Dependencies

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 \(w\) of positive length, then \(C_j^s=E_j^s\) for every representative \(j\) and every marked index \(s\).

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 \(w\) of positive length, 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 ] .

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 ] .

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

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 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 8.6.1), 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 9.1.1 only in the chosen normality predicate. Theorem 9.2.5 proves the implication between the two forms.

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 \(D_k\), nonzero 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:

Every weight \(\mu _k\) is nonzero. This is the reading of [ CPGSV16 , Section 2.3 ] adopted here: the remark in line 219 there places the degenerate case in the block dimensions (\(D_k=0\)), and lines 224–225 choose each \(\mu _k\) so that the transfer map of \(\mu _kA_k\) has spectral radius one, which a vanishing weight cannot satisfy. The reading loses nothing, since if \(\mu _{k_0}=0\) then \(\mu _{k_0}A_{k_0}^i=0\) is the zero matrix of size \(D_{k_0}\) and

so \(A^i\) is the same coisometric image of the direct sum over the remaining indices, with the \(D_{k_0}\) coordinates of the omitted summand moved to the zero complement. No ordering or separation condition is imposed.

The basis-of-normal-tensors refinement is Definition 9.1.1; it is not part of canonical form itself.

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

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, (5) 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.

An MPO tensor \(M\) has the grouped-corner Gram-dressing property if the following holds. Let \(A\) be a tensor, let \(V_X,V_Y\) be two physical maps, let \(X,Y\) be invertible matrices, and let \(c_X,c_Y{\gt}0\). Suppose that, for every vertical letter \(v\), the following two corner identities hold:

Then, for every vertical letter \(v\),

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

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

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

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

\( = \)

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 ] .

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

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

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 17.4.49 or Theorem 17.4.50; each uses a single noncommuting length supplied by the contrapositive of Lemma L.

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 8.7.3 — 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 17.4.39, 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 8.7.3 — 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 17.4.47); the passage from the displacement to the all-length non-commutation family is not needed for the periodic-sector contradiction below.

A matrix \(P\in M_{d}(\mathbb {C})\) is a sector projector for the weight \(\mu \in \mathbb {C}\) and the matrix \(E_\alpha \in M_{D}(\mathbb {C})\) if \(P\) is an orthogonal projector and inserting \(P\) into the single-site vertical loop selects the sector contribution, \(E_P=\mu E_\alpha \). In the vertical decomposition

of the vertically viewed tensor, carried by the physical space in the proof of [ CPGSV16 , Proposition 4.13 ] , the projector \(P_{\alpha ,k}\) selecting the sector \((\alpha ,k)\) has this property with \(\mu =\mu _{\alpha ,k}\) and \((E_\alpha )_{ab}=\operatorname{tr}(M_\alpha ^{(a,b)})\), independently of \(k\). Indeed, cyclicity of trace cancels the internal similarity \(X_{\alpha ,k}M_\alpha ^{(a,b)}X_{\alpha ,k}^{-1}\). Applied to the physical isometry of a grouped normal sector, the gauge-corner identity constructs the projector and proves its loop identity.

Let \(P\in M_{d}(\mathbb {C})\) and write doubled indices as \(p=(a,b)\) and \(q=(c,e)\). The first-site matrices corresponding respectively to \(PH^{(N)}\), \(H^{(N)}P\), and \(PH^{(N)}P\) are

Closing the vertical indices of a single site of an MPO tensor \(M\) gives the matrix \(E=\sum _i M^{ii}\in M_{D}(\mathbb {C})\), and inserting a matrix \(P\in M_{d}(\mathbb {C})\) into the loop gives \(E_P=\sum _{i,j}P_{ji}M^{ij}\), the closed loop of the vertically viewed product \(PM\), equivalently of \(MP\) by cyclicity of the single-site trace; inserting the identity recovers \(E\). On a chain of length \(N+1\), the compression of the density operator by \(P\) at the first site is \(P_1H^{(N+1)}P_1\) with \(P_1=P\otimes \mathbb {1}^{\otimes N}\), and for a matrix \(E_\alpha \in M_{D}(\mathbb {C})\) the chain trace with one marked loop is \(d_\alpha ^{(N+1)}=\operatorname{tr}(E_\alpha E^N)\). For a representative tensor \(M_\alpha \) whose letters are indexed by horizontal bond pairs, its closed vertical loop is defined entrywise by \((E_\alpha )_{ab}=\operatorname{tr}(M_\alpha ^{(a,b)})\). In this case the chain trace is the scalar \(d_\alpha \) displayed in the proof of [ CPGSV16 , Proposition 4.13 ] , resolved by chain length: one loop of \(M_\alpha \) followed by loops of \(M\), closed by the horizontal trace.

View the MPO tensor \(M\) vertically as the family of physical-space operators \(M_{ab}\) of Definition 17.4.4, indexed by the virtual indices, with matrix elements \((M_{ab})_{ij}=M^{ij}_{ab}\). A matrix \(P\in M_{d}(\mathbb {C})\) then acts on the tensor by

and on an \((N+1)\)-site chain by the first-spin operator \(P\otimes \mathbb {1}^{\otimes N}\). On the letters of the vertically viewed tensor these actions are the left and right matrix products, \((PM)_{ab}=PM_{ab}\) and \((MP)_{ab}=M_{ab}P\). The first-spin operators compose site by site, \(P_1Q_1=(PQ)_1\).

The adjoint tensor \(M^\dagger \) of an MPO tensor \(M\) exchanges the ket and bra physical indices and conjugate-transposes each virtual matrix: \((M^\dagger )^{ij}=(M^{ji})^\dagger \). A tensor equals its adjoint tensor, \(M^\dagger =M\), precisely when it is Hermitian.

For words \(u=(i_1,\ldots ,i_N)\) and \(v=(j_1,\ldots ,j_N)\) in \(\{ 0,\ldots ,d{-}1\} ^N\), the word evaluation is

The empty pair of words evaluates to the identity, \(M^{\varnothing ,\varnothing }=\mathbb {1}_D\), and word pairs of different lengths evaluate to \(0\).

The operator generated by \(M\) on \(N\) sites is the matrix \(\rho ^{(N)}(M)\) indexed by configurations \(\sigma ,\tau \in \{ 0,\ldots ,d{-}1\} ^N\) and given by

An MPO tensor is Hermitian if \(M^{ij}=(M^{ji})^\dagger \) for all \(i,j\in \{ 0,\ldots ,d{-}1\} \).

The cyclic shift of configurations on \(N\) sites is the bijection that relabels each site by one position, \(\sigma \mapsto \sigma \circ r\), where \(r\) is the cyclic rotation \(i\mapsto i+1\) of \(\{ 0,\ldots ,N-1\} \).

Placing one MPO layer on top of another and contracting the shared physical index gives the layer product of two MPO tensors \(M\) and \(M'\) with the same physical dimension, \((MM')^{ij}=\sum _kM^{ik}\otimes M'^{kj}\), an MPO tensor on the tensor product of the virtual spaces. The \(p\)-fold vertical stack \(M^{[p]}\) of \(M\) is the iterated layer product of \(p\) copies of \(M\); the empty stack \(M^{[0]}\) is the trivial-bond tensor with letters \(\delta _{ij}\), which generates the identity operator at every system size.

An MPO tensor is a family of matrices \(\{ M^{ij}\} _{i,j=0}^{d-1}\) with \(M^{ij} \in M_{D}(\mathbb {C})\), indexed by a ket index \(i\) and a bra index \(j\). Diagrammatically,

By combining the pair \((i,j)\) into a single physical index in \(\{ 0,\ldots ,d^2-1\} \), one obtains an MPS tensor with physical dimension \(d^2\) and bond dimension \(D\), with \(M^{\mathrm{MPS}}_{(i,j)}=M^{ij}\), identifying \(\{ 0,\ldots ,d{-}1\} \times \{ 0,\ldots ,d{-}1\} \) with \(\{ 0,\ldots ,d^2{-}1\} \) via the product encoding.

The transfer map associated to \(M\) is the linear map \(\mathcal{E}_M:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) defined by

For a finite family of blocks, put \(j \sim k\) when there is a nonzero scalar factor \(\zeta \) such that \(|V^{(N)}(B_k)\rangle =\zeta ^N|V^{(N)}(B_j)\rangle \) for every positive length \(N\). The associated tuple enumerates the finite quotient, chooses one representative per class, enumerates that representative first in its class, records the scalar relating the representative to each member, and proves that distinct representatives are not MPV-phase equivalent. The pairs \((j,q)\), with \(j\) a phase class and \(q\) a copy in that class, are in bijection with the original block indices.

An MPS tensor \(A\) has no nontrivial \(p\)-periodic vectors if, for every isometry \(V\) with \(V^\dagger V=\mathbb {1}\) and \(A^iV=VB^i\) for all \(i\), with \(B\) irreducible (equivalently, for every restriction \(B\) of \(A\) to a minimal invariant subspace), and for every \(\rho {\gt}0\) and \(r{\gt}0\) with \(\mathcal{E}_B(\rho )=r\rho \), every eigenvalue of \(\mathcal{E}_B\) of modulus \(r\) equals \(r\). The eigenvalue \(r\) is the spectral radius of \(\mathcal{E}_B\) (Theorem  [ LTC26 , “Spectral radius is the Perron eigenvalue” ] ), so the condition states that no block of the invariant-subspace decomposition of  [ CPGSV16 , Section 2 ] , rescaled to spectral radius one, has a peripheral eigenvalue \(e^{2\pi iq/p}\neq 1\). Eigenvectors with such eigenvalues are the \(p\)-periodic vectors of  [ CPGSV16 , Section 2 ] .

A tensor \(A\) is normal [ CPGSV21 ] if there exists a positive integer \(L_0\) such that \(A\) is \(L_0\)-block injective.

After identifying the bond space of a copy with that of its representative, define its normalized physical map by

A tensor \(A\) of physical dimension \(d\) and bond dimension \(D\) is a normal tensor if, after the spectral-radius normalization of  [ CPGSV16 ] , (i) \(A\) admits no nontrivial invariant orthogonal projection and (ii) the associated CPM \(\mathcal{E}_A(X)=\sum _i A^iX(A^i)^\dagger \) has spectral radius \(r(\mathcal{E}_A)=1\) and peripheral spectrum \(\sigma _\partial (\mathcal{E}_A)=\{ 1\} \).

Let \((A_j)_{j=0}^{g-1}\) be the basis representatives of a sector decomposition. They have representative word-tuple separation at length \(L\) when

They have eventual representative word-tuple separation when this equality holds at every sufficiently large length. Notice that the family is indexed by the minimal representatives \(j\), and not by the repeated copies lying over each \(j\).

For tensors \(A\) and \(B\) of possibly different bond dimensions, we write \(\mathcal{V}^+(A) = \mathcal{V}^+(B)\) if \(|V^{(N)}(A)\rangle = |V^{(N)}(B)\rangle \) for every \(N {\gt} 0\).

Given a sector decomposition \(P\), these conditions define the normalized, left-canonical, aperiodic BNT surface used below. This is stronger than the abstract BNT of Definition 9.1.1 and the literal coefficient-form BNT of Definition 9.1.2: besides spanning and eventual linear independence, it fixes irreducible left-canonical representatives, their self-overlap normalization, gauge-phase separation, and the canonical-form weight normalization. It imposes no equal-modulus factorization or strict ordering on the raw weights. Explicitly, each basis block has positive bond dimension, is irreducible, and is left-canonical; the normalized self-overlap of every basis block converges to \(1\); the basis MPV family is eventually linearly independent; distinct same-dimension basis blocks are not gauge-phase equivalent after identifying equal bond dimensions; and the [ CPGSV16 , Section 2.3 ] canonical-form normalization holds,

The MPV expansion takes the raw two-layer form

matching [ CPGSV16 , Section 2.3 ] and [ CPGSV21 , Definition IV.2 ] . The canonical-form normalization admits every example of the source paper (in particular \(C \oplus D\), \(C \oplus (-C)\), \(C \oplus (1/2) C\), and \(C \oplus (-C) \oplus (1/2) C\)) and is not derivable from the per-block normality hypotheses alone.

Let \(P\) be a sector decomposition with representatives \(A_j\), multiplicities \(r_j\), and copy weights \(\mu _{j,q}\). Its physical \(p\)-site blocking has the same representative and copy indices, with

The two-layer decomposition is [ CPGSV16 , lines 283–301 ] . Blocking preserves the copy index sets and bond dimensions, replaces each flattened representative by its \(p\)-site blocking, and raises each flattened weight to the \(p\)-th power.

A sector-weight tuple over \(g\) basis blocks consists of multiplicities \(r_j {\gt} 0\), nonzero weights \(\mu _{j,q}\) for \(q \in \{ 1,\ldots ,r_j\} \), and the sector coefficients

A sector decomposition consists of the basis blocks \((A_j)_{j=1}^g\) together with these multiplicities and weights. If \(A_j\) has bond dimension \(D_j\), then the flattened copy family has total bond dimension

The transfer map associated to a tensor \(A\) is the linear map \(\mathcal{E}_A : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) defined by

Diagrammatically, the transfer map is the double-layer contraction

in which the upper node denotes \(A\), the lower node denotes \(A^\dagger \), and the physical index is summed over between them.

An MPO tensor \(M\) has a grouped vertical BNT decomposition with physical isometries if there are positive bond dimensions \(d_k\), positive weights \(\mu _k\), normal tensors \(B_k\), and isometries \(V_k\colon \mathbb {C}^{d_k}\to \mathbb {C}^d\) with pairwise orthogonal ranges such that

Let \(k=(\alpha ,q)\) be the phase-class enumeration. There are equalities of the representative and copy bond dimensions, invertible gauges \(X_{\alpha ,q}\), and nonzero phases \(\zeta _{\alpha ,q}\) of unit modulus, with \(X_{\alpha ,0}=\mathbb {1}\) and \(\zeta _{\alpha ,0}=1\), such that

The representatives \(M_\alpha \) form a basis of normal tensors for the weighted block tensor and are pairwise inequivalent up to gauge and phase. Every effective coefficient \(c_{\alpha ,q}=\mu _{\alpha ,q}\zeta _{\alpha ,q}\) is nonzero and positive.

For \(P_{\alpha ,q}=V_{\alpha ,q}V_{\alpha ,q}^\dagger \), the triple consisting of \(P_{\alpha ,q}\), \(c_{\alpha ,q}\), and the representative loop of \(M_\alpha \) satisfies the sector-projector identity of Definition 17.4.60. Moreover, for every \((\alpha ,q)\) there is a finite chain length for which the first-site compression by \(P_{\alpha ,q}\) is nonzero. Finally, the grouped physical maps remain isometries with pairwise orthogonal ranges, satisfy both grouped intertwining identities and the grouped corner identity, and reconstruct each vertical letter exactly:

An MPO tensor is in vertical canonical form if there are a basis of normal tensors \(\{ M_\alpha \} _\alpha \) for its vertically viewed tensor \(\widetilde M\), positive diagonal matrices \(\mu _\alpha =\operatorname{diag}(\omega _{\alpha ,0},\ldots ,\omega _{\alpha ,r_\alpha -1})\) with \(r_\alpha \geq 1\), and a matrix \(U\) from the physical space onto the retained nonzero sector space such that

This is the conclusion of [ CPGSV16 , Proposition 4.13, lines 1863–1870 ] . In this orientation \(UU^\dagger =I\) on the retained space, while \(U^\dagger U\) is its support projection in the original physical space. The reconstruction identity says that every letter of \(\widetilde M\) is supported on this projection, so the discarded orthogonal complement and the off-diagonal corners vanish. The orthogonal complement may be a zero sector: the source’s general canonical-form construction explicitly permits the sum of the nonzero block dimensions to be strictly smaller than the original dimension [ CPGSV16 , lines 214–225 ] . Since each \(\mu _\alpha \) is diagonal, the summand \(\mu _\alpha \otimes M_\alpha \) is the direct sum \(\bigoplus _{k=0}^{r_\alpha -1}\omega _{\alpha ,k}M_\alpha \) of weighted copies of \(M_\alpha \), so the right-hand side is a block-diagonal tensor over the pairs \((\alpha ,k)\) in which the multiplicities \(r_\alpha \) and the diagonal entries \(\omega _{\alpha ,k}\) stay explicit; the coefficients \(m_\alpha =\operatorname{tr}(\mu _\alpha )=\sum _{k=0}^{r_\alpha -1}\omega _{\alpha ,k}\) appearing in the renormalization fixed-point characterization are recovered from them.

Let \(W_k:\mathbb {C}^{D_k}\to \mathbb {C}^d\) be a finite family of rectangular matrices. Their block-row map \(U:\mathbb {C}^d\to \bigoplus _k\mathbb {C}^{D_k}\) is defined by \(Ux=(W_k^\dagger x)_k\). This realizes the coisometry in the conclusion of [ CPGSV16 , Proposition 4.13, lines 1863–1870 ] once the sector isometries have been normalized.

The vertical direction of an MPO tensor exchanges the notion of physical and virtual indices [ CPGSV16 , line 943 ] : the vertically viewed tensor \(\widetilde M\) has its letters indexed by the horizontal bond pair \((a,b)\), and each letter is the operator on the physical space with matrix elements \((\widetilde M_{ab})_{ij}=M^{ij}_{ab}\). Concatenating \(\widetilde M\) vertically generates the matrix product vectors of the vertical direction.

If every ordered pair of distinct blocks admits a common-length pairwise block-separating equation, then for each \(k\) there are coefficients \(c_\omega ^{(k)}\) with

Hence common block injectivity plus pairwise separation implies the finite-length blockwise physical–virtual correspondence of the block-injectivity proposition in [ CPGSV16 , Section 2.3 ] and the block-injective canonical-form trace-separation condition.

Let \(\bigoplus _k \mu _k A_k\) be a block-diagonal tensor whose blocks are injective, left-canonical, weighted by nonzero scalars, and jointly block-injective in the sense that for some blocking length \(L\) the tuple of trace pairings against length-\(L\) block words is faithful across all blocks simultaneously. For \(Y\in M_{d}(\mathbb {C})\), write

for the physical-index insertion of \(Y\) into block \(A_k\), and for a configuration \(\sigma =(i_1,\ldots ,i_N)\) write

for \(Y\) acting on the first physical index of the MPV. If \(Y,Z\in M_{d}(\mathbb {C})\) satisfy \(Y_1V^{(N)}(A)=Z_1V^{(N)}(A)\) for every \(N\geq 1\), then \(YA_k=ZA_k\) for every block \(k\).

Scope restriction (per-block separation). The source’s Lemma L separates blockwise contributions at the level of a minimal basis of normal tensors, grouping gauge-equivalent copies of one representative before the trace-separation step; the block-injective hypothesis stated here instead demands trace separation simultaneously across every block, including gauge-equivalent copies, and so excludes decompositions with repeated normal tensors. Documented in [ con26u ] .

Let \((A_j)_{j=1}^g\) be irreducible left-canonical tensors with normalized self-overlaps and with no gauge-phase-equivalent distinct blocks. Suppose \(g\geq 2\), \(L_0{\gt}0\), and every \(A_j\) is block-injective at lengths \(L_0\), \(L_0+1\), and \(3(L_0+1)\). Then

This is the direct-sum induction length in [ PGVWC07 , lines 1346–1408 ] .

Let \(A\) and \(A'\) have the same matrix product vectors at every positive length. If \((A_j)_{j=1}^{g}\) is a basis of normal tensors for \(A\), then the same family is a basis of normal tensors for \(A'\).

In the setting of Theorem D.3.5, every transfer eigenvalue \(\mu \) of \(B\) becomes the transfer eigenvalue \(\mu /r\) of the rescaled unital gauge: the gauge conjugates the transfer map by the congruence with \(\rho ^{1/2}\) and divides it by \(r\). In particular, the peripheral eigenvalues \(e^{2\pi iq/p}\) described in [ CPGSV16 , Section 2.3 ] arise from the transfer eigenvalues of modulus \(r\).

For any word \(w\), the matrix product of the diagonal tensor along \(w\) equals the operator word evaluation with equal ket and bra words, since both multiply the matrices \(M^{w_kw_k}\) site by site.

This auxiliary statement supplies the identity needed to combine the block-injectivity argument [ CPGSV16 , lines 318–345 ] with Appendix C.3, Lemma L [ CPGSV16 , lines 1835–1858 ] .

Let \(A\) be a tensor, and let \(Y^{[L+1]}\) be the physical-block action induced by \(Y\in M_{d}(\mathbb {C})\). For \(I=(i_0,\ldots ,i_L)\), its insertion into the blocked tensor satisfies

Hence equality of the first-site actions of \(Y,Z\) on the MPV family of \(A\) implies equality of the first-block actions of \(Y^{[L+1]},Z^{[L+1]}\) on the MPV family of \(A^{[L+1]}\).

The first-site actions of \(Y,Z\) agree at every length if and only if

for every physical index \(i\) and every finite word \(w\).

Conversely, suppose that \(A\) is \(L\)-block injective. If the two induced insertion tensors on \(A^{[L+1]}\) are equal, then \(YA=ZA\).

Let \(B\) be an irreducible MPS tensor of positive bond dimension with \(\mathcal{E}_B(\rho )=r\rho \) for some \(\rho {\gt}0\) and \(r{\gt}0\), and suppose every eigenvalue of \(\mathcal{E}_B\) of modulus \(r\) equals \(r\). Then \(r^{-1/2}B\) is a normal tensor.

Let \(B\) be an irreducible MPS tensor of positive bond dimension, not all of whose matrices vanish. Then there are \(\rho {\gt}0\) and \(r{\gt}0\) with \(\mathcal{E}_B(\rho )=r\rho \).

Let \(P_0,\ldots ,P_{m-1}\) be orthogonal projections summing to the identity and cyclically permuted by a Kraus map, \(\sum _vK_vP_{k+1}K_v^\dagger =P_k\). Then the projections are mutually orthogonal, none of them vanishes, and each Kraus operator intertwines consecutive projections:

For every marked index \(s\) and every unmarked word \(w\),

The exponent \(|w|+1\) includes the weight at the marked site.

Let \(\widehat M\) be the doubled-index tensor associated with \(M\). Then insertion of \(Y_L\), \(Y_R\), and \(Y_C\) gives, respectively, the doubled-index tensors associated with \(P\widetilde M\), \(\widetilde MP\), and \(P\widetilde MP\).

For ket and bra configurations \(\sigma ,\tau \), \(V^{(N)}(\widehat M)_{(\sigma ,\tau )} =H^{(N)}(M)_{\sigma ,\tau }\). If \(p_0=(a,b)\) is the first doubled index and \(\sigma ',\tau '\) are the ket and bra configurations on the remaining \(N\) sites, contraction with \(Y_L\), \(Y_R\), and \(Y_C\) gives, respectively,

Summing the diagonal word evaluations of an MPO tensor over all length-\(N\) configurations closes the vertical loops site by site,

and hence \(\operatorname{tr}H^{(N)}=\operatorname{tr}(E^N)\).

Under the hypotheses of Theorem 17.4.100, the retained vertical decomposition has at least one sector and hence at least one phase class.

For words \(u_1,v_1\) of equal length and arbitrary words \(u_2,v_2\), word evaluation factors as \(M^{u_1u_2,\, v_1v_2}=M^{u_1,v_1}M^{u_2,v_2}\).

Let \(a,b\in \{ 0,\ldots ,d{-}1\} \), and let \(u,v\) be words of equal length. Writing \(au,bv\) for adding the letters at the beginning and \(ua,vb\) for adding them at the end, one has \(\operatorname{tr}\! \left(M^{au,\, bv}\right)=\operatorname{tr}\! \left(M^{ua,\, vb}\right)\).

The transfer map of an MPO tensor agrees with the transfer map of its doubled-index MPS view: \(\mathcal{E}_M=\mathcal{E}_{M^{\mathrm{MPS}}}\).

For an MPO tensor \(M\) and a configuration \(\sigma \), the matrix product vector of the diagonal tensor \(i\mapsto M^{ii}\) equals the matrix product vector of the doubled-index tensor \(\widehat{M}\) evaluated on the diagonal-paired configuration \(k\mapsto (\sigma _k,\sigma _k)\): \(V^{(N)}(i\mapsto M^{ii})(\sigma ) =V^{(N)}(\widehat{M})(k\mapsto (\sigma _k,\sigma _k))\).

Suppose the doubled-index tensor \(\widehat{M}\) shares its positive-length matrix product vector family with a block-diagonal direct sum \(\bigoplus _k\mu _kA_k\) of blocks \(A_k\) on the doubled index, as in a horizontal canonical form. Then the diagonal tensor of \(M\) generates the same positive-length matrix product vector family as the block-diagonal direct sum \(\bigoplus _k\mu _kA_k^{\mathrm{diag}}\) on the single-spin index, where \(A_k^{\mathrm{diag}}(i)=A_k(i,i)\) is the diagonal restriction of each block. For every \(N\geq 1\),

For an MPO tensor \(M\) and a configuration \(\sigma \), the matrix product vector of the diagonal tensor equals the diagonal entry of the generated operator:

Both sides of (64) equal \(\operatorname{tr}(M^{\sigma _1\sigma _1}\cdots M^{\sigma _N\sigma _N})\).

If \(\rho ^{(N)}(M)\) is positive semidefinite, then \(V^{(N)}(i\mapsto M^{ii})(\sigma )\geq 0\) for every configuration \(\sigma \).

A nonempty finite family of blocks has at least one MPV phase class.

Let \(g\) be a block count, \(\dim ,\operatorname{mult}:\{ 0,\ldots ,g-1\} \to \mathbb {N}\), weights \(\omega _{\alpha j}\in \mathbb {C}\) for \(0\leq j{\lt}\operatorname{mult}_\alpha \), and blocks \(A_\alpha \) of bond dimension \(\dim _\alpha \). The vertical direct-sum tensor is the block-diagonal direct sum over the flattened index \((\alpha ,j)\) that repeats \(A_\alpha \) with weight \(\omega _{\alpha j}\), written \(\bigoplus _\alpha \bigoplus _j\omega _{\alpha j}A_\alpha \). Its matrix product vector decomposes as

For a sector decomposition \(S\) with associated tensor \(A_{\mathrm{tot}}\), the MPV satisfies

Let \(J\) be a finite set of blocks, and let \((A_j)_{j\in J}\) be a family of tensors such that every \(A_j\) is \(L\)-block injective. Suppose that for each block \(k\) there are coefficients \(c_\omega ^{(k)}\), indexed by words of length \(S\), such that

Then the simultaneous length-\((L+S)\) block-word evaluations span \(\prod _{j\in J} M_{D_j}(\mathbb {C})\).

Under the BNT canonical form, the [ CPGSV16 , Section 2.3 ] normalization provides indices \((j_*,q_*)\) with \(|\mu _{j_*,q_*}|=1\).

Let \((A_j)_{j=0}^{g-1}\) be a family with \(g=1\). If its unique block is \(L\)-block injective, then the simultaneous length-\(L\) word evaluations span the full one-block product algebra.

The support projection \(\pi \) of a positive semidefinite matrix \(\rho \) factors as \(\pi =\rho W\) for a spectral pseudo-inverse \(W\), witnessing that the range of \(\pi \) is the range of \(\rho \). For a rectangular matrix \(Y\), the support projection \(\pi \) of \(YY^\dagger \) fixes \(Y\), \(\pi Y=Y\); and whenever \(AY=YG\) for some matrix \(G\)—that is, \(A\) maps the range of \(Y\) into itself—the one-sided invariance \((\mathbb {1}-\pi )A\pi =0\) holds.

Let \((B_k)_{k\in I}\) be a finite family of blocks with weights \(\nu _k\), and let \((A_\alpha )_{\alpha \in J}\) be a finite family of representative blocks. Suppose there is a bijection

such that, writing \(e(k)=(\alpha ,j)\), one has \(\nu _k=\omega _{\alpha j}\) and, for every positive length \(N\), \(V^{(N)}(B_k)=V^{(N)}(A_\alpha )\). Then

for every positive length \(N\). This is the finite-sum reindexing after the vertical sectors have been grouped in [ CPGSV16 , Proposition 4.13 ] .

Adjunction exchanges the two oriented horizontal bond indices in the vertical view:

Let \((A_k)_{k}\) be a finite family of block tensors of physical dimension \(d\), and fix a length \(L\); write \(A_k^w\) for the length-\(L\) word evaluation of \(A_k\) along a word \(w \in \{ 0,\ldots ,d{-}1\} ^L\). The finite-length spanning hypothesis is that the tuple-valued evaluations span the full product algebra of block matrices,

The corresponding finite-length separation property is that some length \(L\) makes the trace pairing against block words nondegenerate: there is a length \(L\) such that, for every tuple \((\Delta _k)_k\) of block matrices,

Spanning at length \(L\) gives separation at the same length: a trace functional that vanishes on every tuple \((A_k^w)_k\) vanishes on their span, hence on the whole product algebra, so nondegeneracy of the product trace pairing forces every \(\Delta _k=0\). Combined with injectivity, left-canonicality, and nonzero block weights, this finite-length separation supplies the older per-copy trace-separation hypotheses for the block-diagonal tensor.

The spanning hypothesis has an equivalent entrywise form. For a block index \(k\) and matrix entry \((a,b)\), let \(w\longmapsto (A_k^w)_{a,b}\) be the corresponding scalar word family, indexed jointly by the triple \((k,a,b)\). Linear independence of this scalar family already implies the spanning condition, hence the separation property, and hence the older per-copy trace-separation hypotheses directly.

The block-injectivity proposition of [ CPGSV16 , lines 340–345 ] gives a concrete sufficient criterion for the spanning condition: a common length \(L\) with every block \(L\)-block-injective, together with a length \(S\) at which, for every \(k\), there is a coefficient family \((c_w)_{w\in \{ 0,\ldots ,d{-}1\} ^S}\) with

Concatenating the injective prefix with this finite selector family produces the spanning condition at length \(L+S\), hence the separation property and the older per-copy trace-separation hypotheses.

Let \(M\) generate an MPDO whose doubled-index tensor \(\widehat M\) generates the same positive-length MPV family as a BNT canonical form \(\bigoplus _{j,q}\mu _{j,q}A_j\), and let \(P=P^\dagger \) satisfy \(PM=PMP\) on the vertically viewed tensor. Then on every minimal representative \(A_j\) the insertions of \(MP\) and \(PMP\) agree; equivalently, \((\mathbb {1}-P)MP=0\) on every representative.

Let \(\bigoplus _{j,q}\mu _{j,q}A_j\) be a BNT canonical form and let \(Y_L,Y_C,Y_R\in M_{d}(\mathbb {C})\), with \(Y_1V^{(N)}(A)\) denoting \(Y\) acting on the first physical index of the MPV. If, for every \(N\geq 1\),

then \(Y_RA_j=Y_CA_j\) for every minimal representative \(A_j\). This is the representative-grouped application of Lemma L and anticipates the concrete instantiation with \(Y_L,Y_C,Y_R\) the physical-index insertions of \(P\), \(PHP\), \(HP\) in Theorem 17.4.31.

Let the doubled-index tensor \(\widehat M\) generate the same positive-length MPV family as a BNT canonical form \(\bigoplus _{j,q}\mu _{j,q}A_j\). If, entrywise at every positive length,

then for every minimal representative \(A_j\) the insertions associated with \(H^{(N)}P\) and \(PH^{(N)}P\) agree. Repeated copies over \(A_j\) are grouped through the coefficient \(\sum _q\mu _{j,q}^{N}\).

If \((A_j)_{j=1}^{g}\) is a coefficient-form basis of normal tensors for \(A\), then \((A_j)_{j=1}^{g}\) is an algebraic basis of normal tensors for \(A\). Only this implication from spectral normality to algebraic normality is asserted.

Let

be literal CPSV canonical-form data. There are phase classes \(j\), copy multiplicities \(r_j\), and an equivalence

Each class has a representative \(C_j\). If the copy \((j,q)\) corresponds to the block index \(k\), then its bond dimension equals that of \(C_j\) and there are \(\zeta _k\in \mathbb {C}\), with \(|\zeta _k|=1\), and \(X_k\in \mathrm{GL}_{D_k}(\mathbb {C})\) such that

The copy retains the original coefficient \(\mu _k\), and the representatives \((C_j)_j\) form a basis of normal tensors for \(A\).

Enumerate \(\mathcal G\). The induced equivalence \(\Phi \) between the dependent grouped block coordinates and the original flattened listed coordinates gives

For \(x=(j,q)\), let \(\widehat\mu _x\) be the original weight of that copy and let \(\widehat B_x^i=\zeta _xC_j^i\), with the required dimension identification. Define the grouped tensor by

If \(G=\bigoplus _kX_k\) is the block-diagonal gauge, then, letter by letter,

and

Let \(T_{\mathrm{grp}}\) be the full grouped tensor of Theorem 9.6.1.9, and let \(P_{\mathrm{rep}}\) be its representative sector decomposition. For a marked family \((C_j^s)_j\), define the marked tensor \(\widehat C_{\mathrm{grp}}\) on all grouped listed coordinates as follows. The copy \((j,q)\) carries the marked block \(\lambda _{j,q}C_j^s\), where \(\lambda _{j,q}=\nu _{k(j,q)}\zeta _{k(j,q)}\).

For every marked letter \(s\) and every unmarked word \(w\), one has

The exponent \(|w|+1\) includes the factor at the marked site.

Let \(M\) be an MPO tensor whose doubled-index MPS tensor is in literal CPSV canonical form. If \(Q^2=Q\) and \(Q\widetilde M\ne Q\widetilde M Q\), then there is an integer \(N\geq 0\) such that, for \(Q_1=Q\otimes \mathbb {1}^{\otimes N}\),

Thus a displaced idempotent has a noncommuting chain length. The proof of [ CPGSV16 , Proposition 4.13, lines 1888–1893 ] only requires this existential conclusion, not noncommutation at every length.

Consider the actual phase-class grouping furnished under literal CPSV canonical form. For a class \(\alpha \) and copy \(q\), identify its normal representative with the bond space of that copy using the equality of dimensions. Then for every vertical letter \(v\),

In the actual phase-class grouping under literal CPSV canonical form, every gauge \(X_{\alpha ,q}\) has a positive real Gram scalar \(\omega _{\alpha ,q}\) such that

Consequently \(U_{\alpha ,q}=\omega _{\alpha ,q}^{-1/2}X_{\alpha ,q}\) is unitary. This proves the gauge-normalization step of [ CPGSV16 , Proposition 4.13, lines 1903–1921 ] and supplies the positive Gram scalars used to normalize the physical sector maps.

In the phase-class grouping obtained from literal CPSV canonical form, let \(Q_{\alpha ,q}=\omega _{\alpha ,q}^{-1/2}X_{\alpha ,q}\) and \(W_{\alpha ,q}=V_{\alpha ,q}Q_{\alpha ,q}\), after identifying the bond space of each copy with that of its representative. Then

where \(c_{\alpha ,q}=\nu _{\alpha ,q}\zeta _{\alpha ,q}{\gt}0\). Moreover, every vertical letter is reconstructed exactly:

Let \(M\) be an MPO tensor whose doubled-index MPS tensor is in the literal CPSV canonical form, and suppose that \(M\) generates positive semidefinite operators on every nonempty chain. Let \(A\) be a tensor, let \(X\) and \(Y\) be invertible matrices, and suppose that two vertical corners satisfy

Then the Gram-dressed tensors agree:

for every vertical letter \(v\).

Let \(M\) be an MPO tensor whose doubled-index MPS tensor is in the literal CPSV canonical form, and suppose that \(M\) generates positive semidefinite operators on every nonempty chain. Let \(A\) be normal, let \(X\) and \(Y\) be invertible matrices, and suppose that

Then there is a real number \(\omega {\gt}0\) such that

Let \(M\) be an MPO tensor whose doubled-index MPS tensor is in literal CPSV canonical form, and let \(P\in M_{d}(\mathbb {C})\). If one vertical letter has a nonzero two-sided corner, then some finite-chain first-site compression is nonzero:

where \(P_1=P\otimes \mathbb {1}^{\otimes N}\). This is the separation assertion at [ CPGSV16 , Proposition 4.13, line 1898 ] , obtained from Lemma L in [ CPGSV16 , Appendix C.3, lines 1835–1858 ] . It uses neither positivity of \(H^{(N+1)}\) nor a grouping of the vertical sectors.

Under the same hypotheses, partition the normalized vertical sectors by equality of their matrix product vector families up to a fixed nonzero scalar per site. There are normal representatives \(M_\alpha \), an enumeration \(k=(\alpha ,q)\) of the original sectors, invertible matrices \(X_{\alpha ,q}\), and complex numbers \(\zeta _{\alpha ,q}\) of unit modulus, with \(X_{\alpha ,0}=\mathbb {1}\) and \(\zeta _{\alpha ,0}=1\), such that

The representatives are pairwise gauge-phase distinct and form a basis of normal tensors for the weighted block tensor \(\bigoplus _k\nu _kB_k\). Every grouped coefficient \(\nu _{\alpha ,q}\zeta _{\alpha ,q}\) is positive. With the physical isometries from Theorem 17.4.98 retained unchanged, one has both reducing identities, exact compression, and the literal reconstruction

For each pair \((\alpha ,q)\), the range projector of \(V_{\alpha ,q}\) obeys the representative-loop identity and gives a nonzero compression at some finite length. This is the decomposition and positivity argument of [ CPGSV16 , Proposition 4.13, lines 1898–1902 ] . The relative Gram identity and proportional-unitary normalization of lines 1903–1908 are not asserted here.

Let \(A\) and \(B\) be normal tensors of positive, possibly different, bond dimensions. Suppose that, for some nonzero \(\zeta \in \mathbb {C}\), \(|V^{(N)}(B)\rangle =\zeta ^N|V^{(N)}(A)\rangle \) for \(N\geq 0\). Then the bond dimensions are equal and there is an invertible matrix \(X\) and a scalar \(\eta \) of unit modulus such that \(B^i=\eta X A^iX^{-1}\) for every physical letter \(i\). This is the gauge recovery used in [ CPGSV16 , Appendix A ] .

Let \(A\) be a CPSV normal tensor of positive bond dimension. Then \(A^i\ne 0\) for some physical letter \(i\).

Let \((A_j)_{j=1}^g\) be a finite family of positive-bond-dimension normal tensors, no two of which are gauge-phase equivalent after identifying equal bond dimensions. Then the matrix product vectors \(\{ |V^{(N)}(A_j)\rangle \} _{j=1}^g\) are linearly independent for all sufficiently large \(N\), as in [ CPGSV16 , Appendix A ] .

Let \(A\) be in literal CPSV canonical form. For canonical-form data and its BNT refinement, write

where \(B^i=GT_{\mathrm{grp}}^iG^{-1}\) and \(G\) is the listed block gauge. If marked matrices satisfy \(C=V^\dagger EV\), then for every word \(w\), including the empty word,

The marked matrix makes the closed chain nonempty; no equality of unmarked empty-word matrix product vectors is asserted.

For a linear physical mark \(f\), the linear marking of \(T_{\mathrm{grp}}\) is exactly its full grouped marked tensor. Each copy \((j,q)\) carries \(\lambda _{j,q}f(C_j)\), where \(\lambda _{j,q}=\nu _{k(j,q)}\zeta _{k(j,q)}\). For two linear marks \(f\) and \(g\), put \(F_A^u=\sum _i f_{ui}A^i\) and \(G_A^u=\sum _i g_{ui}A^i\). If

for every marked letter \(u\) and every word \(w\) of positive length, then \(F_A=G_A\) on the original bond space.

Thus the separation conclusion does not require the marked trace identity for the empty tail word. This is the positive-tail form of the argument in [ CPGSV16 , Appendix C.3, lines 1835–1858 ] : the source assumes first-site agreement at every positive system size, while its proof separates the representatives at a suitable nonzero tail length.

In the notation of Theorem 9.6.1.9, let \(k(j,q)\) be the listed block corresponding to the class-copy pair \((j,q)\). The representatives \(C_j\) define a sector decomposition \(P_{\mathrm{rep}}\) with copy weights

Every copy weight in \(P_{\mathrm{rep}}\) is nonzero. The representatives have simultaneous word-tuple separation at every sufficiently large length.

For every positive length \(N\), the grouped tensor \(T_{\mathrm{grp}}\) of Theorem 9.6.1.9 and the representative sector tensor generate the same matrix product vector. The original tensor has the same positive-length matrix product vectors as both:

Hence \(P_{\mathrm{rep}}\) is a BNT sector presentation of \(A\) whenever at least one listed block occurs.

Let \((C_j^s)_j\) and \((E_j^s)_j\) be two marked families on the representatives. Form their full-coordinate marked tensors. If

for every \(s\) and every word \(w\) of positive length, then \(C_j=E_j\) for every representative \(j\).

In particular, let \(Y,Z\in M_{d}(\mathbb {C})\). If their first-site actions agree on the positive-length matrix product vectors of \(T_{\mathrm{grp}}\), then

for every representative \(j\) and every physical letter \(i\).

There is a left-canonical, injective doubled-index tensor with a nonzero block weight satisfying the older per-copy finite-length separation hypothesis whose restriction to the diagonal physical pairs is not normal. On bond space \(\mathbb {C}^2\), take

Then the four matrices \(A^{(a,b)}\) span \(M_{2}(\mathbb {C})\) and \(\sum _{a,b}(A^{(a,b)})^*A^{(a,b)}=\mathbb {1}\). For a single block, injectivity also gives finite-length block separation. The diagonal restriction, however, consists only of

Every product of these two matrices is diagonal, so no blocking length spans \(M_{2}(\mathbb {C})\).

Suppose a phase-class grouping has positive effective coefficients \(c_{\alpha ,q}\), corner identities

and distinguished gauges \(X_{\alpha ,0}=\mathbb {1}\). Fix a class \(\alpha \) and a copy \(q\). After identifying the representative bond space with that of copy \(q\), there are a physical map \(W_{\alpha ,q}\) and a scalar \(c_{\alpha ,0}{\gt}0\) such that, for every vertical letter \(v\),

Let \(M_\alpha \) be a normal tensor, and let \(X\) and \(Y\) be invertible gauges whose Gram matrices induce the same conjugation on each letter \(M_\alpha ^v\). Then there is a positive real number \(\omega \) such that

and \(\omega ^{-1/2}XY^{-1}\) is unitary. Theorem 17.4.70 gives the Figure 8 comparison in [ CPGSV16 , Proposition 4.13, lines 1909–1921 ] . This comparison supplies precisely the relative hypothesis for a sector and the distinguished sector. In the normalization \(Y=\mathbb {1}\), the resulting unitary is \(\omega ^{-1/2}X\). The common-target declarations attached to this entry are conditional algebraic corollaries, not the reflected marked-chain statement itself.

Let \(X\) be invertible and suppose that \(B\) and a target \(C\) satisfy

Then conjugation by the Gram matrix carries \(B\) to \(C\),

Hence any two gauges satisfying this conditional identity with the same target have equal Gram conjugations, and the ratio of their Gram matrices commutes with \(B\). This gives the relative identity from an abstract common target in [ CPGSV16 , Proposition 4.13, lines 1909–1919 ] . The reflected marked-chain argument in the source does not identify the target separately for either sector. The same-letter choice \(C=B^\dagger \) and the further specialization \(B^\dagger =B\) are useful corollaries, but the latter is not assumed by the source.

Suppose a phase-class grouping has positive effective coefficients \(c_{\alpha ,q}\) and corner identities

Assume that the distinguished copy in each class has identity gauge and that \(M\) has the grouped-corner Gram-dressing property. Then for every class \(\alpha \), copy \(q\), and vertical letter \(v\),

Under the hypotheses of Theorem 17.4.73, suppose in addition that every representative \(A_{\alpha ,q}\) is normal and has positive bond dimension. Then for every class \(\alpha \) and copy \(q\) there is a real number \(\omega _{\alpha ,q}{\gt}0\) such that

Let \(H\) and \(P\) be Hermitian matrices. If \(PH=PHP\), then \((\mathbb {1}-P)HP=0\).

Every MPO tensor in normalized BNT-refined horizontal form has the associated positive-length BNT matrix-product-vector representation.

Every LPDO tensor is an MPDO.

Under the hypotheses of the preceding conditional theorem, define

Then \(U_\gamma \) is unitary and, for every physical index \(i\),

More generally, the same conclusion follows from any exact invertible sector gauge whose Gram matrix is a positive scalar multiple of the identity. This is the inverse-square-root normalization used in [ CPGSV16 , Proposition 4.13, lines 1903–1908 ] , applied at Appendix C.4, lines 2053–2057.

Scope restriction (conditional reflected target): The physical-letter coefficients are supplied by Theorem 21.2.3.24, while their common positive-tail reflected target is an assumption of this theorem. Appendix C.4, lines 2048–2057, does not itself spell out the star-compatible comparison for the raw representative. The mixed-prefix theorem below supplies it under the standing assumptions, as recorded in [ con26n ] .

Let \(M\) be an MPO tensor whose doubled-index tensor is injective, and let \(Q\) be an idempotent on the physical space (Hermiticity is not needed). If \(Q_1H^{(2)}=H^{(2)}Q_1\), then \(Q\widetilde M=Q\widetilde MQ\).

Let \(M\) generate matrix product density operators. If displaced projectors of \(M\) never commute, then periodic vectors of \(M\) supply cyclic projectors, and the vertically viewed tensor \(\widetilde M\) has no nontrivial \(p\)-periodic vectors. This narrows the hypothesis of the periodic-sector step in the proof of [ CPGSV16 , Proposition 4.13, lines 1888–1893 ] to the canonical-form implication of lines 1874–1887: every other ingredient of the step is proved.

Let \(M\) generate matrix product density operators. If periodic vectors of \(M\) supply cyclic projectors, then they supply periodic-sector projectors, and the vertically viewed tensor \(\widetilde M\) has no nontrivial \(p\)-periodic vectors. This reduces the hypothesis of the periodic-sector step in the proof of [ CPGSV16 , Proposition 4.13, lines 1888–1893 ] to the cyclic decomposition of the peripheral spectrum of the vertical transfer map.

Let \(M\) be an MPO tensor and let \(\widetilde M\) carry a nontrivial \(p\)-periodic vector, presented as in Definition 8.7.3: an irreducible restriction \(B\) of \(\widetilde M\) to a minimal invariant subspace, reached through an isometry \(V\), with a positive definite transfer eigenvector \(\rho \) of eigenvalue \(r{\gt}0\) and a transfer eigenvalue \(\mu \ne r\) of modulus \(r\). Then there are an exponent \(q\ge 1\) and an orthogonal projector \(Q\) on the physical space such that \(QA_w=QA_wQ\) for every product \(A_w\) of \(q\) letters of \(\widetilde M\), while \(Q\widetilde M\ne Q\widetilde MQ\) in the sense of Definition 17.4.32. This realizes the projector that the proof of [ CPGSV16 , Proposition 4.13, lines 1889–1891 ] draws from the general theory of the peripheral spectrum.

For every matrix \(P\in M_{d}(\mathbb {C})\) and every chain length \(N+1\),

The marked site closes to the loop with \(P\) inserted, and the remaining \(N\) sites close to plain loops. If \(P\) is idempotent, then also \(\operatorname{tr}(P_1H^{(N+1)}P_1)=\operatorname{tr}(E_P E^N)\).

Let \(M\) generate an MPDO and let \(P=P^\dagger \) satisfy \(PM=PMP\) on the vertically viewed tensor. For every total length \(L\geq 1\), write \(P_1=P\otimes \mathbb {1}^{\otimes (L-1)}\). Then \(P_1H^{(L)}=P_1H^{(L)}P_1=H^{(L)}P_1\). This is the equality chain in the proof of [ CPGSV16 , Proposition 4.13 ] ; the proposition assumes \(P\) is an orthogonal projector, but only self-adjointness of \(P\) is used.

Let \(M\) generate positive semidefinite operators on every nonempty chain, and suppose that one vertical corner is a positive scalar multiple of a similarity of a tensor \(A\),

Then the marked-chain coefficient of the Gram-dressed tensor \((X^\dagger X)A^v(X^\dagger X)^{-1}\) equals the reflected-adjoint marked-chain coefficient of \(A\) in the adjoint tensor, with the two tail words reversed. This is the single-corner, open-index conjugation step from Figures 7–8 of [ CPGSV16 , Proposition 4.13, lines 1909–1919 ] .

Let \(M\) be in normalized BNT-refined horizontal form and let \(P\) be a matrix for which \(P\widetilde M_vP\) is nonzero for some vertical letter \(v\). Then the first-site compression \(P_1H^{(N+1)}P_1\) is nonzero for some \(N\). This is the separation step used in [ CPGSV16 , Proposition 4.13, line 1898 ] .

The normalized BNT-refined horizontal hypothesis is stronger than the literal CPSV canonical-form hypothesis. The literal separation statement is Theorem 17.4.61; it does not by itself construct the grouped vertical sectors used here.

Consider the grouped vertical decomposition furnished by Theorem 17.4.100. For a phase class \(\alpha \) and a copy \(q\), transport its representative tensor to the bond space of that copy. If \(X_{\alpha ,q}\) is the corresponding gauge, then for every vertical letter \(v\),

This is Figure 8 for the actual witnesses of the grouped decomposition.

In the grouped vertical decomposition of a matrix-product density operator in normalized BNT-refined horizontal form, let \(X_{\alpha ,k}\) be the gauge of a copy of the normal representative \(M_\alpha \). Fix the distinguished gauge to \(X_{\alpha ,0}=\mathbb {1}\). Then there is a positive real number \(\omega _{\alpha ,k}\) such that

and \(U_{\alpha ,k}\) is unitary. This is the normalization in [ CPGSV16 , Proposition 4.13, lines 1903–1921 ] .

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

Let \(M\) generate matrix product density operators, and suppose that its doubled-index MPS tensor is in literal CPSV canonical form. Then there are positive integers \(d_k\), positive scalars \(\nu _k\), normal tensors \(B_k\) of bond dimension \(d_k\), and isometries \(V_k:\mathbb {C}^{d_k}\to \mathbb {C}^d\) with pairwise orthogonal ranges. For every sector \(k\) and every vertical letter \(v\),

Every vertical letter is reconstructed by

The omitted orthogonal complement is an all-zero sector. This is the normal-corner intermediate between the periodic exclusion at lines 1889–1893 and the grouped vertical representation used at line 1898 of [ CPGSV16 , Proposition 4.13 ] .

Let \(M\) generate matrix product density operators. Then \(M\) admits no periodic-sector projector.

In the grouped vertical decomposition, put

and identify the bond space of each copy with that of its chosen representative. Then

Writing \(c_{\alpha ,q}=\nu _{\alpha ,q}\zeta _{\alpha ,q}{\gt}0\), the normalized maps satisfy

and retain the literal reconstruction

These are the normalized sector identities in [ CPGSV16 , Proposition 4.13, lines 1895–1921 ] .

Let \(H\) be a nonempty-chain density operator generated by an MPDO, and let \(P=P^\dagger \). If \(PH=PHP\), then \((\mathbb {1}-P)HP=0\). This is the operator identity used in the proof of [ CPGSV16 , Proposition 4.13 ] .

Let \(M\) be in normalized BNT-refined horizontal form and generate positive semidefinite operators on every nonempty chain. Suppose two vertical corners are positive scalar multiples of similarities of the same tensor \(A\):

Then the two Gram conjugations agree for every vertical letter,

Normality of \(A\) is not required for this equality; it enters only in the subsequent proportionality of the Gram matrices. This is the pairwise conclusion of Figures 7–8 in [ CPGSV16 , Proposition 4.13, lines 1909–1919 ] .

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

Let \(M\) be an MPO tensor of physical dimension \(d\) and bond dimension \(D\). Let \(A\) be a CPSV normal tensor of positive bond dimension \(n\) and physical dimension \(D^2\), let \(V\in \mathbb {C}^{d\times n}\) be an isometry, let \(X\in \mathbb {C}^{n\times n}\) be invertible, and let \(\omega \) be nonzero. Suppose that, for every vertical letter \(v\),

Then the range projector \(VV^\dagger \) has a nonzero vertical corner:

for some \(v\). This is the algebraic nonvanishing step used in [ CPGSV16 , Proposition 4.13, line 1898 ] .

Let \(M\) generate positive semidefinite operators on every nonempty chain, let \(A\) be a vertical tensor, and suppose that a matrix \(V\) selects the vertical corner \(V^\dagger \widetilde M_vV=cA^v\), where \(c{\gt}0\). For vertical indices \(r,s\), write \((\mathcal{R}(A)^{rs})_{ab}=(A^{(a,b)})_{rs}\), and set \((A^\sharp )^v=(A^{v^{\mathrm{op}}})^\dagger \). Then, for every pair of equal-length tail words \(\sigma ,\tau \),

Thus the adjoint exchanges the oriented horizontal bond pair and reverses the unmarked tail. In particular, this is not a same-tail identity and does not identify \(A\) with \(A^\sharp \) letterwise.

Let \(H^{(N)}\), \(N\geq 1\), be a density operator generated by an MPDO and let \(P=P^\dagger \). Then \(PH^{(N)}P\ge 0\), and if \(PH^{(N)}P\ne 0\) its trace is a positive real. In particular this holds for the first-site compressions \(P_1H^{(N+1)}P_1\). Applied to the sector projectors \(P_{\alpha ,k}\) of the vertical decomposition, this is the positivity \(\operatorname{tr}(P_{\alpha ,k}H^{(N)}P_{\alpha ,k}){\gt}0\) from which the proof of [ CPGSV16 , Proposition 4.13 ] concludes \(\mu _{\alpha ,k}d_\alpha {\gt}0\); the identification of the trace with \(\mu _{\alpha ,k}d_\alpha \) is Theorem 17.4.64.

Let \(P\) be a sector projector for the weight \(\mu \) and the matrix \(E_\alpha \). Then for every chain length \(N+1\),

Once spectral normalization and grouping identify the reducing projectors with sector projectors \(P_{\alpha ,k}\) satisfying Definition 17.4.60, this specializes to the trace identification \(\operatorname{tr}(P_{\alpha ,k}H^{(N)}P_{\alpha ,k})=\mu _{\alpha ,k}d_\alpha \) used in the proof of [ CPGSV16 , Proposition 4.13 ] ; the source states it at the particular length where the compression does not vanish.

Let \(M\) generate an MPDO, let \(Q\) and \(P\) be sector projectors for the weights \(\mu _{\alpha ,0}\) and \(\mu _{\alpha ,k}\) and a common matrix \(E_\alpha \), and suppose \(\mu _{\alpha ,0}{\gt}0\). If the compression of \(H^{(N+1)}\) by \(P\) at the first site is nonzero, then

Granted such sector projectors, this yields the conclusion of the trace argument in the proof of [ CPGSV16 , Proposition 4.13 ] : since w.l.o.g. \(\mu _{\alpha ,0}{\gt}0\), the positive trace forces \(d_\alpha {\gt}0\) and hence \(\mu _{\alpha ,k}{\gt}0\).

Under the same hypotheses, partition the normalized vertical sectors by equality of their matrix product vector families up to a fixed nonzero scalar per site. There are normal representatives \(M_\alpha \), an enumeration \(k=(\alpha ,q)\) of the original sectors, invertible matrices \(X_{\alpha ,q}\), and complex numbers \(\zeta _{\alpha ,q}\) of unit modulus, with \(X_{\alpha ,0}=\mathbb {1}\) and \(\zeta _{\alpha ,0}=1\), such that

The representatives are pairwise gauge-phase distinct and form a basis of normal tensors for the weighted block tensor \(\bigoplus _k\nu _kB_k\). Every grouped coefficient \(\nu _{\alpha ,q}\zeta _{\alpha ,q}\) is positive. With the physical isometries from Theorem 17.4.97 retained unchanged, one has both reducing identities, exact compression, and the literal reconstruction

For each pair \((\alpha ,q)\), the range projector of \(V_{\alpha ,q}\) obeys the representative-loop identity and gives a nonzero compression at some finite length. This is the decomposition and positivity argument of [ CPGSV16 , Proposition 4.13, lines 1898–1902 ] . The relative Gram identity and proportional-unitary normalization of lines 1903–1908 are not asserted here.

Under the same hypotheses, there are orthogonal projections \(P_k\) that are complete and pairwise orthogonal and, for every vertical letter \(v\), reduce the corresponding letter:

Let \(M\) be an MPO tensor and put \(T=\widetilde M\). Let \(B_k\) have bond dimension \(d_k\), let \(\mu _k\in \mathbb {C}\), and let \(V_k:\mathbb {C}^{d_k}\to \mathbb {C}^d\) be an isometry for \(0\leq k{\lt}r\). Assume, for every vertical letter \(v\), that

Partition the tensors \(B_k\) into matrix-product-vector phase classes and choose one representative \(A_\alpha \) from each class. If \(\{ A_\alpha \} _\alpha \) is a basis of normal tensors in the spectral sense for the weighted direct sum \(A_{\mathrm{ret}}=\bigoplus _{k=0}^{r-1}\mu _kB_k\), then \(\{ A_\alpha \} _\alpha \) is an algebraic basis of normal tensors for \(T\).

This is the basis-of-normal-tensors conclusion in [ CPGSV16 , Proposition 4.13, lines 1863–1921 ] , stated for the grouped decomposition before the final coisometry is formed.

Under the hypotheses of Theorem 17.4.94, there are positive integers \(d_k\), tensors \(B_k\) of bond dimension \(d_k\), and isometries \(V_k:\mathbb {C}^{d_k}\to \mathbb {C}^d\) such that

Every \(B_k\) is irreducible, and for every vertical letter \(v\),

Moreover, \(\widetilde M\) and the unit-weight direct sum \(\bigoplus _kB_k\) have the same matrix product vectors at every positive length.

Let \(M\) generate matrix product density operators and suppose that periodic vectors of \(M\) supply projectors. Then the vertically viewed tensor \(\widetilde M\) has no nontrivial \(p\)-periodic vectors. This is the periodic-sector step in the proof of [ CPGSV16 , Proposition 4.13 ] , lines 1888–1893, granted the projector that the source draws from the general theory of the peripheral spectrum.

Let \(M\) be in normalized BNT-refined horizontal form. If an idempotent \(Q\) satisfies \(Q\widetilde M\ne Q\widetilde MQ\), then for some \(N\ge 0\), \(Q_1H^{(N+1)}\ne H^{(N+1)}Q_1\). Consequently, if \(M\) generates matrix product density operators, then \(\widetilde M\) has no nontrivial periodic vectors.

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

Let \(M\) generate matrix product density operators and let \(M\)’s doubled-index tensor be injective. Then the vertically viewed tensor \(\widetilde M\) has no nontrivial periodic vectors, unconditionally.

Let \(M\) generate matrix product density operators, and suppose that its doubled-index MPS tensor is in literal CPSV canonical form. Then the vertically viewed tensor \(\widetilde M\) has no nontrivial periodic vectors.

Under the same hypotheses, there are positive integers \(d_k\), positive scalars \(\nu _k\), normal tensors \(B_k\) of bond dimension \(d_k\), and isometries \(V_k:\mathbb {C}^{d_k}\to \mathbb {C}^d\) with pairwise orthogonal ranges. For every sector \(k\) and every vertical letter \(v\),

Moreover, every vertical letter is reconstructed literally by

Let \(M\) be in normalized BNT-refined horizontal form and generate matrix product density operators. If an orthogonal projection \(P\) satisfies \(P\widetilde M_v=P\widetilde M_vP\) for every vertical letter \(v\), then it also satisfies \(\widetilde M_vP=P\widetilde M_vP\) for every vertical letter \(v\). Thus the vertically viewed tensor has invariant-projector closure.

Let \(M\) generate matrix product density operators, and suppose that its doubled-index MPS tensor is in literal CPSV canonical form. If an orthogonal projection \(P\) satisfies \(P\widetilde M_v=P\widetilde M_vP\) for every vertical letter \(v\), then

for every \(v\). Thus the vertically viewed tensor has invariant-projector closure.

If an MPO tensor \(M\) is in normalized BNT-refined horizontal form, then its vertical reshaping is nonzero: \(\widetilde M\ne 0\). This is the nonvanishing observation used in [ CPGSV16 , Proposition 4.13, lines 1895–1902 ] .

The adjoint tensor generates the adjoint operator family up to spatial reflection: entrywise,

where \(\mathrm{rev}\) reverses the site order. For an MPDO and \(N\geq 1\), the conjugation disappears:

and the adjoint tensor is again an MPDO.

For every system size \(N\), the operator generated by a layer product is the product of the density operators generated by the layers, \(H^{(N)}(MM')=H^{(N)}(M)\, H^{(N)}(M')\). Consequently the \(p\)-fold vertical stack generates \(H^{(N)}(M^{[p]})=[H^{(N)}(M)]^p\) for all \(N\) and \(p\); this is the reading of the powers in the displayed identity of [ CPGSV16 , Proposition 4.13, lines 1890–1891 ] as density operators of stacked layers. In particular, the vertical stack of a tensor generating matrix product density operators again generates matrix product density operators on every nonempty chain.

The vertically viewed letters of a layer product are the products of the layers’ vertically viewed letters, \((MM')_{(a,a'),(b,b')}=M_{ab}M'_{a'b'}\). Hence every vertically viewed letter of the \(p\)-fold stack \(M^{[p]}\) is the evaluation of a length-\(p\) word of \(\widetilde M\). In particular, if a matrix \(Q\) satisfies the one-sided invariance \(QA_w=QA_wQ\) for every product \(A_w\) of \(p\) letters of \(\widetilde M\), then \(QM^{[p]}=QM^{[p]}Q\) in the sense of Definition 17.4.32.

If \(X\ge 0\), then \(\mathcal{E}_M(X)\ge 0\).

The operator \(\rho ^{(N)}(M)\) is invariant under the simultaneous cyclic shift of its bra and ket configurations: \(\rho ^{(N)}(M)_{r\sigma ,\, r\tau }=\rho ^{(N)}(M)_{\sigma ,\tau }\).

The MPV of a block-diagonal tensor decomposes as

Equivalently, for each configuration \(\sigma \) one has

Let \(M_\alpha \) be a normal tensor. Any matrix commuting with every matrix of \(M_\alpha \) is a scalar multiple of the identity. In particular, if \(X\ne 0\) and \(X^\dagger X\) commutes with every matrix of \(M_\alpha \), then \(X^\dagger X=\omega \mathbb {1}\) for a necessarily positive constant \(\omega \), and \(\omega ^{-1/2}X\) is unitary. This is the normalized form \(X_{\alpha ,k}^\dagger X_{\alpha ,k}=\omega _{\alpha ,k}\mathbb {1}\) of the Gram-matrix proportionality in the proof of [ CPGSV16 , Proposition 4.13 ] , with \(X_{\alpha ,0}=\mathbb {1}\) and \(U_{\alpha ,k}=\omega _{\alpha ,k}^{-1/2}X_{\alpha ,k}\). Deriving the commutation from self-adjointness of the sector compressions is a separate step.

If \(\omega {\gt}0\) and \(X^\dagger X=\omega \mathbb {1}\), then \(\omega ^{-1/2}X\) is unitary.

Let a phase-class grouping have gauges \(X_{\alpha ,q}\), physical maps \(V_{\alpha ,q}\), and effective coefficients \(c_{\alpha ,q}\). Suppose that, for every class \(\alpha \) and copy \(q\), \(\omega _{\alpha ,q}{\gt}0\) and

Suppose also that the maps \(V_{\alpha ,q}\) are isometries with pairwise orthogonal ranges, they intertwine \(\widetilde M\) with the corresponding gauged representatives, and their grouped corner sum reconstructs every letter of \(\widetilde M\). Then there are maps \(W_{\alpha ,q}\) such that, for every class \(\alpha \) and copy \(q\),

For distinct pairs \((\alpha ,q)\ne (\beta ,p)\),

For every vertical letter \(v\), these maps also satisfy

Let \(P\) be a BNT canonical form with basis representatives \((A_j)_{j=0}^{g-1}\), and suppose that the representatives have already been blocked so that every \(A_j\) is injective. Then representative word-tuple separation holds at every length \(L\geq 6(g-1)+1\). Consequently, if the first-site actions of \(Y,Z\in M_{d}(\mathbb {C})\) agree on the positive-length MPV family of the tensor represented by \(P\), then \(YA_j=ZA_j\) for \(0\leq j{\lt}g\). The same conclusion holds when the first-site equality is given for any tensor with the same positive-length MPV family as the tensor represented by \(P\).

More generally, let \(P\) be a BNT canonical form before physical blocking, let \(p{\gt}0\), and suppose that every \(p\)-blocked representative \(A_j^{[p]}\) is injective. Then the blocked sector decomposition \(P^{[p]}\) has representative word-tuple separation at every blocked length \(L\geq 6(g-1)+1\).

Without changing the physical alphabet, a BNT canonical form also has eventual representative word-tuple separation. If \(p{\gt}0\) is a common block-injectivity length, then one may take pairwise separating words of length \(6p\), form selectors of length \(6p(g-1)\), and append homogeneous words of any length at least \(p\).

Finally, the BNT canonical-form hypotheses imply that there is a common \(L{\gt}0\) for which every unblocked representative \(A_j\) is \(L\)-block injective. Indeed, irreducibility and left-canonicality give a finite peripheral period \(m_j\). The self-overlap tends to \(m_j\) along multiples of \(m_j\), but by normalization it also tends to \(1\); hence \(m_j=1\). Thus every representative is normal, and finiteness gives a common positive block-injectivity length. If the first-site actions of \(Y,Z\) agree on the tensor represented by \(P\), then \(YA_j=ZA_j\) for \(0\leq j{\lt}g\). The same conclusion holds when the first-site equality is given for any tensor with the same positive-length MPV family. This composes the blocking conclusion of [ CPGSV16 , lines 318–345 ] with Appendix C.3, Lemma L [ CPGSV16 , lines 1835–1858 ] .

No additional common-length hypothesis is required.

Suppose \(A\) has invariant-projector closure and no nontrivial \(p\)-periodic vectors. Then there are normal tensors \(A_1,\ldots ,A_r\) of positive bond dimensions \(D_1,\ldots ,D_r\), with \(\sum _kD_k\le D\), and nonzero weights \(\mu _1,\ldots ,\mu _r\) such that

Moreover, there are isometries \(V_k:\mathbb {C}^{D_k}\to \mathbb {C}^D\) with pairwise orthogonal ranges such that

This is the sufficient condition for canonical form of  [ CPGSV16 , Section 2 ] , with that paper’s convention \(\sum _kD_k\le D\) allowing zero blocks: a corner of \(A\) carrying the zero tensor cannot be rescaled to spectral radius one and is omitted because it vanishes at every positive length.

Suppose \(A\) has invariant-projector closure. Then there are isometries \(V_1,\ldots ,V_r\) with \(V_k^\dagger V_k=\mathbb {1}\), pairwise orthogonal ranges, and \(\sum _kV_kV_k^\dagger =\mathbb {1}\), such that \(A^iV_k=V_kA_k^i\) for the corner tensors \(A_k^i=V_k^\dagger A^iV_k\), every corner tensor \(A_k\) is irreducible, and \(\mathcal{V}(A)=\mathcal{V}(\bigoplus _{k=1}^rA_k)\).

If a block family admits the finite-length data described in the block-injectivity proposition of [ CPGSV16 , lines 340–345 ] , then it is in block-injective canonical form.

A matrix commuting with a nonzero power of a nonempty-chain density operator generated by an MPDO commutes with the density operator itself. This is the final operator implication in the periodic-sector contradiction in the proof of [ CPGSV16 , Proposition 4.13 ] .

Let \(H\geq 0\), let \(p\geq 1\), and let \(QH^p=H^pQ\). Then \(QH=HQ\).

A nonzero positive semidefinite matrix has positive trace: the trace is a positive real.

Let \(A_{\mathrm{tot}}\) be the tensor of a sector decomposition with basis representatives \((A_j)_{j=0}^{g-1}\), multiplicities \(r_j{\gt}0\), nonzero weights \(\mu _{j,q}\), and grouped coefficients \(c_j^{(N)}=\sum _{q=0}^{r_j-1}\mu _{j,q}^{N}\). Suppose that the first-site actions of \(Y,Z\in M_{d}(\mathbb {C})\) agree on the MPV family of \(A_{\mathrm{tot}}\). If representative word-tuple separation holds at length \(L\) and \(c_{j_0}^{(L+1)}\neq 0\), then \(YA_{j_0}=ZA_{j_0}\). Consequently, under eventual representative word-tuple separation, \(YA_j=ZA_j\) for every representative \(j\). The same conclusion holds if the first-site equality is initially given for any tensor with the same positive-length MPV family as \(A_{\mathrm{tot}}\).

The representative grouping is the argument of [ CPGSV16 , Appendix C.3, lines 1835–1858 ] .

In the first auxiliary assertion, eventual representative word-tuple separation is an explicit assumption. The later post-blocked theorem derives the needed finite representative separation from the full BNT canonical-form hypotheses. The sharp \(3D^5\) representative-separation bound is Theorem 9.6.1.22; its detailed source comparison is recorded in [ con26u ] .

Let \((C_j^s)\) and \((E_j^s)\) be two marked families over the same sector decomposition. Fix a tail length \(L\), and suppose that, for every marked index \(s\) and every unmarked word \(w\) of length \(L\),

If representative word-tuple separation holds at length \(L\) and \(c_{j_0}^{(L+1)}\neq 0\), then \(C_{j_0}=E_{j_0}\). Under eventual representative word-tuple separation, agreement for every marked index and every positive-length unmarked word implies \(C_j=E_j\) for every \(j\).

Lemma L is stated for marks induced by physical operators on the first site. After its closed-chain identity has been expanded, the separation calculation uses only the matrices in the marked slot. The theorem is the resulting algebraic extension of [ CPGSV16 , Appendix C.3, lines 1848–1858 ] ; it is not attributed to the paper as a separately stated arbitrary-mark theorem.

Let \(M\) be in normalized BNT-refined horizontal form. For doubled-index matrices \(Y,Z\), suppose that \(Y_1V^{(N)}(M)=Z_1V^{(N)}(M)\) for every \(N\geq 1\). Then the corresponding inserted tensors agree on the original bond space, \(YM=ZM\).

Now suppose in addition that \(M\) generates an MPDO, \(Q=Q^\dagger \), and that \(Q\widetilde M=Q\widetilde M Q\) on the vertically viewed tensor. Then, on every representative, the insertions corresponding to \(\widetilde M Q\) and \(Q\widetilde M Q\) agree: \(Y_R(Q)A_j=Y_C(Q)A_j\) for \(0\leq j{\lt}g\). This equality extends through every repeated weighted sector and the direct sum of the copy gauges, so that the original MPO tensor satisfies \(\widetilde M Q=Q\widetilde M Q\). The specialized assertion follows from the two entrywise identities \(Q_1H^{(N)}=Q_1H^{(N)}Q_1\) and \(Q_1H^{(N)}=H^{(N)}Q_1\) at every positive length.

The corresponding literal-CF assertion also holds. Let \(A\) be in literal CPSV canonical form. If the first-site actions of \(Y,Z\in M_{d}(\mathbb {C})\) agree on every positive-length matrix product vector of \(A\), then

on the original bond space. In particular, suppose that \(M\) generates matrix product density operators, \(P=P^\dagger \), and \(P\widetilde M=P\widetilde MP\). Positivity gives agreement of the first-site actions associated with \(\widetilde MP\) and \(P\widetilde MP\) at every positive length. Hence (32) gives the opposite one-sided identity on the original bond space.

This is the representative-indexed conclusion of the invariant-projection step in [ CPGSV16 , Proposition 4.13, lines 1873–1887 ] . Copies with a common normal representative are grouped before separation, exactly as in Appendix C.3, Lemma L. The source assumes that \(Q\) is an orthogonal projector; this implication only uses its Hermiticity.

Let \(B\) be an irreducible tensor with a positive definite transfer eigenvector, \(\mathcal{E}_B(\rho )=r\rho \), where \(\rho {\gt}0\) and \(r{\gt}0\). Then the rescaled unital gauge \(K^v=r^{-1/2}\rho ^{-1/2}B^v\rho ^{1/2}\) is a unital Kraus family, is an irreducible tensor with an irreducible transfer map, and its adjoint map has a positive definite fixed point. These are the hypotheses of the cyclic decomposition of the peripheral spectrum, [ Wol12 , Theorem 6.6 ] .

Let \(M\) be an MPO tensor whose doubled-index MPS tensor is in the literal CPSV canonical form. If \(M\) generates matrix product density operators, then \(M\) is in canonical form in the vertical direction. More precisely, there is a basis of normal tensors \(\{ M_\alpha \} _\alpha \) for the vertically viewed tensor \(\widetilde M\), together with positive diagonal matrices \(\mu _\alpha \) and a coisometry \(U\) from the physical space onto the retained nonzero sectors. Writing \(B=\bigoplus _\alpha \mu _\alpha \otimes M_\alpha \), one has

The source explicitly displays (82) and calls \(U\) an isometry [ CPGSV16 , Proposition 4.13 ] . In this rectangular orientation, however, \(U\) maps the physical space onto the retained sector space and satisfies (81); thus \(U\) is a coisometry and \(U^\dagger \) is the isometric inclusion. The reverse identity (83) is not displayed separately in the proposition. It records the exact reconstruction implicit in the preceding assertion that \(\widetilde M\) is itself in canonical form, including the permitted zero orthogonal complement.

Let \(\{ A_\alpha \} _{\alpha =0}^{g-1}\) be a BNT for the vertically viewed tensor \(\widetilde M\). For each \(\alpha \), let \(r_\alpha {\gt}0\), let \(\omega _{\alpha ,q}{\gt}0\) for \(0\leq q{\lt}r_\alpha \), and let \(W_{\alpha ,q}:\mathbb {C}^{D_\alpha }\to \mathbb {C}^d\) satisfy

Suppose that, for every vertical letter \(v\) and every pair \((\alpha ,q)\), \(\widetilde M_vW_{\alpha ,q} =W_{\alpha ,q}(\omega _{\alpha ,q}A_\alpha ^v)\), and that the letters have the exact sector decomposition

Then \(M\) is in vertical canonical form. This is the final finite-dimensional implication in [ CPGSV16 , Proposition 4.13, lines 1863–1870 ] .

If \(M\) has a grouped vertical BNT decomposition with physical isometries in the sense of Definition 17.4.99, and \(M\) has the grouped-corner Gram-dressing property of Definition 17.4.71, then \(M\) is in vertical canonical form.

A tensor in normalized BNT-refined horizontal form that generates matrix product density operators is also in canonical form in the vertical direction: there exist a basis of normal tensors \(\{ M_\alpha \} _\alpha \) for the vertically viewed tensor \(\widetilde M\), positive diagonal matrices \(\mu _\alpha \), and a coisometry \(U\) from the physical space onto the retained nonzero sectors. Writing \(B=\bigoplus _\alpha \mu _\alpha \otimes M_\alpha \), one has

The conclusion includes the basis-of-normal-tensors condition in [ CPGSV16 , Proposition 4.13 ] , together with the positive weights and the coisometric realization. Its hypothesis is the normalized BNT-refined form of Definition 17.4.1, rather than the literal CPSV canonical form assumed in Theorem 17.4.111.

Suppose that

and, for every letter \(v\) and every sector \(k\), \(T_vW_k=W_kB_k^v\). Then the block-row map \(U\) is a coisometry and conjugates \(T\) to the direct sum of the sector tensors:

If the letters also satisfy the literal sector decomposition

then

This last equality is a conditional consequence of (78); no additional assertion from the source is used.