Dependencies

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

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 \(A^{[L]}\).

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

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

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

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).

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

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

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

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

Equivalently, its Kraus map is trace-preserving.

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

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

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.

Two blocks \(A\) and \(B\) are MPV-phase equivalent if there is a nonzero scalar \(\zeta \) such that, for every positive system size \(N\) and every physical word, \(V^{(N)}(B)_\sigma =\zeta ^N V^{(N)}(A)_\sigma \). The two bond dimensions are allowed to differ.

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.

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

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

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.

Two tensors \(A\) and \(B\) (of the same bond dimension) generate the same MPV family if \(|V^{(N)}(A)\rangle = |V^{(N)}(B)\rangle \) for every system size \(N\).

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 \(N {\gt} 0\).

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 the sectors of two decompositions \(P\) and \(Q\) be matched by \(\beta :J(Q)\simeq J(P)\), with sector phases \(\zeta _k\). A copy-weight matching consists of copy permutations \(\tau _k\) such that, for all sectors \(k\) and copies \(q\), \(\nu _{k,q}=\zeta _k^{-1}\mu _{\beta (k),\tau _k(q)}\). This is the copy-level conclusion of [ CPGSV16 , Appendix MPV proof, line 1188 ] .

The construction takes as input a sector matching \(\beta :J(Q)\simeq J(P)\), nonzero sector phases \(\zeta _k\), and the eventual matched-sector coefficient identities

It returns the copy-weight matching obtained by choosing, in every sector, a copy permutation \(\tau _k\) satisfying \(\nu _{k,q}=\zeta _k^{-1}\mu _{\beta (k),\tau _k(q)}\). This is the sectorwise construction in [ CPGSV16 , Appendix MPV proof, line 1188 ] .

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.

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 \(I\) be a finite index set and let \((A_i)_{i \in I}\) be a finite family of tensors such that \(O_{A_iA_i}(N) \to 1\) for each \(i\), and \(O_{A_iA_j}(N) \to 0\) whenever \(i \neq j\). Then the MPV states \(\{ |V^{(N)}(A_i)\rangle \} _{i \in I}\) are linearly independent for all sufficiently large \(N\).

Let \((A_j)_{j=1}^{g_A}\) and \((B_k)_{k=1}^{g_B}\) be finite families of tensors. Suppose the overlaps inside each family are asymptotically orthonormal, and suppose every mixed overlap \(O_{A_jB_k}(N)\) tends to \(0\). Then the combined MPV family

is linearly independent for all sufficiently large \(N\).

If \(B^i = X A^i X^{-1}\) for all \(i\), then \(B^w = X A^w X^{-1}\) for every word \(w\).

Let \(\zeta A\) denote the tensor with matrices \(\{ \zeta A^i\} _{i=0}^{d-1}\). Then, for any word \(w\), \((\zeta A)^w = \zeta ^{|w|} A^w\), where \(|w|\) is the length of \(w\).

If a finite family of vectors is linearly independent for all sufficiently large \(N\), and two finite linear combinations of that family are equal for all sufficiently large \(N\), then the corresponding coefficients are equal for all sufficiently large \(N\).

Let \(w_1,\ldots ,w_n\) be nonzero complex numbers. If \(\sum _i c_i w_i^N=0\) for every sufficiently large \(N\), then the same equality holds for every non-negative exponent.

If \(A\) is a primitive MPS tensor with PSD fixed point \(\rho \), then every fixed point \(\sigma \) of \(\mathcal{E}_A\) is proportional to \(\rho \): \(\sigma =\frac{\operatorname{tr}(\sigma )}{\operatorname{tr}(\rho )}\rho \).

If \(A\) is a primitive MPS tensor with positive-definite fixed point \(\rho \), then the transfer map \(\mathcal{E}_A\) is irreducible.

The proof uses fixed-point uniqueness (Lemma C.5.5): every PSD fixed point is proportional to \(\rho \), so Wolf’s criterion for irreducibility (a positive-definite fixed point together with uniqueness of positive fixed points implies irreducibility) applies directly.

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 \(A\) and \(B\) have the same bond dimension. If \(\| \langle V^{(N)}(A) | V^{(N)}(B) \rangle \| \to 1\), then the mixed transfer map has spectral radius at least \(1\).

Let \(\sigma \) be positive definite. The transfer map of the Perron gauge determined by \(\sigma \) is primitive if and only if the original transfer map is primitive.

Suppose every sector of \(Q\) has an equal-dimensional, gauge-phase-equivalent partner in \(P\) with non-decaying overlap, and the converse holds from \(P\) to \(Q\). Then the forward partners may be chosen as a bijection between the two finite sector sets, preserving all three properties.

Let \(P\) be a BNT canonical form, let \(Q\) be a sector decomposition, and assume \(P_{\mathrm{tot}}\) and \(Q_{\mathrm{tot}}\) have the same MPV family at every positive length. Suppose that there is a bijection \(\beta :J(Q)\simeq J(P)\) and scalars \(\zeta _k \in \mathbb {C}\) such that, for every \(k \in J(Q)\), every positive length \(N\), and every spin configuration \(\sigma \) of length \(N\),

Then for every \(k \in J(Q)\) there is \(N_0\) such that, for all \(N{\gt}N_0\), \(c_N^{(\beta (k))}(P)=\zeta _k^N\, c_N^{(k)}(Q)\). This is the coefficient-comparison step of [ CPGSV16 , Appendix MPV proof, lines 1187–1188 ] in full-basis form.

Let \(P\) and \(Q\) each be a BNT canonical form, and suppose every mixed overlap \(O_{P_j Q_k}(N)\) tends to \(0\). Then the union family

is linearly independent for all sufficiently large \(N\). This is [ CPGSV16 , Appendix A ] .

If \(P\) is a BNT canonical form and \(j \neq k\), then \(O_{P_j P_k}(N) \to 0\).

Let \(P\) and \(Q\) be BNT canonical forms whose total tensors have the same MPV family at every positive length. Then there are a bijection \(\beta :J(Q)\simeq J(P)\), bond-dimension equalities, unit-modulus phases \(\zeta _k\), and block gauges \(X_k\) such that \(Q_k^i=\zeta _k X_kP_{\beta (k)}^iX_k^{-1}\). For the same phases there is a copy-weight matching: after a permutation of the copies inside each matched sector, \(\nu _{k,q}=\zeta _k^{-1}\mu _{\beta (k),\tau _k(q)}\).

Under the hypotheses of Lemma 9.7.1.1, the per-sector multiplicities agree and there is a permutation \(\tau \) of the per-copy index set identifying the individual weights up to the gauge phase: for every \(q\), \(\nu _{\tilde k_0, \tau (q)} = \zeta ^{-1} \cdot \mu _{j_0, q}\). This is the per-copy realisation of [ CPGSV16 , Appendix A ] : the multiset equality from Lemma 9.7.1.1 is upgraded to a concrete indexing between the matched \(P\)- and \(Q\)-copies.

Let \(P, Q\) be sector decompositions with weights \(\mu _{j,q}\) and \(\nu _{k,q'}\), and fix indices \(j_0\), \(\tilde k_0\), and a nonzero scalar \(\zeta \in \mathbb {C}\setminus \{ 0\} \). Suppose that for every \(N {\gt} N_0\),

Then the per-sector multiplicities agree, \(r_{j_0}(P) = r_{\tilde k_0}(Q)\), and the rescaled per-copy weight multisets coincide. This is [ CPGSV16 , Appendix A ] read on a single matched sector.

For any sector decomposition \(P\) and any sector \(j\), the coefficient \(c_{P,j}^{(N)}=\sum _q\mu _{j,q}^N\) is not eventually zero in \(N\). More generally, for every positive integer \(p\), the subsequence \(c_{P,j}^{(pN)}\) is not eventually zero.

If \(A\) is a primitive MPS tensor with positive-definite fixed point \(\rho \), then it satisfies the strengthened condition in Definition 7.1.1.6: \(\mathcal{E}_A\) is irreducible, the fixed point \(\rho \) is positive definite, and the peripheral spectrum of \(\mathcal{E}_A\) is \(\{ 1\} \). The source clause is [ SPGWC10 , Proposition 3(c) ] ; the definition used here records irreducibility separately.

If, for every block \(k\) and physical index \(i\), \(B_k^i=X_kA_k^iX_k^{-1}\), then the weighted direct sums satisfy

where \(X\) is the flattened block-diagonal gauge from Definition 10.3.1. Hence the two assembled tensors are gauge equivalent.

If \(\sigma \) is positive definite and \(B\) is defined by (1), then \(A\) and \(B\) are gauge equivalent.

The trace-adjoint identity of Theorem  [ LTC26 , “Trace adjointness of a Kraus map” ] holds when the Kraus family is the family of matrices of an MPS tensor and the two Kraus maps are written as transfer maps.

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 left-canonical tensor of positive bond dimension. If \(O_{AA}(N)\to 1\), then \(A\) is normal.

Let \(A\) be an irreducible left-canonical tensor of positive bond dimension. If \(O_{AA}(N)\longrightarrow 1\), then the transfer map of \(A\) is primitive and \(A\) is normal.

Let \(\alpha , \beta : \{ 0, \ldots , n-1\} \to \mathbb {C}\). If \(\sum _i \alpha _i^k = \sum _i \beta _i^k\) for \(1 \le k \le n\), then the multisets \(\{ \alpha _i\} \) and \(\{ \beta _i\} \) are equal.

Two variants relax the equal-cardinality assumption. If \(\alpha : \{ 0, \ldots , m-1\} \to \mathbb {C}\) and \(\beta : \{ 0, \ldots , n-1\} \to \mathbb {C}\) have no zero entries and their power sums agree for \(1 \le k \le \max (m,n)\), then \(m=n\) and the multisets are equal. Without a nonzero hypothesis, the list form \(\sum _{k=1}^{x_a} \lambda _{a,k}^N = \sum _{k=1}^{x_b} \lambda _{b,k}^N\) for \(1 \le N \le \max (x_a,x_b)\) gives equality of the nonzero multisets; if it also holds at \(N=0\) then \(x_a=x_b\) and the full lists agree. The nonzero variant is the one applied to the nonzero coefficients of a canonical form, which is how [ CPGSV16 , Lemma A.5 ] is used.

Let \(E\) be a channel with a positive-definite fixed point. If all nonzero positive semidefinite fixed points of \(E\) are proportional, then \(E\) is irreducible.

Let a left-canonical tensor \(A\) commute with a family of orthogonal projections \((P_k)\) summing to \(\mathbb {1}\). Then \(A\) decomposes into compressed sector tensors \((C_k)\), with a unit-weight direct-sum MPV decomposition, corner isomorphisms \(\varphi _k:M_{\dim C_k}(\mathbb {C})\xrightarrow {\sim }P_kM_{D}(\mathbb {C})P_k\), and rectangular isometries \(V_k\) satisfying

The sector letters obey \(\varphi _k(C_k^i)=P_kA^iP_k\).

Every normal tensor has positive bond dimension.

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 \(A\) and \(B\) be normal tensors of positive, possibly different, bond dimensions. Their self-overlaps satisfy

Moreover, either \(O_{AB}(N)\longrightarrow 0\), or their bond dimensions are equal; call the common value \(D\). Then there are \(X\in \mathrm{GL}_D(\mathbb {C})\) and \(\zeta \in \mathbb {C}\), with \(|\zeta |=1\), such that \(B^i=\zeta X A^iX^{-1}\) for every physical letter \(i\). In the latter case, \(|O_{AB}(N)|\to 1\). This is [ CPGSV16 , Lemma A.2 ] .

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 \(A\) 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 \(A\) and \(B\) be normalized MPS tensors. Every eigenvalue \(\mu \) of \(F_{AB}\) satisfies \(|\mu |\leq 1\).

Let \(A\) and \(B\) be normalized MPS tensors of bond dimensions \(D_1\geq 1\) and \(D_2\geq 1\). Every eigenvalue \(\mu \) of \(F_{AB}^{\mathrm{rect}}\) satisfies \(|\mu |\leq 1\).

If \(A\) and \(B\) are gauge equivalent, then they generate the same MPV family.

If \(B^i = \zeta X A^i X^{-1}\) for every physical index \(i\), then for every system size \(N\) and configuration \(\sigma \) one has \(V^{(N)}(B)_\sigma = \zeta ^N V^{(N)}(A)_\sigma \).

Let \(A\) be a normalized tensor of bond dimension \(D_1\) and \(B\) a normalized tensor of bond dimension \(D_2\). Suppose \(\rho _A,\rho _B\) are fixed points of the respective transfer maps and \(S_A,S_B\) are invertible matrices with \(S_AS_A^\dagger =\rho _A\) and \(S_BS_B^\dagger =\rho _B\) (for instance the positive definite Perron–Frobenius fixed points of Theorem 5.3.2 and their square roots, as in the right-canonical gauge of Theorem  [ LTC26 , “Right-canonical (unital) gauge” ] ). Gauge to unital families \(A'^i=S_A^{-1}A^iS_A\) and \(B'^i=S_B^{-1}B^iS_B\). If \(X\in M_{D_1\times D_2}(\mathbb {C})\setminus \{ 0\} \) satisfies \(F_{AB}^{\mathrm{rect}}(X)=\mu X\) with \(|\mu |=1\), then \(X'=S_A^{-1}X(S_B^\dagger )^{-1}\) is nonzero, the families \(A'\) and \(B'\) are unital, and, for every \(i\),

Let \(A\) and \(B\) be tensors of the same bond dimension. Suppose invertible gauges take them to \(A'\) and \(B'\), and suppose there are an invertible matrix \(X'\) and a scalar \(\mu \) with \(|\mu |=1\) such that \(A'^iX'=\mu X'B'^i\) for every physical index \(i\). Then \(A\) and \(B\) are gauge-phase equivalent.

If \(A\) is an injective MPS tensor, then its transfer map \(\mathcal{E}_A\) is irreducible.

If a finite matrix family satisfies

then every blocked family satisfies the same identity.

For every \(X\in M_{D}(\mathbb {C})\) and \(N\geq 0\),

where \(A^\sigma =A^{\sigma _1}\cdots A^{\sigma _N}\) denotes the word evaluation of Definition 2.1.2.

Let \(A\) and \(B\) be injective normalized MPS tensors. If \(\rho _{\operatorname{spec}}(F_{AB})\geq 1\), then \(A\) and \(B\) are gauge-phase equivalent.

Let \(A\) and \(B\) be irreducible trace-preserving normalized MPS tensors of bond dimension \(D\). If \(\rho _{\operatorname{spec}}(F_{AB})\geq 1\), then \(A\) and \(B\) are gauge-phase equivalent.

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 irreducible MPS tensors of the same bond dimension with \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\) and \(\sum _i(B^i)^\dagger B^i=\mathbb {1}\). If \(\| \langle V^{(N)}(A) | V^{(N)}(B) \rangle \| \to 1\), then \(A\) and \(B\) are gauge-phase equivalent.

Let \(A\) and \(B\) be irreducible trace-preserving 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 irreducible and trace-preserving normalized, then \(O_{AB}(N)\to 0\) as \(N\to \infty \).

For tensors \(A\) and \(B\) of bond dimension \(D\geq 1\),

For tensors \(A\) of bond dimension \(D_1\geq 1\) and \(B\) of bond dimension \(D_2\geq 1\),

Let \(D{\gt}0\) and \(m{\gt}0\). Let \(A\) be a left-canonical irreducible MPS tensor with bond dimension \(D\), and suppose that the peripheral eigenvalues of its transfer map are exactly \(\{ z\in \mathbb {C}:z^m=1\} \). Then \(O_{AA}(mn)\longrightarrow m\).

Let \(A\) be an irreducible MPS tensor with \(D {\gt} 0\) and some \(A^i \neq 0\). Then there exist a positive definite matrix \(\sigma \) and a positive real \(r {\gt} 0\) such that \(\mathcal{E}_A^\dagger (\sigma ) = r\sigma \).

This combines the eigenvector-existence part of [ Wol12 , Theorem 6.5 ] with the irreducible upgrade to positive definiteness ( [ Wol12 , Theorem 6.3(2)–(3) ] ); the application to the adjoint transfer map follows  [ CPGSV16 , Appendix A ] .

Let \(\alpha , \beta : \{ 0, \ldots , n-1\} \to \mathbb {C}\). If \(\sum _i \alpha _i^k = \sum _i \beta _i^k\) for all \(k \ge 1\), then the multisets \(\{ \alpha _i\} \) and \(\{ \beta _i\} \) are equal.

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.

Let \(A\) be a normalized MPS tensor with a nonzero PSD fixed point \(\rho \) of the transfer map. If \(\rho _{\operatorname{spec}}(\mathcal{E}_A-P){\lt}1\), where \(P\) is the fixed-point projection, then \(O_{AA}(N)\to 1\) as \(N\to \infty \).

Assume \(D \ge 1\). Let \(A\) be an MPS tensor with \(\sum _i (A^i)^\dagger A^i = \mathbb {1}\), so that \(\mathcal{E}_A\) is trace-preserving. Then there exists \(\rho \ge 0\), \(\rho \neq 0\), with \(\mathcal{E}_A(\rho ) = \rho \).

Let \(A\) be an injective MPS tensor and let \(\rho \ge 0\), \(\rho \neq 0\), be a fixed point of \(\mathcal{E}_A\). Then \(\rho \) is positive definite.

Let \(A\) be an injective MPS tensor. If \(\rho ,\sigma \geq 0\) are nonzero fixed points of \(\mathcal{E}_A\), then \(\sigma =c\rho \) for some \(c\in \mathbb {C}\).

Let \(A\) be an MPS tensor whose transfer map is irreducible. If \(\rho ,\sigma \geq 0\) are fixed points of \(\mathcal{E}_A\) and \(\rho \neq 0\), then \(\sigma =c\rho \) for some \(c\in \mathbb {C}\).

Let \(A\) be irreducible and suppose that the spectral radius of \(\mathcal{E}_A\) is one. There is a positive definite matrix \(\sigma \) satisfying

The tensor

is left-canonical and irreducible, and \(A\) and \(B\) are related by a pure invertible similarity.

Assume \(D \ge 1\). Let \(A\) be an injective MPS tensor with \(\sum _i (A^i)^\dagger A^i = \mathbb {1}\). Then the transfer map \(\mathcal{E}_A\) has a unique positive semidefinite fixed point \(\rho \) up to scaling, and \(\rho \) is positive definite.

Let \(A\) and \(B\) be irreducible trace-preserving tensors of positive, possibly different, bond dimensions. If \(|O_{AB}(N)|\longrightarrow 1\), then their bond dimensions are equal.

Let \(P, Q\) be two BNT canonical forms with the same MPV family at every positive length. Then there is a bijection \(\beta : J(Q) \simeq J(P)\) such that for every \(k \in J(Q)\) the bond dimensions match, the basis tensors \(P_{\beta (k)}\) and \(Q_k\) are gauge-phase equivalent after identifying the matched bond dimensions, and the basis cross-overlap does not tend to zero.

Let \(P\) and \(Q\) be BNT canonical forms whose total tensors have the same MPV family at every positive length. Then there exist:

• a bijection \(\beta :J(Q)\simeq J(P)\),

• bond-dimension equalities \(\dim P_{\beta (k)}=\dim Q_k\),

• copy-number equalities \(r_{\beta (k)}(P)=r_k(Q)\) and copy permutations \(\tau _k\),

• unit-modulus phases \(\zeta _k\) with \(|\zeta _k|=1\) and block gauges \(X_k\),

such that, for every sector \(k\), copy \(q\), and physical index \(i\),

Moreover, if \(X=\bigoplus _k (\mathbf{1}_{r_k}\otimes X_k)\) is the flattened block-diagonal gauge, then, for every physical index \(i\), the flattened \(Q\)-coordinates satisfy

This is the explicit direct-sum gauge construction of [ CPGSV16 , Appendix MPV proof, lines 1189–1192 ] .

Let \(P, Q\) be two BNT canonical forms with the same MPV family at every positive length. Then for every sector \(k\) of \(Q\) there exist a sector \(j\) of \(P\) with matched bond dimension \(\dim _P(j) = \dim _Q(k)\), a gauge-phase equivalence after identifying the matched bond dimensions, and a non-decaying cross-overlap.

Suppose that the sectors of \(P\) and \(Q\) are matched, with bond-dimension equalities, nonzero phases, and block gauges satisfying \(Q_k^i=\zeta _k X_kP_{\beta (k)}^iX_k^{-1}\). If the same phases also give a copy-weight matching, then the flattened \(Q\)-tensor is obtained from the matched-coordinate \(P\)-tensor by the direct-sum gauge \(X=\bigoplus _k(\mathbf{1}_{r_k}\otimes X_k)\). This is only a reformulation of Theorem 10.3.2.

Suppose that the sectors of two sector decompositions \(P\) and \(Q\) are matched by a bijection \(\beta :J(Q)\simeq J(P)\), with bond-dimension equalities, copy permutations \(\tau _k\), nonzero phases \(\zeta _k\), and block gauges \(X_k\). If, for every sector \(k\), copy \(q\), and physical index \(i\),

then, in the flattened \(Q\)-coordinates,

This is the direct-sum gauge construction of [ CPGSV16 , Appendix MPV proof, lines 1189–1192 ] , separated from the coefficient-comparison step that supplies the weight identities.

Let \(P\) and \(Q\) be BNT canonical forms whose total MPV families are eventually nonzero and proportional. For every sector \(j\) of \(P\), there is a sector \(k\) of \(Q\) with equal bond dimension, gauge-phase-equivalent basis tensor after the dimension identification, and non-decaying cross-overlap.

Under the same hypotheses, every sector of \(Q\) has an equal-dimensional, gauge-phase-equivalent partner in \(P\) with non-decaying cross-overlap.

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.

A nonzero multiplicative \(\mathbb {C}\)-linear endomorphism of \(M_{D}(\mathbb {C})\) is bijective.

For normalized tensors, \(\rho _{\operatorname{spec}}(F_{AB})\leq 1\).

Let \(A\) be a tensor normalized on the support of an orthogonal projection \(P\), so that \(PA^iP=A^i\) and \(\sum _i(A^i)^\dagger A^i=P\). Then there are a corner tensor \(C\), a linear isomorphism \(\varphi :M_{\dim C}(\mathbb {C})\xrightarrow {\sim }PM_{D}(\mathbb {C})P\), and a rectangular isometry \(V:\mathbb {C}^{\dim C}\to \mathbb {C}^D\) such that

Moreover, \(\varphi (C^i)=A^i\) for every physical letter, and for every word \(\sigma =(\sigma _1,\ldots ,\sigma _N)\),

Let \(\sigma \) be positive definite and satisfy \(\sum _i (A^i)^\dagger \sigma A^i = \sigma \). Define

Then \(B\) is trace-preserving: \(\sum _i (B^i)^\dagger B^i = \mathbb {1}\).

Let \(E\) be trace-preserving with fixed point \(\rho \), where \(\operatorname{tr}(\rho )\neq 0\), and let \(P\) be the fixed-point projection of the corresponding definition in the companion quantum-channel volume  [ LTC26 , “Fixed-point projection” ] . If \(\rho _{\operatorname{spec}}(E-P){\lt}1\), then \(\operatorname{Tr}(E^n)\to 1\) as \(n\to \infty \).

If \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\), then the transfer map \(\mathcal{E}_A(X)=\sum _i A^iX(A^i)^\dagger \) is a quantum channel (CPTP).

The transfer map \(\mathcal{E}_A(X)=\sum _i A^iX(A^i)^\dagger \) is completely positive.

If an MPS tensor \(A\) is irreducible, then its transfer map \(\mathcal{E}_A\) is irreducible.

Let \(A\) and \(B\) be irreducible trace-preserving normalized MPS tensors of the same bond dimension that are not gauge-phase equivalent. Then \(\rho _{\operatorname{spec}}(F_{AB}){\lt}1\).

Let \(A\) and \(B\) be irreducible trace-preserving normalized tensors of bond dimensions \(D_1\) and \(D_2\), respectively, with \(D_1\neq D_2\). Then \(\rho _{\operatorname{spec}}(F_{AB}^{\mathrm{rect}}){\lt}1\).

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.