Dependencies

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

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

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.

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

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

Equivalently, its Kraus map is trace-preserving.

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.

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 \(\mathbb {C}^{d^N}\).

Two tensors \(A\) and \(B\) (of possibly different bond dimensions) generate nonzero proportional MPV families if for every \(N{\gt}0\) there is a nonzero scalar \(c_N \in \mathbb {C}\) such that \(|V^{(N)}(A)\rangle = c_N |V^{(N)}(B)\rangle \). This is the projective reading of the proportionality hypothesis in [ CPGSV16 , Theorem 2.10 ] .

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

Given nonzero weights \((\mu _k)_{k=1}^r\) and blocks \((B_k)_{k=1}^r\), the single-copy sector decomposition has \(r\) sectors, one basis block \(B_k\) per sector, multiplicity \(1\), and sector weight \(\mu _k\). As a finite sum,

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 \(v_1, \ldots , v_g\) and \(w_1, \ldots , w_g\) be two linearly independent families in a complex vector space with the same span. Then there exists an invertible matrix \(U \in \mathrm{GL}_g(\mathbb {C})\) such that \(w_j = \sum _{i=1}^g U_{ij} v_i\) for every \(j\).

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

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 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 \(N\).

A tensor with one-dimensional bond space and identity transfer map is a normal tensor.

Every tensor with a one-dimensional bond space is irreducible.

Suppose CF summands are grouped by BNT representative, every original copy weight is nonzero, and every summand assigned to a representative has that representative’s bond dimension. Then the total bond dimension of the resulting BNT sector decomposition is the sum of the original block dimensions.

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

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 \(a_1,\ldots ,a_m\) and \(b_1,\ldots ,b_n\) be nonzero complex numbers. If their power sums agree for every sufficiently large exponent, then \(\sum _i a_i^k=\sum _j b_j^k\) for every non-negative exponent \(k\).

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.

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

Let \(F_k\) be a finite family of square matrices indexed by the original blocks. If \(k=(j,q)\) enumerates the copies in the MPV phase classes, then \(\sum _j\sum _q F_{j,q}=\sum _k F_k\).

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

Nonzero proportionality implies eventual nonzero proportionality, is symmetric, and equality of MPV families is the special case with scalar \(1\), at every positive length.

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.

In a prepared family of positive-bond-dimension trace-preserving primitive irreducible NTs, every CF summand assigned to a BNT representative has that representative’s bond dimension.

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.

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 \(P\) 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 ] .

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.

The single-copy sector decomposition of \((\mu _k,B_k)_k\) represents the same MPS family as \(\bigoplus _k\mu _kB_k\).

For nonzero weights \((\mu _k)_{k=1}^r\), blocks \((B_k)_{k=1}^r\), and any \(p\ge 1\), the blocking identity (6) holds, where \(\sim \) denotes generation of the same MPV family at every length.

If the self-overlaps of a finite block family tend to \(1\) and the cross-overlaps of distinct blocks tend to \(0\), then the MPV states are linearly independent for all sufficiently large system sizes.

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 \(A\) 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 ] .

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 \(V^{(N)}(B)_\sigma = \zeta ^N V^{(N)}(A)_\sigma \).

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 ] .

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.

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 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 \(\alpha , \beta : \{ 0, \ldots , n-1\} \to \mathbb {C}\). 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 \(A\) 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 \(P\) 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.

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 \(A\) be a left-canonical MPS tensor with \(D{\gt}0\). Suppose that \(A\) is tensor-irreducible and that its transfer map is primitive. Then there exists a positive blocking length \(L\) such that \(A^{[L]}\) is one-site injective.

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.