Dependencies

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 ] .

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

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

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

Equivalently,

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

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

The coefficient form of a basis of normal tensors: each block \(A_j\) is a CPSV normal tensor (Definition 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.

A BNT sector presentation of a tensor \(A\) is a nonempty sector decomposition \(P\) such that every representative has a positive number of copies, every copy has nonzero weight, every representative \(C_j\) is normal, the representative matrix product vectors are eventually linearly independent, and, for every \(N{\gt}0\),

This records the nonzero canonical summands constructed at [ CPGSV16 , lines 217–246 ] : repeated copies and their weights remain explicit.

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.

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

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

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.

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.

For two finite nonzero-weight block families, a common MPV phase cover consists of a third finite family of blocks, a map from each nonzero-weight block family onto that common family, and MPV-phase equivalences between each block and its common representative. Both maps are required to be surjective, so the common family contains no unused representative.

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.

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.

We regard \(|V^{(N)}(A)\rangle \) as an element of \(\mathbb {C}^{d^N}\) indexed by configurations \(\sigma \in \{ 0,\ldots ,d{-}1\} ^N\), with \(\sigma \)-component \(V^{(N)}(A)_\sigma \) as in Definition 2.1.7. This allows us to use the standard Euclidean inner product on \(\mathbb {C}^{d^N}\).

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

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

For a tensor \(A\), its single-sector decomposition has \(A\) as its only basis tensor, with one copy of weight \(1\).

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 \((A_j)_{j=1}^g\) 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 \(A^1,\ldots ,A^r\) be irreducible left-canonical representatives with normalized self-overlaps, and suppose that distinct same-dimensional representatives are not gauge-phase equivalent. Fix \(L_0{\gt}0\) such that every block is injective at lengths \(L_0\), \(L_0+1\), and \(3(L_0+1)\). Then each ordered pair \(k\ne j\) is trace-separated at the common length \(S=3(L_0+1)\): the map

is onto \(M_{D_k}(\mathbb {C})\oplus M_{D_j}(\mathbb {C})\). In particular, one may prescribe an arbitrary matrix on either block while making the same word combination vanish on the other block.

If two finite families of blocks have asymptotically orthonormal overlaps within each family, and if their MPV spans agree for all sufficiently large system sizes \(N\), then for all sufficiently large \(N\) there exists an invertible matrix \(U_N\) expressing the MPV states of one family as linear combinations of the MPV states of the other.

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 \((A_j)_{j=1}^g\) 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 \((A_j)_{j=1}^{g_A}\) and \((B_k)_{k=1}^{g_B}\) be two families of injective tensors of positive bond dimension satisfying the TP normalization \(\sum _i (A_j^i)^\dagger A_j^i = \mathbb {1}\) and \(\sum _i (B_k^i)^\dagger B_k^i = \mathbb {1}\), whose self-overlaps converge to \(1\) and whose cross-overlaps within each family converge to \(0\). If, for all sufficiently large system sizes \(N\),

then \(g_A = g_B\), and there is a permutation \(\pi \) of the common block index set such that \(A_j\) and \(B_{\pi (j)}\) have the same bond dimension and are gauge-phase equivalent for each \(j\).

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 \(N\).

Every block in a coefficient-form basis of normal tensors has positive bond dimension.

For any words \(w_1, w_2\), \(A^{w_1 \cdot w_2} = A^{w_1} A^{w_2}\).

If \(w = (J_1, \ldots , J_m)\) is a word in the blocked alphabet, where each \(J_k\) is an \(L\)-tuple in \(\{ 0, \ldots , d{-}1\} ^L\), then concatenating the block words gives a word \(\widetilde{w}\) of length \(mL\) in the original alphabet and \((A^{[L]})^w = A^{\widetilde{w}}\).

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

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 each pair \((A_k, B_k)\) is gauge equivalent, then the weighted block-diagonal tensors \(\bigoplus _k \mu _k A_k\) and \(\bigoplus _k \mu _k B_k\) are gauge equivalent.

If \(A\) and \(B\) are gauge equivalent and \(A\) is \(L\)-block injective, then \(B\) is \(L\)-block injective.

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 \(\mathcal{E}_B(\rho )=r\rho \).

Multiplying every block weight in a block-diagonal MPS tensor by a scalar \(c\) multiplies every length-\(N\) MPV coefficient by \(c^N\).

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

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.

For prepared trace-preserving primitive irreducible NTs of positive bond dimension with all copy weights nonzero, grouping CF summands by BNT representative preserves the total bond dimension: the collapsed sector decomposition has total bond dimension equal to the sum of the original block dimensions.

Let \(X\) and \(Y\) be trace-preserving, primitive, irreducible NTs of positive bond dimension. If their MPV families differ by a nonzero scalar power, then \(X\) and \(Y\) have the same bond dimension.

Let \(X\) and \(Y\) be trace-preserving, primitive, irreducible NTs of positive bond dimensions. Fix \(\zeta \in \mathbb {C}\setminus \{ 0\} \) and suppose that, for every positive length \(N\) and every physical word \(\sigma \),

Then, after identifying the bond spaces, there exists \(G\in \mathrm{GL}_{D_Y}(\mathbb {C})\) such that, for every physical index \(i\),

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.

For every sector decomposition \(P\), length \(N\), and basis index \(j\), \(c_N^{(j)} = \sum _{q=1}^{r_j} \mu _{j,q}^N\). This is [ CPGSV16 , Section 2.3 ] .

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 \(O_{P_j P_k}(N) \to 0\).

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.

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

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

Let \((A_j)_{j=1}^g\) and \((B_j)_{j=1}^g\) be injective families of positive bond dimension satisfying trace-preserving normalization, with self-overlaps tending to \(1\) and off-diagonal overlaps tending to \(0\) within each family. If the spans of the MPV states agree for all sufficiently large system sizes \(N\), then there is a permutation \(\pi \) of \(\{ 1, \ldots , g\} \) such that \(A_j\) and \(B_{\pi (j)}\) have the same bond dimension and are gauge-phase equivalent.

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 \(P\) be a BNT canonical form of total bond dimension \(D\), with representatives \(A_j\) of bond dimensions \(D_j\). For every \(0\leq j{\lt}g\), the representative \(A_j\) is \(D^4\)-block injective: \(S_{D^4}(A_j)=M_{D_j}(\mathbb {C})\).

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 \(A\) 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.

Let \(A\) and \(B\) be MPS tensors, possibly with different bond dimensions. Suppose that \(U\) satisfies \(UU^\dagger =\mathbb {1}\) and \(A^i=U^\dagger B^iU\) for every physical index \(i\). Then \(A\) and \(B\) generate the same matrix product vectors at every positive length.

Let \((A_j)_{j=1}^g\) be a finite family of positive-bond-dimension, left-canonical normal tensors. There is a positive integer \(L\) such that every \(A_j\) is \(L\)-block injective.

Let \((B_k)_{k=1}^r\) be a finite family of positive-bond-dimension MPS tensors, each trace-preserving, primitive, and tensor-irreducible. Then there is a single \(L\ge 1\) such that each \(B_k^{[L]}\) is one-site injective, and this blocking preserves trace preservation, primitivity, and irreducibility for every block.

Let \(P\) and \(Q\) be BNT canonical forms (Definition 9.6.1.13) whose total tensors \(P_{\mathrm{tot}}\), \(Q_{\mathrm{tot}}\) generate the same MPV family for every positive system size \(N\). Every listed sector has a positive number of copies and every copy weight is nonzero. Then their total bond dimensions coincide; writing the common value as \(D\), there is one invertible matrix \(X\in \mathrm{GL}_D(\mathbb {C})\) such that, for every physical index \(i\),

Let \((A_j)_{j=1}^g\) be a basis of normal tensors, with bond dimensions \(D_j\). There is a positive integer \(L\) such that the one-letter evaluations of the blocked tensors span the full product algebra:

Moreover, there is \(L_0\) such that, for every \(L\geq L_0\), the unblocked word tuples have the simultaneous span

This is the existence form of simultaneous block injectivity associated with [ CPGSV16 , lines 317–345 ] ; no numerical bound on \(L\) or \(L_0\) is asserted here.

Distinct members of a basis of normal tensors are not gauge-phase equivalent after identifying equal bond dimensions.

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 literal canonical-form data of ambient bond dimension \(D\) have phase classes \(j\) and chosen representatives of dimensions \(D_j\). Then

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.

Two BNT presentations of the same positive-length matrix product vector family have equally many representatives. After a permutation, corresponding representatives have equal bond dimension and are related by an invertible gauge and a unit-modulus phase.

Let \((A_j)_{j=1}^{g}\) be a basis of normal tensors with bond dimensions \(D_j\). If \(K{\gt}0\) and \(\sum _jD_j\leq K\), then there is a positive integer \(L\leq 3K^5\) such that

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.

Every tensor with literal CPSV canonical-form data admits a finite basis of normal tensors \((B_j)_{j=1}^{g}\), each of positive bond dimension. The family is obtained by choosing one representative from each matrix-product-vector phase class of the blocks \(A_1,\ldots ,A_r\).

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 \(A\) and \(B\) have the same ambient bond dimension \(D\). Suppose each tensor has CPSV canonical-form data, both data satisfy the separate weight normalization condition (26), neither tensor is zero, and \(\mathcal{V}^+(A)=\mathcal{V}^+(B)\). Then \(A\) and \(B\) are gauge equivalent: there is \(G\in \mathrm{GL}_D(\mathbb {C})\) such that, for every physical index \(i\),

Unused ambient coordinates are allowed. The conclusion requires the common ambient dimension \(D\) and makes no claim for canonical forms with different ambient bond dimensions.

Let \(A\) be in literal CPSV canonical form with positive ambient bond dimension \(D\). There is a basis of normal tensors \((A_j)_{j=1}^{g}\) and a positive integer \(L\leq 3D^5\) such that

Hence \(A\) is in block-injective canonical form after blocking at most \(3D^5\) sites. This is the proposition at [ CPGSV16 , lines 342–345 ] .

Every normal tensor has positive bond dimension.

Every normal tensor in the sense of [ CPGSV16 , Section 2.2 ] is injective after blocking by some finite length; its positive bond dimension follows from normality.

Let \(A\) and \(B\) be normal tensors of positive, possibly different, bond dimensions. Then either \(O_{AB}(N)\to 0\), or there is a phase \(\zeta \in \mathbb {C}\), with \(|\zeta |=1\), such that \(|V^{(N)}(B)\rangle =\zeta ^N|V^{(N)}(A)\rangle \) for every chain length \(N\). This is [ CPGSV16 , Corollary A.3 ] .

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 \(A\) 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 \((A_j)_{j=1}^g\) 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 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.

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 \(A\) have CPSV canonical-form data satisfying the separate normalization condition (26), and assume \(A\ne 0\). Then there is a BNT canonical-form sector decomposition \(P\) such that

Thus the dimension is the total dimension of the blocks, not the ambient bond dimension \(D\). This statement records positive-length MPV equality; the exact coisometric reconstruction is stated next.

Let \(A\) have CPSV canonical-form data satisfying the separate normalization condition (26), and assume \(A\ne 0\). There are a BNT canonical-form sector decomposition \(P\), a coisometry \(U\in \mathbb {C}^{D_{\mathrm{tot}}(P)\times D}\), and \(X\in \mathrm{GL}_{D_{\mathrm{tot}}(P)}(\mathbb {C})\). The first two identities below hold, and the third holds for every physical index \(i\):

Let \(A\) and \(B\) be tensors, possibly with different bond dimensions, whose self-overlaps tend to one. Suppose that, for all sufficiently large \(N\), there is \(c_N\in \mathbb {C}\) such that \(|V^{(N)}(A)\rangle =c_N|V^{(N)}(B)\rangle \). Then \(|O_{AB}(N)|\to 1\).

Let \(A\) be a left-canonical normal MPS tensor with bond dimension \(D{\gt}0\). Then \(S_{D^4}(A)=M_{D}(\mathbb {C})\); equivalently, \(A\) is \(D^4\)-block injective.

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 \(e:\{ 1,\ldots ,r\} \simeq \{ 1,\ldots ,r'\} \) satisfy \(D_k=D'_{e(k)}\). Let \(Q_k\) be the coordinate permutation whose rows are indexed by the primed block and whose columns are indexed by the unprimed block. Suppose that, for every block and physical index,

The induced equivalence \(E\) of flattened dependent block coordinates has a permutation matrix \(Q_E\), with rows indexed by the primed coordinates and columns indexed by the unprimed coordinates. For every physical index \(i\),

The MPV of a block-diagonal tensor decomposes as

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

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

Every normal tensor is gauge equivalent to a tensor whose single-sector decomposition is a BNT canonical form. The original tensor and the total tensor of this decomposition generate the same matrix product vectors at every positive length.

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 trace-preserving blocks whose self-overlaps converge to \(1\). If they are not gauge-phase equivalent, then for every lower bound \(N_0\) there is \(N \ge N_0\) such that \(|V^{(N)}(A)\rangle \) is not proportional to \(|V^{(N)}(B)\rangle \). Consequently, in an already separated same-dimensional BNT block family, any two distinct blocks have such non-proportional MPV states at arbitrarily large lengths.

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

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 injective and normalized, 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 \(N\to \infty \).

Let \((B_k)_{k=1}^r\) have positive bond dimensions and suppose that every \(B_k\) is trace-preserving, primitive, and irreducible. Assume \(\mu _k\ne 0\) and \(|\mu _k|\le 1\) for every \(k\), and \(|\mu _{k_*}|=1\) for some \(k_*\). Partition the blocks into matrix-product-vector phase classes, choose one representative from each class, and retain every class member as a copy of its representative with the corresponding phase-adjusted weight. The resulting sector decomposition is in BNT canonical form.

Let \(A\) be an MPS tensor whose matrix-product family has a nonzero coefficient at some positive length. Then there exist a blocking length \(p\geq 1\), a positive real \(m\), and a sector decomposition \(P\) over the blocked alphabet in the BNT canonical form of Definition 9.6.1.13, such that for every positive length \(N\),

The scalar \(m^{N}\) is the per-site normalization cost of the line-246 weight choice, which is realized rather than assumed. The nonzero-family hypothesis is the standing convention of [ CPGSV16 , Section 2.3 ] : the unit-modulus weight chosen there presupposes a nonempty weight family. The normalization and the proportional-MPV comparison are documented in [ con26h ] .

Let \(r \in \mathbb {N}\), and for each \(k = 1,\ldots ,r\) let \(D_k \in \mathbb {N}\), \(\mu _k \in \mathbb {C}\), and let \(B_k\) be an MPS tensor of physical dimension \(d\) and bond dimension \(D_k\). Assume:

1. positive bond dimensions: \(D_k {\gt} 0\) for all \(k\),

2. TP normalization: \(\sum _i (B_k^i)^\dagger B_k^i = \mathbb {1}\),

3. primitivity: \(\mathcal{E}_{B_k}\) is primitive for each \(k\),

4. irreducibility: each \(B_k\) is irreducible,

5. nonzero weights: \(\mu _k \neq 0\) for all \(k\),

6. bounded weights: \(|\mu _k| \le 1\) for all \(k\),

7. global unit witness: \(|\mu _k| = 1\) for some \(k\).

Then there exists a sector decomposition \(P\) such that \(P\) and \(\bigoplus _{k=1}^{r} \mu _k B_k\) generate the same MPV family at every positive length, and \(P\) satisfies the basis-of-normal-tensors conditions (Definition 9.1.2) together with the multiplicity, span, and weight-bound conditions of Definition 9.6.1.13.

Under the hypotheses of Theorem 9.6.3.2, there are a BNT canonical-form sector decomposition \(P\), an equivalence \(E\) of flattened bond coordinates, and \(X\in \mathrm{GL}_{D_{\mathrm{tot}}(P)}(\mathbb {C})\). Write \(Q_E\) for the permutation matrix whose rows are indexed by the coordinates of \(P\) and whose columns are indexed by the original block coordinates. The first identity below holds, and the second holds for every physical index \(i\):

Under the hypotheses of Theorem 9.6.3.2, there is a BNT canonical-form sector decomposition \(P\) such that

For nonzero weights \(\mu _1,\ldots ,\mu _r\) with \(r\geq 1\) there is a positive real \(m\) such that every normalized weight satisfies

and at least one normalized weight has unit modulus, \(|\mu _{k_0}/m|=1\). This is the normalization that [ CPGSV16 , line 246 ] chooses for the canonical-form weights.

Let \(A\) have positive bond dimension and satisfy \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\). If \(A\) is tensor-irreducible and the transfer map is primitive, then \(O_{AA}(N)\to 1\).

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 \(A\) be an MPS tensor. Then there is a blocking length \(p\ge 1\), nonzero weights \(\mu _k\), and blocks \(B_k\) over the \(p\)-blocked alphabet such that:

1. each block has positive bond dimension;

2. each block is trace-preserving, primitive, tensor-irreducible, and one-site injective;

3. \(A^{[p]}\) and \(\bigoplus _k \mu _k B_k\) generate the same MPV family at every positive length.

No weight-modulus normalization is asserted at this stage.

Let \(X\) and \(Y\) be trace-preserving, primitive, irreducible NTs of positive bond dimensions. If their matrix-product-vector families differ by a nonzero scalar power, then their bond dimensions are equal and, after identifying the bond spaces, \(X\) and \(Y\) are gauge-phase equivalent. Neither positivity of the scalar nor unitarity of the gauge is asserted.

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.

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 the representative \(A_j\) have bond dimension \(D_j\), and let \(D\) be the bond dimension of the flattened sector tensor. Then \(D_j\leq D\). If every \(D_j\) is positive and there are \(g\) representatives, then also \(g\leq D\).

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 \(P\) be a BNT canonical form and let \(p\geq 1\). Then the blocked sector decomposition \(P^{[p]}\) is again a BNT canonical form.

If \(P\) is in BNT canonical form, then its basis blocks form an algebraic basis of normal tensors for the total tensor associated with \(P\). Only this implication from canonical form to algebraic BNT is asserted.

Let \(P, Q\) be two BNT canonical forms with the same MPV family (at all positive lengths). For every sector \(j\) of \(P\) there exist a sector \(k\) of \(Q\) 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. No per-sector unit-modulus hypothesis is required.

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.

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.

Physical blocking preserves the copy sets and bond dimensions. On the flattened copy index, it raises the weight to the \(p\)-th power and replaces the representative by its \(p\)-site blocking.

The coefficient of the blocked decomposition satisfies

Moreover, blocking the tensor assembled from \(P\) gives the same MPV family as the tensor assembled from the blocked sector decomposition.

Let \(P\) be a BNT canonical form with total bond dimension \(D\). There is a positive integer \(N\leq 3D^5\) such that its representatives have the simultaneous fixed-length span

This is the block-injectivity estimate in [ CPGSV16 , lines 340–345 ] .

Let \(A\) have positive bond dimension, be irreducible and left-canonical, and satisfy \(O_{AA}(N)\longrightarrow 1\). Then its single-sector decomposition is a BNT canonical form.

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 \(A\) 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 \((A_j)_{j=1}^g\) and \((B_j)_{j=1}^g\) be tensor families with the same physical dimension and the same bond dimension \(D_j\) in each labelled block. Suppose that for every \(j\) there is an arbitrary invertible matrix \(X_j\in \mathrm{GL}_{D_j}(\mathbb {C})\) such that, for every physical index \(i\),

Then, at every fixed length \(L\),

Both directions hold for general invertible gauges; no unitarity condition is imposed. In particular, a separation certificate proved after passing to left-canonical representatives can be transported back to a source unital gauge carrying a positive dual fixed point. This supplies only the simultaneous-span transport; the subsequent source-gauge boundary and parent-Hamiltonian arguments are separate.

For every \(n\),

Consequently, if \(L{\gt}0\) and \(A\) is \(L\)-block injective, then \(A\) is \(m\)-block injective for every \(m\ge L\). This auxiliary length-shift statement combines the block-injectivity discussion in  [ CPGSV16 , Section II ] with [ CPGSV16 , Appendix C.3, Lemma L ] ; it is not stated there as a separate lemma.