Dependencies

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 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.

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.

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 (26) remains a separate condition on the underlying canonical-form data.

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.

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 \(i\),

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 \(\mathrm{GL}_{\sum _k D_k}(\mathbb {C})\).

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 8.1.1.1).

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 ] .

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

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.

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

Equivalently, its Kraus map is trace-preserving.

A (translation-invariant, PBC) MPS tensor with physical dimension \(d\) and bond dimension \(D\) is a collection of matrices \(\{ A^i\} _{i=0}^{d-1}\), where \(A^i \in M_{D}(\mathbb {C})\), indexed by a physical index \(i \in \{ 0, \ldots , d{-}1\} \). Such a tensor defines an MPV family. Diagrammatically,

The black node denotes the tensor \(A\), the horizontal legs are virtual, and the upper leg is the physical index \(i\).

The matrix product vector (MPV) of a tensor \(A\) at system size \(N\) is the vector

Equivalently, for a configuration \(\sigma = (i_1, \ldots , i_N) \in \{ 0,\ldots ,d{-}1\} ^N\), we write

The coefficient function \(\sigma \mapsto V^{(N)}(A)_\sigma \) gives the components of the vector in (2); the displayed ket is the corresponding vector in \((\mathbb {C}^d)^{\otimes N}\). For a general word \(w\), we also write \(c_w(A) := \operatorname{tr}(A^w)\). The MPV family generated by \(A\) is the collection \(\mathcal{V}(A) = \bigl\{ |V^{(N)}(A)\rangle \bigr\} _{N \ge 1}\). The coefficient \(V^{(N)}(A)_\sigma \) is the periodic contraction

of \(N\) copies of the local tensor \(A\), with the outer virtual legs closed by the trace.

The MPV overlap between two tensors \(A\) (of bond dimension \(D_1\)) and \(B\) (of bond dimension \(D_2\)) at system size \(N\) is

Diagrammatically, the overlap contracts the MPV ring of \(A\) against the conjugate ring of \(B\) along their shared physical indices, both virtual rings closed by the trace:

This pairing is linear in the first MPV coefficient and conjugate-linear in the second. By Lemma 2.7.4, it is the complex conjugate of the Hilbert-space inner product from Definition 2.7.2.

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.

This is a one-copy canonical-form condition: each listed block is one representative and repeated copies are not encoded as separate blocks. A family is in normal canonical form with BNT separation if it satisfies Definition 8.9.2 and distinct blocks of the same bond dimension are not gauge-phase equivalent. The weight moduli are non-increasing but need not be strictly decreasing, so equal-modulus blocks are allowed; the grouping into a BNT is governed by gauge-phase equivalence, not by distinctness of moduli. Repeated equal-modulus copies of one basis tensor are not separate blocks here; they appear as the weights \(\mu _{j,q}\) in \(c_N^{(j)} = \sum _q \mu _{j,q}^N\) in the sector decomposition below.

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 \(\sigma _\partial (\mathcal{E}_A)=\{ 1\} \).

A left-canonical irreducible MPS tensor \(A\) of bond dimension \(D\) is \(m\)-periodic (for some \(m\ge 1\)) if the peripheral spectrum of its transfer map \(\mathcal{E}_A\) is exactly the \(m\)-th roots of unity \(\{ e^{2\pi i k/m}:0\le k{\lt}m\} \). This is the normalized form-II convention used later in the source. A \(1\)-periodic tensor is exactly an irreducible tensor with primitive transfer map (the corresponding definition in the companion quantum-channel volume  [ LTC26 , “Primitive map” ] ).

An unnormalized positive-length PGVWC07 witness for a tensor \(A\) consists of finitely many blocks \((A_k)_{k=1}^r\) of positive bond dimensions and positive real weights \((a_k)_{k=1}^r\). Each block is unital, has only scalar transfer-map fixed points, and has a diagonal positive-definite fixed point \(\Lambda _k\) for the adjoint transfer map. Moreover,

and the retained bond dimensions satisfy \(\sum _k\dim (A_k)\le D\).

Let \(A=(A^i)_{i=0}^{d-1}\) have bond dimension \(D\), let \((Q_u)_{u\in \mathbb Z/p\mathbb Z}\) be cyclic projections, and fix a starting sector \(u\). For each \(0\le j{\lt}N\), the corresponding chain of length \(N\) has local matrices

where sector indices are read modulo \(p\). It has the same bond dimension \(D\) as \(A\) and repeats with period \(p\).

\(A\) has primitive irreducible cyclic sectors of period \(m\ge 1\) if there is a family \((C_k)_{k=1}^m\) with \(A^{[m]}\sim \bigoplus _{k=1}^mC_k\) with unit weights, each \(C_k\) left-canonical, with primitive transfer map, tensor-irreducible, and of positive bond dimension.

Assume \(D{\gt}0\). A tensor \(A\) together with a matrix \(\rho \in M_{D}(\mathbb {C})\) is primitive if

where \(P\) is the fixed-point projection associated to \(\rho \). When the choice of \(\rho \) is irrelevant, we simply say that \(A\) is a primitive MPS tensor. This condition combines a complementary spectral gap with a nonzero positive semidefinite fixed point. It is not the paper definition in Definition 7.1.1.3, which is the uniform spreading condition (1). With the additional hypothesis \(\rho {\gt}0\), the complementary-gap condition implies strong irreducibility and hence paper primitivity by Theorem 7.1.1.13.

For tensors \(A\) and \(B\) of possibly different bond dimensions \(D_1\) and \(D_2\), we write \(\mathcal{V}(A) = \mathcal{V}(B)\) if \(|V^{(N)}(A)\rangle = |V^{(N)}(B)\rangle \) for every word length \(N\), including the empty word. This algebraic relation is used for exact direct-sum identities.

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 \(N {\gt} 0\).

We call the tensor strongly irreducible here if there is a matrix \(\rho {\gt}0\) such that

and \(\mathcal{E}_A\) is irreducible. The condition in [ SPGWC10 , Proposition 3(c) ] requires that \(1\) be the only peripheral eigenvalue and that its eigenspace be one-dimensional, generated by a positive-definite fixed point. The definition above strengthens this condition by also requiring irreducibility explicitly.

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.

A unitary matrix \(U\) determines an invertible matrix whose inverse is \(U^\dagger \).

If \(\mu \in \mathbb {C}\) is nonzero and \(p\ge 1\), then \(\mu ^p\neq 0\); blocking preserves the nonzero-weight hypothesis of a weighted block decomposition.

Let \(A\) be left-canonical, tensor-irreducible, and have primitive transfer map. Then \(A\) has primitive fixed-point data \((\rho ,P_\rho ,N)\) with \(\rho {\gt}0\) and \(N^n\to 0\). For every blocking length \(P\), the tensor \(A^{[P]}\) is left-canonical, its transfer map is \(\mathcal{E}_A^P\), and \(\rho \) is fixed by this blocked map. For every \(n\ge 1\),

Finally, a channel with a positive-definite fixed point whose positive semidefinite fixed points are all proportional is irreducible as a map; irreducibility of a transfer map implies tensor irreducibility.

Let \((A_j)_{j=1}^g\) be a finite family of tensors such that \(O_{A_jA_j}(N) \to 1\) for each \(j\) and \(O_{A_iA_j}(N) \to 0\) for \(i \neq j\). Then the MPV states \(\{ |V^{(N)}(A_j)\rangle \} _{j=1}^g\) are linearly independent for all sufficiently large \(N\).

Let \((\mu _k, A_k)_{k=1}^r\) be a weighted block family. Assume each \(A_k\) is injective and left-canonical, \(\sum _i (A_k^i)^\dagger A_k^i = \mathbb {1}\), that \(O_{A_kA_k}(N) \to 1\) for each \(k\), and that distinct equal-dimension blocks are not gauge-phase equivalent. Then the blocks form a basis of normal tensors for the block-diagonal tensor \(\bigoplus _k \mu _k A_k\).

If the letters of \(A\) act irreducibly on \(\mathbb {C}^D\), then \(A\) has no nontrivial invariant orthogonal projection.

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 \(A\) be an MPS tensor and let \(P\) be an orthogonal projection. If \((\mathbb {1}-P)A^iP=0\) for every physical index \(i\), then \(P\mathcal{E}_A(PXP)P=\mathcal{E}_A(PXP)\) for every \(X\in M_{D}(\mathbb {C})\).

Let \(V_1,\ldots ,V_r\) be isometries, \(V_k^\dagger V_k=\mathbb {1}\), with \(\sum _kV_kV_k^\dagger =\mathbb {1}\), and suppose \(A^iV_k=V_kB_k^i\) for some tensors \(B_1,\ldots ,B_r\). Then \(\mathcal{V}(A)=\mathcal{V}(\bigoplus _{k=1}^rB_k)\).

On this one-copy surface, if a family is in normal canonical form with BNT separation and each block is also normal in the algebraic sense, then the blocks form a basis of normal tensors for the associated block-diagonal tensor. The blocks here are distinct representatives.

Every orthogonal projection \(P\) on \(\mathbb {C}^D\) factors through an isometry: there are \(n\in \mathbb {N}\) and \(V\in \mathbb {C}^{D\times n}\) such that

Let \(A\) have invariant-projector closure, and let \(V\) be an isometry, \(V^\dagger V=\mathbb {1}\), whose range projection \(VV^\dagger \) commutes with every \(A^i\). Then the corner tensor \(B^i=V^\dagger A^iV\) again has invariant-projector closure.

The spectral normality of Definition 8.9.2 and the algebraic normality of Definition 2.4.4 agree: the Wielandt theory of Chapter 7 supplies the implication from primitivity to eventual block injectivity used here.

Let \(A\) have positive bond dimension, suppose that \(\mathcal{E}_A\) is irreducible, and assume \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\). Let \(Q\) be an orthogonal projection. If \((\mathcal{E}_{A^\dagger })^m\) preserves the corner \(QM_{D}(\mathbb {C})Q\), then

Let \(A\) be left-canonical and let \((P_k)\) be orthogonal projections summing to \(\mathbb {1}\) and fixed by the adjoint transfer map. Then the same sector decomposition and support isometries of Theorem D.5.2 exist. In particular, the adjoint fixed-point condition \(\mathcal{E}_A^\dagger (P_k)=P_k\) gives \(P_kA^i=A^iP_k\), which gives the corner-letter identity \(\varphi _k(C_k^i)=P_kA^iP_k\).

Let \(A\) be a left-canonical irreducible tensor whose adjoint transfer map has peripheral spectrum \(\{ \gamma ^j:0\le j{\lt}m\} \) for a primitive \(m\)th root of unity \(\gamma \), with blocked cyclic-sector decomposition \((C_u)_{u=1}^m\), \((P_u)_{u=1}^m\), and linear isomorphisms \(\varphi _u:M_{\dim C_u}(\mathbb {C})\xrightarrow {\sim }P_uM_{D}(\mathbb {C})P_u\). Assume explicitly that the \(P_u\) are orthogonal projections summing to \(\mathbb {1}\) and satisfying \(\mathcal{E}_{A^\dagger }(P_{u+1})=P_u\), that each sector has nonzero bond dimension, and that the compression maps satisfy

Here \(P_uA^{[m]}\) denotes the blocked tensor with letters \(P_u(A^{[m]})^\alpha \). Then every sector tensor satisfies \(\sigma _\partial (\mathcal{E}_{C_u})=\{ 1\} \) and is tensor-irreducible.

Let \(A\) be an irreducible MPS tensor with \(D{\gt}0\) and \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\). Then there exists a blocking length \(p{\gt}0\) such that \(\sigma _\partial (\mathcal{E}_{A^{[p]}})=\{ 1\} \).

Suppose \(A\) generates the same MPV family as \(\bigoplus _{k=1}^rB_k\), where each \(B_k\) is irreducible with at least one nonzero Kraus operator. Then there are nonzero weights \((\mu _k)_{k=1}^r\) and blocks \((C_k)_{k=1}^r\) such that, for every \(N\) and every \(\sigma \),

each \(C_k\) is irreducible and left-canonical, \(\sum _i(C_k^i)^\dagger C_k^i=\mathbb {1}\), with positive bond dimension. This is the blockwise form used in the arbitrary-tensor trace-preserving gauge below.

Let \(A\) be an irreducible MPS tensor in TP gauge, \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\), with \(D{\gt}0\). Then there exist a unitary \(U\) and a diagonal positive definite \(\Lambda \) such that:

1. \(B^i:=U^\dagger A^iU\) is TP,

2. \(\mathcal{E}_B(\Lambda )=\Lambda \),

3. \(\Lambda \) is diagonal and positive definite,

4. \(A\) and \(B\) generate the same MPV family.

These are the two normalization identities in the CFII definition of [ CPGSV16 , Appendix A ] . The theorem does not by itself prove that \(A\) is in canonical form or that its block is normal. The unitary conjugation is performed inside TP gauge after TP normalization.

Let \(A\) and \(B\) have common bond dimension \(D\), and let \(C\) and \(E\) have common retained bond dimension \(n\). Suppose that \(U_A,U_B\in \mathbb {C}^{n\times D}\) satisfy \(U_AU_A^\dagger =U_BU_B^\dagger =\mathbb {1}_n\) and, for every physical index \(i\),

If \(C\) and \(E\) are gauge equivalent, then \(A\) and \(B\) are gauge equivalent.

Each sector tensor in the common reblocked cyclic-sector family of Lemma D.5.5 is trace-preserving, \(\sum _i(B^i)^\dagger B^i=\mathbb {1}\), has primitive transfer map, \(\sigma _\partial (\mathcal{E}_B)=\{ 1\} \), is tensor-irreducible, and has positive bond dimension.

The powered nonzero-part decomposition \(\bigoplus _k\mu _k^pB_k^{[p]}\) carries over under the common-alphabet identification of Lemma D.5.8: the common-alphabet sectors have the same nonzero-part decomposition as the direct \(p\)-blocked tensors.

Let \((B_k)_{k=1}^r\) be a finite family of positive-bond-dimension, left-canonical, tensor-irreducible MPS tensors. Then there is a single blocking length \(p{\gt}0\) such that \(\mathcal{E}_{B_k^{[p]}}\) is primitive for every \(k\).

Let \(A\) have a canonical-form MPV expansion

at every positive length, where the tensors \(\widetilde A_k\) are normal tensors of positive bond dimension and every coefficient \(\mu _k\) is nonzero. Let \((A_j)_{j=1}^{g}\) be tensors of positive bond dimension. Then \((A_j)_{j=1}^{g}\) is a basis of normal tensors for \(A\) if and only if:

1. every \(A_j\) is normal;

2. for every \(k\) there are an index \(j\), an invertible matrix \(X_k\), and a phase \(e^{i\phi _k}\) such that \(\widetilde A_k^i=e^{i\phi _k}X_kA_j^iX_k^{-1}\) for every physical index \(i\);

3. no two distinct tensors \(A_j\) and \(A_{j'}\) are related in this way.

This is [ CPGSV16 , Proposition 2.7 ] , read with the nonzero-coefficient convention of Definition 8.6.7.

Let \(A\) have canonical-form data as in (25), and let \((B_j)_{j=1}^{g}\) have positive bond dimensions. The family \((B_j)_{j=1}^{g}\) is a basis of normal tensors for \(A\) if and only if

Each displayed conjugacy also requires the two bond dimensions to agree, and every scalar \(e^{i\phi }\) has modulus one. This is [ CPGSV16 , Proposition 2.7 and Appendix A ] , specialized to the literal canonical-form data.

For canonical-form data as in (25), every positive length \(N\) and every configuration \(\sigma \) satisfy

Equivalently, \(|V^{(N)}(A)\rangle =\sum _{k=1}^{r}\mu _k^N|V^{(N)}(A_k)\rangle \). This is the identity immediately following the canonical-form definition in  [ CPGSV16 , Section 2.3 ] .

For every tensor \(A\), there are a positive integer \(p\), a bond dimension \(D'\), and a tensor \(B\) of physical dimension \(d^p\) and bond dimension \(D'\) such that \(B\in \mathrm{CF}\) and, for every \(N\ge 1\),

Here \(B\) is another tensor generating the same positive-length MPV family. The assertion \(B\in \mathrm{CF}\) is literal in the bond coordinates of \(B\); no assertion is made that \(A^{[p]}\) itself is literally in canonical form in its original bond coordinates.

Let \(A\) be a tensor in CPSV canonical form, with ambient bond dimension \(D\). Then there is a tensor \(B\) of the same bond dimension and an \(G\in \mathrm{GL}(D,\mathbb {C})\) such that, for every physical index \(i\),

Moreover, \(B\) is in canonical form II and the total retained block dimension is unchanged. This is the nonsingular gauge assertion of [ CPGSV16 , Appendix A ] .

Let \(A\) and \(B\) be gauge-equivalent MPS tensors and let \(c\in \mathbb {C}\). Then

If \(B^i=XA^iX^{-1}\) for every physical index \(i\), then there is an invertible matrix \(C\) such that

Let \(A\) have canonical-form data

For every \(L\geq 1\), \(A^{[L]}\) has canonical-form data with the same coisometry \(U\), the blocked normal tensors \(A_k^{[L]}\), and the weights \(\mu _k^L\):

If \(A\) is irreducible, then \(\overline A\) is irreducible.

For a tensor or matrix, an overline denotes entrywise conjugation. If \(A\) and \(B\) are gauge equivalent through \(X\), then \(\overline A\) and \(\overline B\) are gauge equivalent through \(\overline X\). If \(A\) is left-canonical, then \(\overline A\) is left-canonical. If \(A\) is algebraically normal, then \(\overline A\) is algebraically normal. Independently, if \(A\) is a CPSV normal tensor, then \(\overline A\) is a CPSV normal tensor.

Let \(\sigma \) denote entrywise conjugation. For every tensor \(A\), as operators on the matrix space,

evaluating at \(\overline X\) gives, for every matrix \(X\),

If \(\Lambda \) is positive definite and diagonal, then

Let \(X\) be invertible and suppose

for every physical index \(i\). If \(B\) is a CPSV normal tensor, then \(A\) is a CPSV normal tensor.

Let \(A\) be a CPSV normal tensor and let \(L\geq 1\). Then the blocked tensor \(A^{[L]}\) is a CPSV normal tensor.

Let \(A\) be a normal tensor in the sense of [ CPGSV16 , Section 2.2 ] . Its bond dimension is necessarily positive, and there is a positive definite matrix \(\sigma \) such that the gauge obtained from \(\sigma \) is trace-preserving, primitive, and irreducible. No scalar rescaling is needed.

Let \(e:\{ 0,\ldots ,d'-1\} \to \{ 0,\ldots ,d-1\} \) be a bijection and set

The transfer maps of \(A\) and \(e^*A\) agree. In particular, physical relabeling preserves left-canonical normalization and CPSV normality. If \(A\) has canonical-form data

then \(e^*A\) has the same weights and coisometry and the relabeled normal blocks:

For an invertible matrix \(C\), a linear map \(E\) is primitive if and only if

is primitive.

For an invertible matrix \(C\), the linear maps

have the same spectral radius.

Let \((A_j)_{j=1}^r\) be a family in which each \(A_j\) is injective and left-canonical, \(\sum _i (A_j^i)^\dagger A_j^i = \mathbb {1}\), and distinct equal-dimension blocks are not gauge-phase equivalent. Then \(O_{A_jA_k}(N) \to 0\) for every pair \(j \neq k\).

Let \(A\) be a left-canonical irreducible tensor. Then there is a period-removing length \(m\ge 1\) such that \(A^{[m]}\) admits:

1. left-canonical sector tensors \((C_k)_{k=1}^m\) of nonzero bond dimension,

2. a unit-weight direct-sum MPV decomposition \(A^{[m]}\sim \bigoplus _{k=1}^mC_k\),

3. orthogonal projections \((P_k)_{k=1}^m\) summing to \(\mathbb {1}\) with \(\mathcal{E}_{A^\dagger }(P_{k+1})=P_k\),

4. commutation relations \(P_k(A^{[m]})^i=(A^{[m]})^iP_k\) for every blocked letter \(i\),

5. the sector trace formula \(f_{C_k}(\sigma ) =\operatorname{tr}(P_k(A^{[m]})^{\sigma _1}\cdots (A^{[m]})^{\sigma _N})\),

6. linear isomorphisms onto the compression corners \(\varphi _k:M_{\dim C_k}(\mathbb {C})\xrightarrow {\sim }P_kM_{D}(\mathbb {C})P_k\),

7. transfer-intertwining identities \(\varphi _k(\mathcal{E}_{C_k^\dagger }(X)) =\mathcal{E}_{(P_kA^{[m]})^\dagger }(\varphi _k(X))\),

8. multiplicativity and adjoint compatibility \(\varphi _k(XY)=\varphi _k(X)\varphi _k(Y)\) and \(\varphi _k(X^\dagger )=\varphi _k(X)^\dagger \).

This is the cyclic-sector part of the period-removal step of  [ CPGSV16 , Section 2.3 ] ; the projections \(P_k\) come from the peripheral spectrum of the adjoint transfer map ( [ Wol12 , Theorem 6.6 ] ).

Let \(A\) be a left-canonical irreducible tensor, and suppose the adjoint transfer map \(\mathcal{E}_{A^\dagger }\) has a positive-definite fixed point, is irreducible as a positive map, and has peripheral spectrum

where \(\gamma \) is a primitive \(m\)th root of unity and \(m\ge 1\). Then the spectral cyclic-sector decomposition of \(A^{[m]}\) has compression maps which may be chosen with rectangular support isometries \(V_k\) satisfying

These isometries also realize the blocked corner letters \(\varphi _k(C_k^\alpha )=P_k(A^{[m]})^\alpha P_k\).

Let \(A\) be a left-canonical irreducible tensor. Then there is a period \(m\ge 1\) and a unit-weight decomposition of \(A^{[m]}\) into sector blocks \((C_k)_{k=1}^m\), each of which is left-canonical, has primitive transfer map, is tensor-irreducible, and has nonzero bond dimension. Equivalently, the cyclic-sector decomposition of \(A^{[m]}\) already satisfies the trace-preservation, primitivity, and irreducibility hypotheses needed for the common-blocking step (Lemma D.5.5).

With hypotheses as in Lemma D.5.5, and given any common multiple \(p\) of the per-block period-removal lengths \((m_k)\), the common reblocking can be performed at the prescribed length \(p\).

Let \(A\) be an irreducible MPS tensor with \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\) and \(D{\gt}0\). Then there exist a unitary \(U\) and a diagonal positive definite \(\Lambda \) such that, for \(B^i:=U^\dagger A^iU\),

This proves the diagonal Perron normalization required in CFII. It does not prove that \(A\) is in the CPSV canonical form or that its block is normal; those require the corresponding primitivity and canonical-form hypotheses.

Let \(\rho \ge 0\) be a fixed point of \(\mathcal{E}_A\), and let \(P\) be its support projection. Then \((\mathbb {1}-P)A^iP=0\) for all \(i\): the range of \(P\) is invariant under every \(A^i\).

If two tensors generate the same MPV family at every positive length, then the cyclic-sector decompositions on the two sides can be expressed at one common positive blocking length \(p\). For every \(N\ge 1\) and every blocked \(\sigma \), each blocked tensor equals its powered nonzero part:

The two powered nonzero parts agree:

If \(A\) and \(B\) generate the same MPV family at every positive length, then after removing the all-zero blocks and applying the trace-preserving gauge there are weighted nonzero-block tensors \(\bigoplus _j\mu _j^AA_j\) and \(\bigoplus _k\mu _k^BB_k\) such that, for every \(N\ge 1\) and \(\sigma \),

The two nonzero parts agree:

Each nonzero block has primitive irreducible cyclic sectors: for each \(j\), there is \(m_j\ge 1\) with a unit-weight sector decomposition \(A_j^{[m_j]}\sim \bigoplus _{u=1}^{m_j}C^A_{j,u}\), and similarly for \(B_k\).

Suppose that \(A\) has invariant-projector closure and that \(P\) is an orthogonal projection satisfying \((\mathbb {1}-P)A^iP=0\) for all \(i\). Then \(P\) reduces every tensor matrix: \(PA^i=A^iP\) for all \(i\).

Every MPS tensor \(A\) admits a block-diagonal tensor \(\bigoplus _{k=1}^r A_k\) with each \(A_k\) irreducible, \(\mathcal{V}(A)=\mathcal{V}(\bigoplus _k A_k)\), and \(\sum _{k=1}^rD_k=D\).

If \(A\) is irreducible as a tensor (Definition 8.1.1.1), then the Kraus operators act irreducibly on \(\mathbb {C}^D\) (Definition 7.7.3.3).

If \(A\) is irreducible and unital, \(\mathcal{E}_A(\mathbb {1})=\mathbb {1}\), and \(\mathcal{E}_A(X)=X\), then \(X\) is a scalar multiple of \(\mathbb {1}\). This is the final fixed-point assertion in the proof of [ PGVWC07 , Theorem 4 ] .

Let \(D{\gt}0\). If \(A\) is algebraically normal and left-canonical, then

Suppose \(P\) is an orthogonal projection satisfying \((\mathbb {1}-P)A^iP=0\) for every physical index \(i\). Then there exist \(n,m\) with \(n+m=D\) and MPS tensors \(A_1,A_2\) of bond dimensions \(n\) and \(m\) such that \(A\) and the two-block tensor \(A_1\oplus A_2\) generate the same MPV family in the sense of Definition 2.3.3.

The MPV of a block-diagonal tensor decomposes as

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

Every MPS tensor \(A\) admits a finite family of nonzero irreducible blocks \((B_k)_{k=1}^r\), each with at least one nonzero Kraus operator and positive bond dimension. For every \(N\ge 1\) and every \(\sigma \),

Moreover, \(\sum _{k=1}^rD_k\le D\).

If \(A\) is a primitive MPS tensor with positive-definite fixed point \(\rho \), then \(A\) is normal (has eventually full Kraus rank). No identity-in-the-one-step-span condition is needed.

This connects the complementary-gap condition in Definition 7.6.1 directly to normality.

Let \(A\) be a left-canonical MPS tensor with \(D{\gt}0\), tensor-irreducible, and with peripheral spectrum \(\{ 1\} \). Tensor irreducibility implies irreducibility of the completely positive transfer map, so \(A\) is normal.

Let \(A\) and \(B\) be injective normalized tensors of the same bond dimension that are not gauge-phase equivalent. Then \(O_{AB}(N)\to 0\) as \(N\to \infty \).

If \(D_1\neq D_2\) and both tensors are injective and normalized, then \(O_{AB}(N)\to 0\) as \(N\to \infty \).

Suppose that the orthogonal projections satisfy \(\sum _uQ_u=\mathbb {1}\) and \(Q_uA^i=A^iQ_{u+1}\). For a word \(\sigma =(i_0,\ldots ,i_{N-1})\), define the boundary product

At every length \(N\geq 0\),

If \(N{\gt}0\), then \(\operatorname{tr}F_u(\sigma )\) is the coefficient of the explicit component chain from Definition 8.1. At \(N=0\), the boundary product is \(F_u(\varnothing )=Q_u\), so

This is not the sum of the ordinary empty component-chain coefficients: an empty chain evaluates to \(\mathbb {1}\) and does not retain its starting projection.

Let \(A\) be an irreducible MPS tensor in unital gauge \(\sum _iA^i(A^i)^\dagger =\mathbb {1}\), with \(D{\gt}0\). Then there exist a unitary \(U\) and a diagonal positive definite \(\Lambda \) such that, for \(B^i:=U^\dagger A^iU\),

Moreover, \(A\) and \(B\) generate the same MPV family. This is the dual-map fixed-point and unitary-diagonalization step of [ PGVWC07 , Theorem 4 ] .

This is a normalized form of the state-vector conclusion in [ PGVWC07 , Theorem 5 ] . Let \(A\) be an irreducible left-canonical tensor whose peripheral transfer spectrum is the set of \(p\)-th roots of unity. If \(p\mid N\), there are orthogonal cyclic projections \((Q_u)_{u\in \mathbb Z/p\mathbb Z}\) such that \(\sum _uQ_u=\mathbb {1}\), \(Q_uA^i=A^iQ_{u+1}\), and \(Q_{u+N}=Q_u\). For every ring length \(N\geq 0\), its coefficients satisfy the boundary-aware decomposition

For \(N{\gt}0\), this becomes the source’s decomposition into the \(p\) explicit component chains:

Each \(B_u\) is a \(p\)-periodic MPS of bond dimension \(D\) with local matrices given by (1). At \(N=0\), (6) instead uses the boundary traces \(\operatorname{tr}(Q_u)\) from (4); it does not identify them with ordinary empty-chain coefficients. If \(p\nmid N\), then \(|V^{(N)}(A)\rangle =0\).

Let \(W\) be an unnormalized positive-length witness with positive weights \(a_k\). If some positive-length MPV coefficient of the original tensor is nonzero, then \(W\) has at least one block. For any nonempty witness, define

Then \(s{\gt}0\), every \(\nu _k\) is positive, \(|\nu _k|\le 1\), and \(|\nu _k|=1\) for at least one \(k\). For every \(N{\gt}0\),

Every MPS tensor has an unnormalized positive-length PGVWC07 witness. More precisely, the construction has four stages. First, a nonzero irreducible block remains irreducible under nonzero scalar rescaling and can be put in a unital gauge with scalar fixed-point space and a diagonal positive-definite dual fixed point. Second, this construction is applied to each member of a finite nonzero irreducible block family. Third, the nonzero irreducible decomposition of an arbitrary tensor supplies that family and the total bond-dimension bound. Finally, these blocks, weights, and fixed points form a witness in the sense of Definition 8.5.1.

Let \(A\) be a translation-invariant MPS representation of bond dimension \(D\). There is a positive scalar \(s\) and a finite family of blocks \(A_k\), possibly empty, such that the globally rescaled tensor \(s^{-1}A\) has the same MPV coefficients on every positive-length ring as \(\bigoplus _k \mu _k A_k^i\). The weights are positive and satisfy \(1\ge |\mu _k|\); if the family is nonempty, then some weight has norm \(1\). For each block,

Here \(\Lambda _k\) is diagonal, positive, and full-rank, and each block has positive bond dimension. Moreover, the only fixed points of \(\mathcal{E}_{A_k}(X)=\sum _i A_k^iX(A_k^i)^\dagger \) are the scalar multiples of \(\mathbb {1}\): for every \(X\), the identity \(\mathcal{E}_{A_k}(X)=X\) implies that there exists \(c\in \mathbb {C}\) such that \(X=c\mathbb {1}\). The total bond dimension of the decomposition is at most \(D\).

This is the positive-length form of [ PGVWC07 , Theorem 4 ] , with the global spectral-radius normalization step made explicit.

For every MPS tensor \(A\), one of the following alternatives holds:

1. \(V^{(N)}(A)_\sigma =0\) for every \(N{\gt}0\) and every word \(\sigma \);

2. there are a scalar \(s{\gt}0\) and a nonempty normalized block family satisfying all conclusions of Theorem 8.5.4.

Suppose that \(V^{(N)}(A)_\sigma \neq 0\) for some \(N{\gt}0\) and some word \(\sigma \). Then there are a scalar \(s{\gt}0\) and a nonempty finite family of positive-bond-dimension blocks \((A_k)_{k=1}^r\) with positive weights \((\mu _k)_{k=1}^r\) such that \(s^{-1}A\) and \(\bigoplus _k\mu _kA_k\) have identical MPV coefficients at every positive length. The weights satisfy \(|\mu _k|\le 1\), with equality for at least one block, and

for a diagonal positive-definite matrix \(\Lambda _k\). Every fixed point of \(\mathcal{E}_{A_k}\) is a scalar multiple of \(\mathbb {1}\), and \(\sum _k\dim (A_k)\le D\).

Let \(E\) be a linear endomorphism of a finite-dimensional complex inner product space. Then \(E\) is primitive if and only if its adjoint \(E^\dagger \) is primitive. Equivalently, the eigenvalues \(\lambda \) of \(E\) with \(|\lambda |=1\) form the singleton \(\{ 1\} \) if and only if the same holds for \(E^\dagger \), because adjunction sends every eigenvalue to its complex conjugate.

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 \(A\) and \(\bigoplus _k\mu _kB_k\) have the same MPV coefficients at every positive length, then after any positive blocking length \(p\), \(A^{[p]}\) and \(\bigoplus _k\mu _k^{\, p}B_k^{[p]}\) have the same MPV coefficients at every positive length.

If \(\bigoplus _j\mu _j^AA_j\) and \(\bigoplus _k\mu _k^BB_k\) have the same MPV coefficients at every positive length, then after any positive common blocking length \(p\), \(\bigoplus _j(\mu _j^A)^pA_j^{[p]}\) and \(\bigoplus _k(\mu _k^B)^pB_k^{[p]}\) have the same MPV coefficients at every positive length.

If \(U\) is a unitary matrix, then \(A\) and \(U^\dagger A U\) generate the same MPV family.

Let \(A\) be an irreducible MPS tensor with positive bond dimension and \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\). Let \((P_k)_{k\in \mathbb {Z}/m\mathbb {Z}}\) be orthogonal projections such that

If \(X,Y\in P_kM_{D}(\mathbb {C})P_k\) are fixed by \((\mathcal{E}_{A^\dagger })^m\), then

Let \(A\) be a left-canonical irreducible MPS tensor with \(D{\gt}0\), let \(m\) be the period of its cyclic-sector decomposition, and let \((P_u)_{u=1}^m\) be the cyclic projections of the adjoint transfer map with \(\mathcal{E}_{A^\dagger }(P_{u+1})=P_u\) and \(\sum _uP_u=\mathbb {1}\). Then the restriction of \((\mathcal{E}_{A^\dagger })^m\) to each corner \(P_uM_{D}(\mathbb {C})P_u\) is irreducible.

Let \(A\) be an irreducible MPS tensor with \(D {\gt} 0\) and some \(A^i \neq 0\). Then there exist a positive real \(r\), a positive definite matrix \(\sigma \), and a tensor \(B\) gauge-equivalent to \(r^{-1/2}A\) such that

and \(\sum _i (B^i)^\dagger B^i = \mathbb {1}\).

This is the “spectral rescaling \(+\) TP gauge” step of [ CPGSV16 , Appendix A ] . The similarity is generally non-unitary.

For any MPS tensor \(A\), there exist nontrivial blocks \((B_k)_{k=1}^r\) with nonzero weights \((\mu _k)_{k=1}^r\), each block irreducible, left-canonical \((\sum _i(B_k^i)^\dagger B_k^i=\mathbb {1})\), and of positive bond dimension, such that, for every \(N\ge 1\) and every \(\sigma \),

and \(\sum _{k=1}^rD_k\le D\).

For any MPS tensor \(A\), there exist \(p\ge 1\), nonzero weights \((\mu _k)_{k=1}^r\), and blocks \((B_k)_{k=1}^r\) such that, for every blocked \(\sigma \) of positive length \(N\),

The retained block dimensions satisfy \(\sum _{k=1}^{r}D_k\le D\). Each block is left-canonical and primitive:

Each bond dimension is positive.

Let \(A\) have positive bond dimension and be trace-preserving, primitive, and tensor-irreducible. For every \(k{\gt}0\),

and \(A^{[k]}\) is tensor-irreducible. This \(k\) is a later common-refinement or injectivity length, distinct from the period-removal length that produced the sector block.

Let \(A\) be an MPS tensor whose transfer map \(\mathcal{E}_A\) is irreducible. If \(\rho \geq 0\), \(\rho \neq 0\), and \(\mathcal{E}_A(\rho )=\rho \), then \(\rho \) is positive definite.

If \(\mathcal{E}_A\) has a nonzero PSD fixed point \(\rho \) that is not positive definite, then there exist MPS tensors \(A_1,A_2\) of smaller bond dimensions with \(\mathcal{V}(A)=\mathcal{V}(A_1\oplus A_2)\).

Let \(A\) and \(B\) have the same MPV family at every positive length. Then there is a positive blocking length at which both blocked tensors have the same positive-length MPV family as weighted direct sums of trace-preserving, primitive, tensor-irreducible blocks with positive bond dimensions and nonzero weights. The two weighted nonzero-sector families have the same positive-length MPV family.

Let \(A\) be an irreducible MPS tensor with \(D {\gt} 0\) and some \(A^i \neq 0\). Then there exist a positive real \(r\), a positive definite matrix \(\rho \), and a tensor \(B\) gauge-equivalent to \(r^{-1/2}A\) such that

and \(\sum _i B^i(B^i)^\dagger = \mathbb {1}\).

This is the Perron–Frobenius unital-gauge orientation used in [ PGVWC07 , Theorem 4, lines 765–770 ] , stated for one irreducible nonzero block.

Let \(\rho \) be positive definite with \(\mathcal{E}_A(\rho )=r\rho \) for some \(r{\gt}0\). For \(B^i:=r^{-1/2}\rho ^{-1/2}A^i\rho ^{1/2}\),

If \(r=1\), then the unscaled gauge \(B^i:=\rho ^{-1/2}A^i\rho ^{1/2}\) is unital and generates the same MPV family as \(A\). This is the spectral-radius normalization and full-rank fixed-point gauge of [ PGVWC07 , Theorem 4 ] .

Suppose \(A\) is unital, \(\mathcal{E}_A(\mathbb {1})=\mathbb {1}\). If \(X\) is a Hermitian fixed point of \(\mathcal{E}_A\) which is not a scalar multiple of \(\mathbb {1}\), then there is a nonzero PSD fixed point \(\rho \) which is not positive definite, and consequently \(A\) admits a two-block MPV-equivalent decomposition with both block dimensions strictly smaller than \(D\). This is the fixed-point shift and further decomposition step in [ PGVWC07 , Theorem 4 ] .