists of a length-independent diagonal \(\chi _{\alpha ,\beta ,\gamma }\)-family, positivity of every diagonal entry, and the positive-length trace-power identity for the BNT-label coefficients. This is the coefficient statement of [ CPGSV16 , Theorem 4.14(ii) ] . If the coefficient family is chosen canonically from the same \(\chi \)-family, positivity of the diagonal entries alone gives such a witness.

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 ] .