Dependencies

Let \(A\) be an MPS tensor in irreducible form II and let \(U : G \to \mathrm{GL}_d(\mathbb {C})\) be a representation of a group \(G\) on the physical leg, acting unitarily (\(U(g)\, U(g)^{\dagger } = \mathbb {1}\)). Assume the periodic equal-case Fundamental Theorem (Theorem 22.2.2.1) for the dimensions \((d, D)\). If \(A\) is on-site symmetric under \(U\) — that is, the twisted tensor \(\widetilde{A}_g\) has the same MPV family as \(A\) for every \(g \in G\) — then for each \(g \in G\) there exists a positive period \(m_g\) and a \(\mathbb {Z}_{m_g}\)-gauge equivalence between \(A\) and \(\widetilde{A}_g\).

Concretely there are matrices \(Z_g\) and \(Y_g\), with \(Z_g^{m_g} = \mathbb {1}\) and \([A^i, Z_g] = 0\), satisfying \(Z_g A^i = Y_g \widetilde{A}_g^{\, i}Y_g^{-1}\) for every physical index \(i\).

Assume the periodic equal-case theorem: tensors in irreducible form II with equal MPV families are \(Z\)-gauge equivalent. Let \(A\) be in irreducible form II, and suppose \(B:=A_M\) is also in irreducible form II with \(\mathcal{V}(A)=\mathcal{V}(B)\). Then there are \(m{\gt}0\), an invertible matrix \(Y\), and a matrix \(Z\in M_{D}(\mathbb {C})\) such that, for every physical index \(i\),

Hence \(A\) and \(B\) are related by a \(\mathbb Z_m\) virtual gauge.

If \(A\) and \(B\) are \(\mathbb {Z}_m\)-gauge equivalent, then the blocked tensors \(A^{[m]}\) and \(B^{[m]}\) have the same MPV family.

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

For a finite family of tensors \((B_i)_{i \in I}\) with periods \((p_i)_{i \in I}\), the common-period blocked family consists of the blocked tensors \(B_i^{[L]}\) with \(L = \operatorname{lcm}_{j \in I}(p_j)\).

A family of compressed sector tensors \((C_k)_{k=0}^{m-1}\) on corner bond spaces forms an off-diagonal decomposition of the blocked tensor \(A^{(m)}\) if there exist orthogonal projections \(P_0,\ldots ,P_{m-1}\) on the ambient bond space such that \(\sum _k P_k = \mathbb {1}\), the projections satisfy the cyclic-shift relation \(\mathcal{E}_A^\dagger (P_{k+1}) = P_k\) with the index \(k+1\) taken cyclically modulo \(m\), each \(P_k\) commutes with every blocked letter \(P_k A^{(m)}_i = A^{(m)}_i P_k\), and \(V^{(N)}(C_k)_\sigma = \operatorname{tr}(P_k (A^{(m)})^\sigma )\) for every word \(\sigma \), and each block carries a \(\ast \)-algebra isomorphism \(\varphi _k : M_{\dim C_k}(\mathbb {C}) \xrightarrow {\sim } P_k \cdot M_D(\mathbb {C}) \cdot P_k\) (multiplicative and adjoint-preserving) intertwining the compressed adjoint transfer map with the sector adjoint transfer map on the corner of \(P_k\). Retaining the projections and corner isomorphisms as data gives the witness-bearing formulation. This is the inverse cyclic indexing of the off-diagonal convention in [ 191 , Appendix A ] . There the same adjoint transfer map is written \(E^*\), and the source projectors satisfy \(E^*(P_u)=P_{u+1}\) together with \(A^i=\sum _u P_u A^i P_{u+1}\). Here \(P_k\) corresponds to the source sector \(u=-k \pmod m\), which turns the source shift into the displayed adjoint-transfer relation.

The matched flattened data are the functions \(\widetilde\mu \), \(\widetilde P\), and \((k,q)\mapsto X_k\) defined in (4) on the flattened copy coordinates of \(Q\).

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, (8) 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.

A tensor \(A\) admits an irreducible-form representation if it generates the same matrix-product-vector family as the block-diagonal tensor

where each block \(A_j\) is a left-canonical \(m_j\)-periodic tensor. The defining relation is \(|V^{(N)}(A)\rangle =|V^{(N)}(\widehat A)\rangle \) for every \(N\geq 0\), rather than literal equality of the tensors \(A\) and \(\widehat A\). In particular, it includes the empty word and hence equality of the bond-dimension traces. The source comparison is at positive physical lengths and permits the irreducible-form representative to have smaller bond dimension [ 191 , lines 266–271 ] . Following [ 191 , Section II ] , the phases are fixed by the convention \(\mu _j{\gt}0\), so the weights are positive real scalars. This is a normalized matrix-product-vector representation of the source irreducible form, whose displayed decomposition is a literal tensor equality and whose periodic blocks are only required to have irreducible transfer maps of spectral radius one. It extends the normal canonical form (Definition 7.9.2), which requires the stronger condition that every block is \(1\)-periodic.

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

For a finite family of positive integer periods \((p_i)_{i \in I}\), their least common multiple \(\operatorname{lcm}_{i \in I}(p_i)\) defines the shared blocking period.

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

Equivalently, its Kraus map is trace-preserving.

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

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

Let the blocks be indexed by \(k\in \{ 0,\ldots ,r{-}1\} \), let \(n_k\) be their nonnegative dimensions, and put \(n=\sum _{k\in \{ 0,\ldots ,r{-}1\} }n_k\). The standard block-order bijection is

For each \(k\in \{ 0,\ldots ,r{-}1\} \), define \(J_k:\mathbb C^{n_k}\to \mathbb C^n\) by

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 2.8.4, it is the complex conjugate of the Hilbert-space inner product from Definition 2.8.2.

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.

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 [ 13 , Theorem 2.10 ] .

An MPS tensor \(A\) has on-site symmetry under a representation \(U:G\to \mathrm{GL}_d(\mathbb {C})\) if \(\mathcal{V}(A)=\mathcal{V}(\widetilde A_g)\) for every \(g\in G\). This equality of periodic MPV families provides the condition for constructing virtual symmetry gauges.

A linear endomorphism \(\mathcal{E}\) of \(M_D(\mathbb {C})\) is \(p\)-divisible if there exists a CPTP map \(\mathcal{E}'\) of \(M_D(\mathbb {C})\) such that \(\mathcal{E}= (\mathcal{E}')^p\).

Given a tensor \(B\), the family \(\mathcal{V}(B)\) can be \(p\)-refined if there exist another tensor \(A\) and an isometry \(W : \mathbb {C}^d \to (\mathbb {C}^d)^{\otimes p}\) such that, for every \(N\),

Equivalently, after identifying a \(p\)-letter with an element of \(\{ 0,\ldots ,d^p-1\} \), the \(p\)-blocked tensor \(A^{[p]}\) satisfies

for every \(N \in \mathbb {N}\) and every \(\tau \in \{ 0,\ldots ,d^p-1\} ^N\).

The bijection between blocked words of length \(L\) over the alphabet of \(d^m\) blocked letters and single-site words of length \(mL\) over the alphabet of \(d\) letters, \([d^m]^L\simeq [d]^{mL}\), is obtained from the iterated blocking bijection. Here \([q]^r\) denotes the set of words of length \(r\) over a \(q\)-letter alphabet.

Let \(P_0,\ldots ,P_{m-1}\) be cyclic sector projections for a tensor \(A\). The one-site corner-transition letters of [ 191 , Appendix A ] are \(A_u^i=P_u A^i P_{u+1}\), with the index \(u+1\) taken cyclically modulo \(m\).

For a word \(\mathbf i=(i_1,\ldots ,i_\ell )\), define

The empty word gives \(F_u^\varnothing =P_u\), and a one-letter word gives \(F_u^{(i)}=A_u^i\). This is the notation of [ 191 , Appendix A ] ; in the Case 3 contraction the length is \(\ell =mN_0\).

Given a positive period \(m\), cyclic projectors \(P_u\) and \(Q_v\), a cyclic offset \(q\), and corner operators \(U_v\), the global gauge is

Write \(\rho _j\) and \(e(j,q)\) for the representatives and the enumeration of the phase-class data of Definition 8.6.1.7, as in Theorem 22.2.3.65. Given nonzero weights \(\mu _k\) and unit-modulus phases \(\xi _{j,q}\), let \(n_j\) be the number of copies in class \(j\) and define

This is the multiplicity form of [ 191 , Section 2.1 ] .

Suppose \(A\) has the same matrix product vectors at every length, including zero, as \(\bigoplus _k\mu _k A_k\), where \(\mu _k{\gt}0\) and \(A_k\) is left-canonical periodic of period \(m_k\), and let \(p{\gt}0\). The blocked presentation of \(A^{[p]}\) is indexed by the pairs \((k,a)\) with \(0\le a{\lt}\gcd (m_k,p)\); the block \(B_{k,a}\) is the \(a\)th prescribed orbit block of \(A_k^{[p]}\), with weight \(\mu _k^p\) and period \(m_k/\gcd (m_k,p)\). It is an irreducible-form presentation of \(A^{[p]}\). This is the blockwise application of [ 191 , Lemma 6, proof of Theorem 4.1 ] .

Let \(A\) be an MPS tensor with physical dimension \(d\) and bond dimension \(D\), and assume it is on-site symmetric under a group action \(U : G \to \mathrm{GL}_d(\mathbb {C})\). The tensor \(A\) satisfies the periodic projective-rigidity hypothesis if one may choose a family of virtual gauges \(Y : G \to \mathrm{GL}_D(\mathbb {C})\) and a scalar \(2\)-cochain \(u : G \times G \to \mathbb {C}^{\times }\) such that

1. every \(Y_g\) realizes the pointwise \(Z\)-gauge intertwining modeled on the symmetry corollary with the twisted tensor \(\widetilde{A}_g\) for some accompanying \(Z_g\) (with \(Z_g^{m_g} = \mathbb {1}\) and \([A^i, Z_g] = 0\)); and

2. the family obeys the projective multiplication law on the bond space: \(Y_h\, Y_g = u(g, h)\, Y_{gh}\).

The source supplies the corresponding intertwining for a single local unitary. The group-indexed coherent choice and the projective multiplication law are additional assumptions: after proving the symmetry corollary, the source paper says that its consequences will be explored elsewhere. The injective case reduces this coherence to scalar uniqueness of gauges (cf. §10.1); in the periodic setting there is a non-trivial \(\mathbb {Z}_m\)-gauge ambiguity, so coherence of the factor system \(u\) is an independent analytic assumption.

Let

be a multiplicity-bearing sector decomposition. For tensors \(B_j\) with the same bond dimension as \(A_j\), its representative replacement is

Thus the index set, bond dimensions, multiplicities, and every entry of the diagonal matrices \(R_j\) remain unchanged. In particular, for every length \(N\) and every representative \(j\), the grouped coefficient remains

Let \(A\) be an MPS tensor, let \(P_0,\ldots ,P_{m-1}\in M_D(\mathbb {C})\), and let \(c_0,\ldots ,c_{m-1}\in \mathbb {C}\). Define \(A'^i=\sum _u c_u P_u A^i\), as in [ 191 , before eq. Aprime-is-cPA ] .

Let \(m{\gt}0\), \(p\ge 0\), \(r=\gcd (m,p)\), and let \(P_u\) be projections indexed modulo \(m\). For \(0\le \alpha {\lt}r\), set

These are the grouped projections of [ 191 , Lemma 6 ] .

A left-canonical irreducible MPS tensor \(A\) of bond dimension \(D\) is \(m\)-periodic (for some \(m\ge 1\)) if the peripheral spectrum of its transfer map \(\mathcal{E}_A\) is exactly the \(m\)-th roots of unity \(\{ e^{2\pi i k/m}:0\le k{\lt}m\} \). This is the normalized form-II convention used later in the source. A \(1\)-periodic tensor is exactly an irreducible tensor with primitive transfer map (the corresponding definition in the companion quantum-channel volume  [ 216 , “Primitive map” ] ).

Fix \((d,D,p)\). This states the combined construction: whenever a blocked tensor \(C\) in irreducible form II has the same MPV family as a \(p\)-blocked tensor \(A^{[p]}\), one can replace \(A\) by a left-canonical tensor \(\widetilde A\) with \(\mathcal{E}_C = \mathcal{E}_{\widetilde A^{[p]}}\).

Fix \((d,D,p)\). This sharpened reconstruction hypothesis asks for a CPTP map \(\mathcal{E}'\) such that \(\mathcal{E}_C = (\mathcal{E}')^p\) and the Choi/Kraus rank of \(\mathcal{E}'\) is at most \(d\), whenever \(C\) is related to \(A^{[p]}\) by a blocked \(\mathbb {Z}_m\)-gauge witness.

Fix \((d,D,p)\). This hypothesis says that if \(m \ge 1\) and a blocked tensor \(C\) in irreducible form II is \(\mathbb {Z}_m\)-gauge equivalent to \(A^{[p]}\), then one can construct a tensor \(\widetilde A\) with \(\sum _i (\widetilde A^i)^\dagger \widetilde A^i = \mathbb {1}\) and \(\mathcal{E}_C = \mathcal{E}_{\widetilde A^{[p]}}\). This channel-level condition is a strictly weaker consequence of the construction in [ 191 , lines 765–810 ] , which additionally supplies block scalars, intermediate matrices, and the full blocked-tensor identity \(\widetilde A^{[p]} = C\).

Fix \((d,D,p)\). This hypothesis says that whenever a blocked tensor \(C\) in irreducible form II has the same MPV family as the \(p\)-blocked tensor \(A^{[p]}\), there exists an integer \(m \ge 1\) and a \(\mathbb {Z}_m\)-gauge equivalence between \(C\) and \(A^{[p]}\).

Given a matrix \(M \in \mathbb {C}^{m\times d}\) and an MPS tensor \(A\) with physical dimension \(d\), the physical-index rotation is the tensor with physical dimension \(m\) given by \((A_M)^i := \sum _j M_{ij}\, A^j\) for \(i=0,\ldots ,m-1\). When the matrix is a unitary \(u\), the rotation is written \(A_u\).

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 \(N {\gt} 0\).

For a scalar \(\zeta \), two tensors \(A\) and \(B\) are similar carrying \(\zeta \) if they have the same bond dimension and there is an invertible matrix \(Y\) with

for every physical index \(i\). This is the repeated-block relation of [ 191 , def. repeated, lines 276–284 ] with the scalar named instead of quantified away, so that one statement can use the same scalar in the similarity and in the multiplicity relation.

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

An irreducible MPS tensor \(A\) is spectrally \(m\)-periodic if \(m\geq 1\), the spectral radius of its transfer map is one, and

No trace-preserving normalization is imposed. This is the form of the periodic blocks before the similarity transformation in [ 191 , lines 248–261 and 313–332 ] .

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.

Given a group representation \(U : G \to \mathrm{GL}_{d}(\mathbb {C})\) on the physical index, the \(g\)-twisted tensor is defined, for each \(i \in \{ 0,\ldots ,d{-}1\} \), by

For matched multiplicity entries \((\mu _i)_i\) and \((\nu _i)_i\) on a finite index type, define the diagonal matrix \(Z:=\operatorname{diag}\! (z(\mu _i,\nu _i))_i\).

Given complex multiplicity entries \(\mu ,\nu \), define the entrywise \(Z\)-gauge ratio by \(z(\mu ,\nu ):=\mu /\nu \).

Let \(A\) be an \(m\)-periodic irreducible tensor of bond dimension \(D\). A global finite-order intertwining relation is given by a matrix \(Z\in M_{D}(\mathbb {C})\) satisfying:

1. \(Z\) commutes with every tensor matrix: \([A^i,Z]=0\) for all physical indices \(i\), and

2. \(Z^m=\mathbb {1}_D\).

Two tensors \(A,B\) are \(Z\)-gauge equivalent if there exists an invertible \(Y\) and a \(Z\)-gauge transformation \(Z\) for \(A\) such that \(ZA^i=YB^iY^{-1}\) for all \(i\). Theorem 22.2.2.1 refines this global relation to block-multiplicity diagonal gauges \(Z_j\).

Let \(Z=\bigoplus _k z_k\mathbb {1}_{D_k}\) act by the scalar \(z_k\) on the bond space of the \(k\)-th block, and let \(X=\bigoplus _k X_k\) be block diagonal for the same block decomposition. Then \(ZX=XZ\).

This is why the multiplicity gauge of [ 191 , line 653 ] survives the blockwise normalizing similarity of [ 191 , lines 313–332 ] unchanged.

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

Let \(R\) be a ring, let \(V\) be an \(R\)-module, and let \(I\) and \(K\) be finite index sets. Let \(T\subseteq I\), and let \(\{ u_i\} _{i\in I}\) and \(\{ v_k\} _{k\in K}\) be vectors in \(V\). Suppose that, for every \(i\notin T\), there are a scalar \(\alpha _i\) and an index \(\phi (i)\in K\) such that \(u_i=\alpha _i v_{\phi (i)}\). If

and the coproduct-indexed family \(w\colon T\amalg K\to V\), defined by \(w(\operatorname {inl}(i))=u_i\) and \(w(\operatorname {inr}(k))=v_k\), is linearly independent, then \(a_i=0\) for every \(i\in T\).

Let \(G\) be a commutative group, and let \(\kappa _0,\ldots ,\kappa _n\in G\). If \(\prod _{k=0}^{n}\kappa _k=1\), then there are \(\varphi _0,\ldots ,\varphi _n\in G\) such that \(\kappa _k=\varphi _k\varphi _{k+1}^{-1}\) for every \(k{\lt}n\), and \(\kappa _n=\varphi _n\varphi _0^{-1}\). Equivalently, on a nonempty finite cyclic index set, a product-one family satisfies \(\kappa _k=\varphi _k\varphi _{k+1}^{-1}\) for every cyclic edge. In particular, a product-one family of complex unit phases admits such a choice with all \(\varphi _k\) again of unit modulus.

Suppose \(B^i=\zeta XA^iX^{-1}\) and \(|\zeta |=1\), with \(X\) invertible. Then \(A\) and \(B\) satisfy the repeated-block relation \(A^i=\zeta ^{-1}X^{-1}B^iX\).

Gauge-phase equivalence is symmetric after identifying equal bond dimensions. Moreover, suppose that \(A\), \(B\), and \(C\) have identified bond dimensions, that \(X_1\) and \(X_2\) are invertible, that \(\zeta _1,\zeta _2\ne 0\), and that, for every physical index \(i\),

Then

so \(B\) and \(C\) are gauge-phase equivalent after the corresponding identification of their bond dimensions.

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 \(B\) be left-canonical irreducible tensors of common positive bond dimension satisfying \(B^i=\zeta XA^iX^{-1}\) for an invertible \(X\) and a nonzero scalar \(\zeta \). There is a unitary matrix \(U\), with the same scalar \(\zeta \), such that \(|\zeta |=1\) and \(B^i=\zeta UA^iU^\dagger \) for every \(i\).

Let \(A\) and \(B\) be left-canonical irreducible tensors of common positive bond dimension that are gauge-phase equivalent. Then the gauge can be taken unitary: there are a unitary matrix \(U\) and a scalar \(\zeta \) with \(|\zeta |=1\) such that, for every physical index \(i\),

Let two sector decompositions have distinguished blocks \(j_0\) and \(k_0\), with multiplicity entries \((\mu _{j_0,q})_q\) and \((\nu _{k_0,q})_q\), and let \(\zeta \) be a nonzero scalar. Suppose that

for every \(n\) beyond some threshold. Then the two blocks carry the same multiplicity, and there is a bijection \(\tau \) between their copy indices with

for every \(q\). This is the form in which the power-sum comparison enters [ 191 , lines 681–684 ] , where the comparison is available only along the multiples of a period; the recovery of the entries from their power sums is [ 191 , lines 537–569 ] .

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 \(A\) be periodic of period \(m\). Let \(P_0,\ldots ,P_{m-1}\) be orthogonal projections satisfying

If

then \(U=0\).

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

From a sector right inverse \(\Omega \) recovering every sector word product \(\bar A_k^{\, W}\) over blocked words \(W\in [d^m]^L\) with \(L{\gt}0\) ( [ 191 , eq. Omegauprop ] ) one obtains an ambient corner right inverse \(\hat\Omega \) over single-site words of length \(mL\). For every corner-supported \(Y\), that is, \(P_kYP_k=Y\),

If \(W\in [d^m]^L\) is a blocked word and \(\widehat W\in [d]^{mL}\) is its image under the flattening bijection, then \(\widehat W\) is exactly the concatenation of the \(L\) decoded length-\(m\) blocks of \(W\).

Suppose the coefficients \(\Omega _{k+1}\) recover each inserted matrix \(X_k\) from the repeated corner product \(F_{k+1}\) over length-\(L\) words: \(\sum _{\mathbf j}(\Omega _{k+1}(X_k))_{\mathbf j}\, F_{k+1}^{\mathbf j}=X_k\) ( [ 191 , eq. Omegauprop ] at \(X_k\)). Then for any sector indices \(\sigma \),

Summing the concatenated chain against these coefficients collapses each repeated product \(F_{k+1}\) to its inserted matrix \(X_k\). The recovery is needed only at the inserted matrices \(X_k\), which in the Case-3 contraction are the corner-supported tensors that \(\Omega _{k+1}\) inverts.

If the sector maps \(P_u\) are orthogonal projections, then for any two words \(\mathbf i\) and \(\mathbf j\), \(F_u^{\mathbf i\mathbf j} =F_u^{\mathbf i}F_{u+|\mathbf i|}^{\mathbf j}\), where indices are taken cyclically modulo \(m\).

Let \(P_0,\ldots ,P_{m-1}\) and \(Q_0,\ldots ,Q_{m-1}\) be orthogonal projections, and suppose the \(m\)-block product identity

holds for every base sector \(u\) and every length-\(m\) word. Then for every word \(W\) of length \(km\) with \(k\ge 1\), \(F_u^{W}=c_u^{\, k} \widehat F_{u+q}^{W}\), where \(\widehat F\) is the repeated corner product built from \(B\). Each of the \(k\) successive length-\(m\) blocks is anchored at the same base sector \(u\), because a full cycle of unit shifts is trivial, \(m\cdot 1=0\) in \(\mathbb Z_m\); hence every block contributes the same scalar \(c_u\).

Under the off-diagonal convention \(P_kA^i=A^iP_{k+1}\), with the \(P_k\) orthogonal projections, the corner-transition product of a single blocked letter is the diagonal blocked corner of [ 191 , eq. Cu ] : \(F_u^{(I)}=P_u\, (A^{[m]})^I\, P_u\). Combined with the corner-isomorphism letter identity, this exhibits the compressed sector letter as a corner-transition product.

Let \(P_k\) be orthogonal projections satisfying the off-diagonal convention \(P_kA^i=A^iP_{k+1}\), and let \(\varphi _k\) be the multiplicative corner isomorphisms with \(\varphi _k(\bar A_k^I)=P_k(A^{[m]})^IP_k\). Then for a nonempty blocked word \(W\), with flattening \(\widehat W\), the corner-transition product is the corner image of the word evaluation,

the content of [ 191 , eq. Fu ] read through \(\varphi \).

Under the off-diagonal convention \(P_kA^i=A^iP_{k+1}\), with the \(P_k\) orthogonal projections, the repeated corner-transition product collapses through its intermediate projectors:

where \(A^{\mathbf i}\) is the ordinary word product and the right projector records the accumulated sector offset.

Let \(A\) be periodic, and let \(B_j\) be its nonzero normalized compressed cyclic sectors, with the corner letters and support isometries of Theorem 22.2.3.26. Two distinct sectors of equal virtual dimension are not gauge-phase equivalent. This is the non-repetition conclusion of [ 191 , Lemma bdcf, lines 404–423 ] .

Let \(m\ge 1\), and let \(A\) and \(B\) carry the cyclic off-diagonal structure

with \(P_u\) and \(Q_v\) families of orthogonal projections summing to the identity. Let \(q\) be the cyclic offset between the matched sector labels, and let each \(U_v\) be a corner partial isometry from the \(Q_v\) corner onto the \(P_{v-q}\) corner, that is,

Assume the per-site proportionality of [ 191 , eq. result ] ,

Then the global gauge \(U=\sum _uP_uU_{u+q}Q_{u+q}\) is unitary and satisfies \(A^i=\zeta UB^iU^\dagger \) for every \(i\), so \(A\) and \(B\) are repeated blocks.

Let \(m\ge 1\). If each corner operator satisfies \(U_v=P_{v-q}U_vQ_v\), then \(P_uU_{u+q}Q_{u+q}=U_{u+q}\), so the global gauge is the plain sector sum \(U=\sum _vU_v\).

Let \(A\) be left-canonical, \(\sum _i A_i^\dagger A_i=1\), with orthogonal sector projectors \(P_k\) satisfying the adjoint shift \(\mathcal E^*(P_{k+1})=P_k\), which gives the inverse-indexed grading \(P_{k+1}A^i=A^iP_k\). After the inverse reindexing \(P_k\mapsto P_{-k}\) the projectors satisfy the off-diagonal convention \(P_{-k}A^i=A^iP_{-(k+1)}\) of [ 191 , eq. Auprop ] , under which the corner-transition products are nonzero.

Let \(\{ A_j\} _{j\in J}\) and \(\{ B_k\} _{k\in K}\) be finite tensor families, and suppose that \(A_j'\) is purely similar to \(A_j\) and \(B_k'\) is purely similar to \(B_k\) for every \(j,k\). Assume that every tensor in each primed family has a non-decaying mixed-overlap partner in the other primed family, and that every non-decaying primed pair is a repeated pair. Then the same three assertions hold for the two original families.

Every tensor is \(1\)-refinable. A linear map is \(1\)-divisible if and only if it is a channel. Thus, at \(p=1\), the forward implication of the printed Theorem 4.1 requires trace preservation of \(\mathcal{E}_B\); see [ 171 ] .

Feeding the ambient corner right inverse into the right-inverse contraction yields the contracted product-tensor identity [ 191 , eq. resultprop ] . For corner-supported inserted matrices \(X_k\),

Let \(C\in M_{D}(\mathbb {C})\) be invertible and let \(E\) be a linear map on \(M_{D}(\mathbb {C})\). If

then \(E_C\) and \(E\) have the same peripheral spectrum.

If \(B^i=XA^iX^{-1}\) for an invertible matrix \(X\), then \(A\) and \(B\) are repeated blocks with equal bond dimension and unit phase.

For a periodic tensor equipped with a cyclic sector decomposition, each nonzero blocked sector tensor is normal.

Under the hypotheses of Lemma 8.7.1.1, 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 [ 13 , Appendix A ] : the multiset equality from Lemma 8.7.1.1 is upgraded to a concrete indexing between the matched \(P\)- and \(Q\)-copies.

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.

Let \(A\) and \(B\) be left-canonical tensors with period \(m\), and let \(\bar A\sim \bigoplus _uA_u\) and \(\bar B\sim \bigoplus _vB_v\) be unit-weight cyclic sector decompositions whose sectors are left-canonical. Suppose \(d_{u_0}=e_{v_0}\), \(d_{u_0}\ne 0\), and, after identifying the virtual spaces, \(A_{u_0}\sim _{\mathrm{gp}}B_{v_0}\). Then for every \(l\in \mathbb {Z}_m\) there is an equality \(d_{u_0+l}=e_{v_0+l}\) and, after this identification, \(A_{u_0+l}\sim _{\mathrm{gp}}B_{v_0+l}\).

Let \(A\) and \(B\) be left-canonical tensors with cyclic sector decompositions \(\{ A_u\} _{u\in \mathbb {Z}_m}\) and \(\{ B_v\} _{v\in \mathbb {Z}_m}\). For every \(u,v\in \mathbb {Z}_m\) and every \(N{\gt}0\), \(O_{A_{u+1},B_{v+1}}(N)=O_{A_u,B_v}(N)\).

Raising every multiplicity entry of a sector decomposition to a fixed power \(m\) leaves the basis blocks and the multiplicities unchanged, and the coefficient of the resulting decomposition at length \(n\) is the original coefficient at length \(mn\):

Let \(A\) and \(B\) be left-canonical tensors with period \(m\), and let \(\bar A\sim \bigoplus _u A_u\) and \(\bar B\sim \bigoplus _v B_v\) be unit-weight cyclic sector decompositions whose sectors are left-canonical. Let the specified cyclic projectors be \(P_u\) and \(Q_v\), and let the specified corner identifications realize the blocked sector tensors by \(A_u^\alpha =P_u\bar A^\alpha P_u\) and \(B_v^\alpha =Q_v\bar B^\alpha Q_v\). Assume \(d_u\ne 0\) and \(A_u\) is normal for every \(u\). Fix \(q\in \mathbb Z_m\) and assume that for every \(u\) there are \(d_u=e_{u+q}\), \(\zeta _u\ne 0\), and \(X_u\in \mathrm{GL}_{d_u}(\mathbb {C})\) such that \(B_{u+q}^\alpha =\zeta _u X_u A_u^\alpha X_u^{-1}\) for every blocked physical index \(\alpha \). The source obtains a phase-only product identity after absorbing these sector gauges into the \(B\)-sectors: with \(Q_v\) the cyclic sector projectors of \(B\), it writes \(\widetilde B_v^i =U_v Q_v B^i Q_{v+1} U_{v+1}^{\dagger }\). The contraction input is then, for every physical block \((i_1,\ldots ,i_m)\),

Choose \(N_0\) so that the \(N_0\)-fold repetition of each cyclic \(m\)-site sector product is injective. For \(\mathbf i=(i_1,\ldots ,i_{mN_0})\), form the repeated product

and choose right inverses \(\Omega _u\) satisfying

Let \(\widetilde F_{u+q+r}^{\mathbf i}\) denote the same cyclic repeated product formed from \(\widetilde B_{u+q+r},\ldots ,\widetilde B_{u+q+r+m-1}\). For every \(r\in \mathbb Z_m\), the block identity gives

Let \(\mathbf i_1,\ldots ,\mathbf i_m\) be independent words of length \(mN_0\), one for each inserted repeated product. Sum each \(\mathbf i_r\) in the contraction by the right-inverse relation for \(\Omega _{u+r}\):

Expand the words \(\mathbf i_1,\ldots ,\mathbf i_m\), and write \(j_{s,t}\) for the physical index in the \(t\)th sector of the \(s\)th consecutive \(m\)-site block of the expanded word, where \(0\leq s\leq mN_0\) and \(1\leq t\leq m\). Since each inserted product \(F_{u+r}^{\mathbf i_r}\) has length \(mN_0\), insertion shifts the cyclic sector position by \(mN_0\equiv 0\pmod m\). Hence every consecutive \(m\)-site block begins at the sector \(u\), and the interleaved word is

Applying the block identity at the sector \(u\) to each of these \(m\)-site blocks gives

For each \(r\in \mathbb Z_m\), substituting \(\mathbf i=\mathbf i_r\) in (5) gives \(\widetilde F_{u+q+r}^{\mathbf i_r} =e^{-iN_0\lambda _{u+q+r}}F_{u+r}^{\mathbf i_r}\). Therefore the right-inverse relation gives the per-sector contraction identity

Since \(r\mapsto u+q+r\) is a permutation of \(\mathbb Z_m\), \(\sum _{r\in \mathbb Z_m}\lambda _{u+q+r} =\sum _{r\in \mathbb Z_m}\lambda _r\). Applying \(\Omega _{u+1}\otimes \cdots \otimes \Omega _{u+m-1}\otimes \Omega _u\) to both sides of (5) and collecting the sector phases gives

Here \(\eta _{u+q}=\lambda _{u+q}(mN_0+1) -N_0\sum _{r\in \mathbb Z_m}\lambda _r\). Then there are a scalar \(\xi \) with \(|\xi |=1\) and a unitary matrix \(U\) such that \(A^i=\xi UB^iU^\dagger \) for every physical index \(i\).

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 9.3.1. Hence the two assembled tensors are gauge equivalent.

Let \(V\) satisfy \(A^iV=VB^i\) for all \(i\). Then, for every \(X\),

If moreover \(V^\dagger V=\mathbb {1}\), then every eigenvalue of \(\mathcal{E}_B\) is an eigenvalue of \(\mathcal{E}_A\).

Let \(A\) be an MPS tensor and let \(w\in \mathbb {C}\). Then, for every bond matrix \(X\),

If \(u\) is unitary, then replacing Kraus operators \(\{ A_i\} \) by their unitary mixtures \(\{ \sum _j u_{ij} A_j\} \) preserves the channel:

Under \(\nu _i\neq 0\) for all \(i\), one has \(Z\operatorname{diag}(\nu )=\operatorname{diag}(\mu )\).

If \(\mu _i^m=\nu _i^m\) and \(\nu _i\neq 0\) for all \(i\), then \(Z^m=\mathbb {1}\).

If \(\nu \neq 0\), then \(z(\mu ,\nu )\nu =\mu \).

If \(\mu ^m=\nu ^m\) and \(\nu \neq 0\), then \(z(\mu ,\nu )^m=1\).

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

If \(A\) generates the same MPV family as \(\bigoplus _k\mu _kB_k\) at every length, then for every \(p\ge 0\) the blocked tensor \(A^{[p]}\) generates the same MPV family as \(\bigoplus _k\mu _k^pB_k^{[p]}\) at every length.

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.

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.

If each tensor \(B_i\) has a primitive transfer map after blocking by its period \(p_i \ge 1\), then each blocked tensor \(B_i^{[L]}\) with \(L = \operatorname{lcm}_j(p_j)\) also has a primitive transfer map.

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 \(\varphi _k(C_k^i)=P_kA^iP_k\).

Assume the periodic projective-rigidity hypothesis (Definition 22.3.4) for the tensor \(A\) and the physical action \(U\). Then the virtual gauge family supplied by that hypothesis defines a projective representation of \(G\) on the bond space: there exist a scalar \(2\)-cochain \(\omega : G \times G \to \mathbb {C}^{\times }\) and virtual gauges \(X : G \to \mathrm{GL}_D(\mathbb {C})\) satisfying \(X(g)X(h) = \omega (g,h)X(gh)\), while for each \(g\) the matrix \(X(g^{-1})\) still realizes the pointwise \(\mathbb {Z}_{m_g}\)-intertwining, encoded in the rigidity hypothesis, between \(A\) and the twisted tensor \(\widetilde{A}_g\).

If the bond dimension \(D\) is positive, the factor system \(\omega \) produced by Theorem 22.3.5 satisfies the multiplicative \(2\)-cocycle identity \(\omega (g,h)\omega (gh,k) = \omega (g,hk)\omega (h,k)\).

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

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

have the same spectral radius.

Under the hypotheses of the cyclic-sector construction for the blocked tensor \(A^{(m)}\), there are sector tensors \(C_k\), projections \(P_k\), and linear corner isomorphisms

whose sector tensors are left-canonical, reproduce the blocked matrix-product vectors, have nonzero bond dimensions, and carry the cyclic adjoint-transfer decomposition of Theorem 22.2.3.24. In addition, for every blocked physical letter \(i\),

This is the corner-letter identity used in [ 191 , Lemma bdcf, eq. Cu ] .

If \(A\) is periodic with period \(m\), then the cyclic-sector decomposition of \(A^{(m)}\) may be chosen with rectangular support isometries \(V_k\) such that

In particular, the same isometries realize the corner-letter identity \(\varphi _k(C_k^i)=P_k(A^{(m)})^iP_k\).

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 \(\varphi _k(C_k^\alpha )=P_k(A^{[m]})^\alpha P_k\).

If \(A\) is periodic with period \(m\), then the blocked tensor \(A^{(m)}\) admits a cyclic sector decomposition. Thus there are projectors \(P_u\) and corner tensors \(C_u=P_u A^{(m)}\) which reproduce the blocked matrix-product vectors, are left-canonical, have nonzero bond dimensions, and realize the cyclic adjoint-transfer relation of Lemma bdcf in [ 191 , lines 404–423 ] . The normality and non-repetition conclusions of Lemma bdcf are recorded separately in the component lemmas below.

If a periodic tensor admits a cyclic sector decomposition, then for every physical letter \(i\) there exists a family \(P_0,\ldots ,P_{m-1}\) such that each \(P_u\) is an orthogonal projection,

and

If a periodic tensor admits a cyclic sector decomposition, then for every \(k\) and every physical letter \(i\) there exists a family \(P_0,\ldots ,P_{m-1}\) such that each \(P_u\) is an orthogonal projection,

and \(P_{k+1}A^i=A^iP_k\).

Under the hypotheses of Theorem 22.2.3.6, assume in addition that \(\sum _{u=0}^{m-1} P_u=\mathbb {1}\). Then every physical letter decomposes as \(A^i=\sum _{u=0}^{m-1}P_{u+1}A^iP_u\). This is [ 191 , eq. Aoffdiag ] , written in the same inverse cyclic indexing convention.

Let \((\mu _i)_{i\in I}\) and \((\nu _i)_{i\in I}\) be two finite lists of multiplicity entries. Suppose that the \(\nu _i\) are nonzero, that \(\mu _i^m=\nu _i^m\) for every \(i\in I\), and that \(\sum _{i\in I}\mu _i^k=\sum _{i\in I}\nu _i^k\) for every positive integer \(k\). Then the two lists determine the same multiset, and there is a diagonal \(Z\) satisfying \(Z^m=\mathbb {1}\) and \(Z\, \operatorname{diag}(\nu )=\operatorname{diag}(\mu )\).

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 \(i\),

Let \(A^i=\bigoplus _j\mu _jA_j^i\) with nonzero weights \(\mu _j\), where every \(A_j\) is irreducible with spectral radius one and has a trace-preserving transfer map. Let \(u\) be a unitary \(d\times d\) matrix with \(u^{\otimes N}|V_N(A)\rangle =|V_N(A)\rangle \) for every \(N{\gt}0\). Then there are a diagonal unitary \(Z\) with \([A^i,Z]=0\) for all \(i\) and \(\mathcal{V}(A)=\mathcal{V}(ZA)\), and a unitary \(U\), such that for every \(i'\)

Moreover \(Z=\bigoplus _jz_j\mathbb {1}\) with \(z_j^{m_j}=1\), where \(m_j\) is the period of \(A_j\). This is the symmetry corollary of [ 191 , lines 834–845 ] ; the diagonal form of the fixed points in irreducible form II is not needed.

If \(uu^\dagger =\mathbb {1}\) and \(A\) is in irreducible form, then \(A_u\) is in irreducible form with the same block structure and rotated blocks.

Every irreducible left-canonical MPS tensor \(A\) of positive bond dimension is \(m\)-periodic for some \(m\ge 1\) [ 191 , lines 248–258 and eq. unital ] .

If \(uu^\dagger =\mathbb {1}\) and \(A\) is irreducible, then \(A_u\) is irreducible.

For any MPS tensor \(A\) and \(m,n\ge 0\),

A tensor of positive bond dimension satisfying \(\sum _iA_i^*A_i=I\) has positive physical dimension. This is the left-canonical normalization of [ 191 , Section 2.1 ] .

If \(uu^\dagger =\mathbb {1}\) and \(A\) is left-canonical, then \(A_u\) is left-canonical.

Let \(A\) have positive bond dimension, and let \(c\in \mathbb {C}\). If both \(A\) and the scalar multiple \(cA\) are left-canonical, then \(|c|=1\).

Let \(A\) and \(B\) be left-canonical tensors with the same positive bond dimension. Suppose that \(|\xi |=1\) and that \(A^i=(\kappa \xi )B^i\) for every physical index \(i\). Then \(|\kappa |=1\).

This is the scalar normalization in [ 191 , Appendix A, lines 1082–1084 ] , after the sectorwise proportionality relation has been obtained.

The inclusions satisfy the following identities, with the second holding for distinct \(k\) and \(\ell \):

If \(B^i=\bigoplus _k\mu _k A_k^i\) is a weighted block-diagonal tensor, then

The MPV of a block-diagonal tensor decomposes as

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

Let \(A\) be left-canonical, and let \(P_0,\ldots ,P_{m-1}\) be orthogonal projections on the bond space. If the adjoint transfer map satisfies \(\mathcal{E}_A^\dagger (P_{k+1})=P_k\) for all \(k\), with indices taken modulo \(m\), then \(P_{k+1}A^i=A^iP_k\) for every physical letter \(i\). This is the single-site grading behind [ 191 , eq. Aoffdiag ] ; the compressed blocks of [ 191 , eq. Auprop ] use the inverse cyclic indexing convention of Definition 22.2.3.5.

Under the hypotheses of Theorem 22.2.3.6, a word \(w=i_1\cdots i_\ell \) satisfies \(P_{k+\ell } A^w=A^w P_k\). Thus the single-site grading accumulates one cyclic step for each letter of the word.

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 \(N\to \infty \).

Assume the blocked equal-case construction. Then every \(p\)-refinable tensor \(B\) in irreducible form II admits a left-canonical tensor \(\widetilde A\) with \(\mathcal{E}_B = \mathcal{E}_{\widetilde A^{[p]}}\).

Let \(B\) be an MPS tensor in irreducible form II. If \(\mathcal{V}(B)\) can be \(p\)-refined, then there exist a tensor \(A\), an isometry \(W : \mathbb {C}^d \to (\mathbb {C}^d)^{\otimes p}\), and a tensor \(C\) defined by \(C^\alpha := \sum _{i=0}^{d-1} W_{\alpha ,i}B^i\) for \(\alpha \in \{ 0,\ldots ,d^p-1\} \), such that \(C\) is again in irreducible form II, \(\mathcal{E}_C = \mathcal{E}_B\), and \(C\) has the same MPV family as \(A^{[p]}\).

Let \(A\) be left canonical with orthogonal cyclic projections \(P_u\) summing to the identity and satisfying \(P_uA^i=A^iP_{u+1}\). Let \(m{\gt}0\), \(p\ge 0\), \(r=\gcd (m,p)\), and \(c_\alpha ^{m/r}=1\). If a matrix \(Z\) satisfies

then there is a left-canonical tensor \(A'\) of the same dimensions with \((A'^{[p]})^I=Z(A^{[p]})^I\) for every \(I\). This is the comparison of the two phase formulas in the proof of [ 191 , Theorem 4.1 ] .

Let \(A\) be periodic of period \(m\), let \(p{\gt}0\), and write \(F=\mathcal{E}_A^\dagger \) and \(r=\gcd (m,p)\). If \(F^p(X)=X\), then \(F^m(X)=X\). Moreover, \(F^p(X)=X\) if and only if \(F^r(X)=X\).

For a finite family \(\{ A_j\} _{j\in J}\) of pairwise non-repeated periodic tensors, in the sense that no two are related by a gauge transformation and a unit-modulus scalar, whose periods divide a common positive integer \(p\), there is \(N_0\) such that for every \(N \ge N_0\), the map

Following [ 191 , lines 589–609 ] , this establishes linear independence of the periodic vectors ( [ 93 ] ).

Let \(A\) be the tensor of a sector decomposition, with representatives \(A_j\) and multiplicity entries \(\mu _{j,q}\). At every length \(N\), the nonzero vectors among \(\{ |V_N(A_j)\rangle \} _j\) span \(|V_N(A)\rangle \). This proves the spanning assertion in [ 191 , Proposition 3.3, lines 604–608 ] ; this assertion does not require periodicity or normalization of the blocks.

Let \(\{ A_j\} _{j\in J}\) be a finite family of pairwise non-repeated periodic tensors, of periods \(m_j\). There is an \(N_0\) such that for every single length \(N\ge N_0\) the vectors

are linearly independent. Since \(|V^{(N)}(A_j)\rangle \) vanishes unless \(m_j\mid N\), this is the assertion of [ 191 , lines 604–608 ] that beyond some length the non-zero members of the family are independent, in the form used in the proof of the equal case at [ 191 , lines 677–679 ] .

Let \(A_j\) and \(B_k\) be left-canonical periodic tensors with periods \(m_j\) and \(n_k\), and let all weights \(\mu _j\) and \(\nu _k\) be nonzero. Repeated blocks are allowed. If \(\bigoplus _j\mu _jA_j\) and \(\bigoplus _k\nu _kB_k\) generate the same vectors at every positive length, then there are complex numbers \(c_j\) with \(c_j^{m_j}=1\) and a matrix \(W\) with \(WW^\dagger =\mathbb {1}\) and \(W^\dagger W=\mathbb {1}\) such that, for every \(i\),

Let \(A_k\) be left-canonical periodic tensors of periods \(m_k\), let \(p{\gt}0\), and put \(r_k=\gcd (m_k,p)\). The blocks \(C_{k,a}\), \(0\le a{\lt}r_k\), of Theorem 22.4.55 for all \(A_k\) admit one unitary \(Y\) with

For every choice of \(c_{k,a}\) with \(c_{k,a}^{m_k/r_k}=1\), there is a left-canonical tensor \(A'\) on the bond space of \(\bigoplus _kA_k\) with

Let \(T=\bigoplus _k\mu _k B_k\), where \(\mu _k\ne 0\) and the \(B_k\) are left-canonical periodic tensors of bond dimension \(D_k\). Suppose that, for \(p{\gt}0\), \(T^{[p]}\) has the same positive-length matrix product vectors as a periodic sector decomposition \(Q\) with a pairwise non-repeated basis and unit-modulus multiplicities. Then \(|\mu _k|=1\) for every \(k\), and \(\sum _k D_k\) equals the total bond dimension of \(Q\). This is the weight normalization in the corrected forward direction of [ 191 , Theorem 4.1, lines 750–756 ] ; the unit-modulus target is the correction of [ 171 ] .

Let \(p{\gt}0\). For any tensor \(A\), the peripheral eigenvalues of \(\mathcal{E}_{A^{[p]}}\) are the \(p\)th powers of those of \(\mathcal{E}_A\). If \(A\) is periodic of period \(m\), put \(q=m/\gcd (m,p)\). Then

Moreover, if \(V^\dagger V=\mathbb {1}_n\), \(VV^\dagger \) commutes with every letter of \(A^{[p]}\), and \(C^I=V^\dagger (A^{[p]})^IV\), every peripheral eigenvalue of \(\mathcal{E}_C\) satisfies \(z^q=1\). This is the ambient spectral calculation in [ 191 , Lemma 6 ] , with the missing power in the printed root set corrected as in  [ 86 ] .

If \(A\) is periodic, then its bond dimension satisfies \(D\geq 1\).

Let \(\mathcal F(X)=\sum _a A^aX(A^a)^*\), and let \(S=\operatorname{span}_{\mathbb {C}}\{ A^a\} _a\). The Choi rank of \(\mathcal F\) is \(\dim _{\mathbb {C}}S\), including when the bond space is zero dimensional.

Let \(A\) be periodic of period \(m\), let \(p{\gt}0\), and put \(r=\gcd (m,p)\) and \(q=m/r\). Suppose \(P_u\) are nonzero orthogonal projections summing to \(\mathbb {1}_D\) with \(P_uA^i=A^iP_{u+1}\). Fix \(\alpha {\lt}r\), an isometry \(V\) with \(V^\dagger V=\mathbb {1}\) and \(VV^\dagger =Q_\alpha \), and set \(C^I=V^\dagger (A^{[p]})^IV\). Then

This is the spectral-set assertion in [ 191 , Lemma 6 ] , with the root notation corrected as in  [ 86 ] .

Let \(A\) be periodic of period \(m\), let \(p{\gt}0\), and let \(P_u\) be orthogonal projections summing to \(\mathbb {1}_D\) with \(P_uA^i=A^iP_{u+1}\). Fix \(\alpha {\lt}\gcd (m,p)\) and an isometry \(V\) with \(VV^\dagger =Q_\alpha \). The compressed tensor \(C^I=V^\dagger (A^{[p]})^IV\) is irreducible. This supplies the irreducibility assertion in [ 191 , Lemma 6 ] .

Suppose \(P_u\) are orthogonal projections summing to \(\mathbb {1}_D\), \(A\) is left canonical, and \(P_uA^i=A^iP_{u+1}\). Let \(p\ge 0\), fix an orbit \(\alpha \), and suppose \(VV^\dagger =Q_\alpha \) and \(C^I=V^\dagger (A^{[p]})^IV\). For \(u=[\alpha +pk]_m\),

This is the compressed form of the final display in the proof of [ 191 , Lemma 6 ] .

Let \(\{ A_j\} _{j\in J}\) and \(\{ B_k\} _{k\in K}\) be finite pairwise non-repeated families. Suppose every tensor in either family has a tensor in the other family whose mixed overlap does not tend to zero, and every such non-decaying pair is repeated. Then there is a bijection \(\pi \colon J\to K\) such that \(A_j\) and \(B_{\pi (j)}\) are repeated for every \(j\in J\).

This is the finite matching step of [ 191 , Theorem 3.4 ] .

Under the hypotheses of Theorem 22.2.2.9, for every \(j\in J\) the multiplicity matrices \(R_j\) and \(S_{\pi (j)}\) have the same size \(r_j\), and there are a reordering \(\tau \) of the diagonal entries of \(S_{\pi (j)}\) and a diagonal matrix \(Z_j\) with

where \(T_j\) implements \(\tau \). This is [ 191 , lines 681–688 ] . The scalar \(\zeta _j\) stays visible: the diagonal entries of \(Z_j\) are \(m_j\)-th roots of unity against the rescaled multiplicity matrix \(\zeta _jR_j\), and the rescaling is in general not trivial even for period one, as the pair of blocks \(B_j=e^{i\theta }A_j\) shows.

Let

be two multiplicity-bearing decompositions whose bases of periodic tensors are matched by a bijection \(\pi \colon J\to K\) and nonzero scalars \(\zeta _j\) with

for every length \(N\), with \(\{ A_j\} _{j\in J}\) pairwise non-repeated and \(A_j\) periodic of period \(m_j\). If \(\mathcal{V}^+(A)=\mathcal{V}^+(B)\), then there is an \(N_0\) such that

for every \(N\ge N_0\) with \(m_j\mid N\) and every \(j\in J\). This is [ 191 , lines 672–680 ] . The scalar \(\zeta _j\) is displayed rather than suppressed: the identity compares the power sums of \(R_j\) with the rescaled power sums of \(S_{\pi (j)}\), which is what the source means at [ 191 , lines 667–671 ] by absorbing the phase into \(S_j\).

Let

be two multiplicity-bearing decompositions, let \(\pi \colon J\to K\) be a bijection with \(D_j=D_{\pi (j)}\), let \(\tau _j\) reorder the multiplicity copies of the matched blocks, let \(\xi _j\) be scalars and \(Y_j\) invertible matrices with

and let \(z_{j,l}\) be scalars with

where \(A_j\) is \(m_j\)-periodic and every \(m_j\) divides \(L\). Writing \(Z_j\) for the diagonal matrix of the \(z_{j,l}\) and \(Z=\bigoplus _{j\in J}Z_j\otimes \mathbb {1}_{D_j}\), one has \(Z^L=\mathbb {1}\), \([A^i,Z]=0\) for every \(i\), \(\mathcal{V}^+(A)=\mathcal{V}^+(ZA)\), and there is an invertible \(Y\) with \(ZA^i=YB^iY^{-1}\) for every \(i\). If every \(Y_j\) is unitary, then \(Y\) is unitary as well.

This establishes the global intertwining relation of [ 191 , lines 689–690 ] , with \(Y\) the direct sum \(\bigoplus _{j\in J} T_j\otimes Y_j\) of the copy reorderings and the blockwise similarities.

Under the hypotheses of Theorem 22.2.2.14, if every multiplicity of \(B\) has modulus one, then every multiplicity of \(A\) has modulus one. This is the multiplicity normalization used in the corrected forward implication of [ 191 , Theorem 4.1, lines 743–756 ] .

Let \(\{ A_j\} _{j\in J}\) be a finite family of periodic tensors whose periods divide a common positive integer \(p\). If

then the vectors \(\{ |V^{(pN)}(A_j)\rangle \} _{j\in J}\) are linearly independent for every sufficiently large \(N\).

Let \(A\) and \(B\) be MPS tensors in irreducible form, written

where \(\{ A_j\} _{j \in J}\) and \(\{ B_k\} _{k \in K}\) are pairwise non-repeated bases of periodic tensors, \(R_j\) and \(S_k\) are diagonal multiplicity matrices of sizes \(r_j\times r_j\) and \(s_k\times s_k\) with nonzero diagonal entries, and \(A_j^i\) and \(B_k^i\) act on bond spaces of dimensions \(D_j\) and \(D_k\). The blocks need only have irreducible transfer maps of spectral radius one ( [ 191 , lines 248–261 ] ); no trace-preserving normalization is imposed, and the periodicity \(m_j\) of each block is derived from the peripheral spectrum ( [ 191 , lines 257–258 ] ). Suppose that \(\mathcal{V}^+(A)=\mathcal{V}^+(B)\).

Then \(|J|=|K|\) and there is a bijection \(\pi :J\to K\) such that \(A_j\) and \(B_{\pi (j)}\) are \(m_j\)-periodic for the same period \(m_j\). Moreover, for each \(j\in J\) there exist:

1. a unit-modulus scalar \(\xi _j\) and an invertible matrix \(Y_j\) with \(A_j^i=\xi _jY_jB_{\pi (j)}^iY_j^{-1}\) for every physical index \(i\) (Definition 22.2.2.8), and

2. a diagonal matrix \(Z_j\) of size \(r_j\times r_j\) with \(Z_j^{m_j}=\mathbb {1}_{r_j}\), so that all its diagonal entries are \(m_j\)-th roots of unity, and a reordering \(T_j\) of the diagonal entries of \(S_{\pi (j)}\) with \(Z_j\left(\xi _jR_j\right)=T_jS_{\pi (j)}T_j^\dagger \).

Setting \(Z=\bigoplus _j Z_j\otimes \mathbb {1}_{D_j}\) one has \(Z^{\operatorname {lcm}_jm_j}=\mathbb {1}\), \([A^i,Z]=0\) for every \(i\), \(\mathcal{V}^+(A)=\mathcal{V}^+(ZA)\), and there is an invertible matrix \(Y\) with \(ZA^i=YB^iY^{-1}\) for all \(i\).

Let \(A\) and \(B\) carry the multiplicity-bearing forms

where \(R_j\) and \(S_k\) are diagonal with nonzero entries and the two displayed families are pairwise non-repeated bases of periodic tensors whose blocks are trace preserving, of periods \(m_j\) and \(n_k\). If \(\mathcal{V}^+(A)=\mathcal{V}^+(B)\), then there is a bijection \(\pi \colon J\to K\) and unit-modulus scalars \(\xi _j\) such that, for every \(j\), one has \(m_j=n_{\pi (j)}\), there is an invertible \(Y_j\) with

for every physical index \(i\), the multiplicity matrices \(R_j\) and \(S_{\pi (j)}\) have the same size, and there are a reordering \(T_j\) of the diagonal entries of \(S_{\pi (j)}\) and a diagonal matrix \(Z_j\) with

Writing \(Z=\bigoplus _{j\in J}Z_j\otimes \mathbb {1}_{D_j}\), one has \(Z^{\operatorname {lcm}_j m_j}=\mathbb {1}\), \([A^i,Z]=0\) for every \(i\), \(\mathcal{V}^+(A)=\mathcal{V}^+(ZA)\), and there is a unitary \(Y\) with \(ZA^i=YB^iY^\dagger \) for every \(i\).

The scalar \(\xi _j\) is the same in the two displayed relations: it is the scalar of the similarity between the matched blocks, and the entries of \(Z_j\) are \(m_j\)-th roots of unity against the rescaled multiplicity matrix \(\xi _jR_j\). That is what the source means at lines 667–671 by absorbing the phase into \(S_j\); the scalar is present, as the period-one pair \(B_j=e^{i\theta }A_j\) shows.

This establishes [ 191 , Theorem 3.8, lines 643–693 ] for tensors whose block maps are trace preserving ( [ 191 , eq. unital, line 316 ] ). The periodic overlap dichotomy and the vanishing of off-period blocks apply directly in this trace-preserving setting, from which Theorem 22.2.2.1 is deduced.

Let \(A\) and \(B\) have multiplicity-bearing irreducible forms

where \(R_j\) and \(S_k\) are diagonal with nonzero diagonal entries and the two displayed block families are pairwise non-repeated bases of periodic tensors, each block being required only to have an irreducible transfer map of spectral radius one, which is the source’s irreducible form at [ 191 , lines 248–261 ] . Suppose that, for every positive \(N\), the vectors \(|V^{(N)}(A)\rangle \) and \(|V^{(N)}(B)\rangle \) are proportional by a nonzero scalar. Then the blocks carry periods \(m_j\) and \(n_k\) for which they are spectrally periodic, and there is a bijection \(\pi \colon J\to K\) such that, for every \(j\in J\), one has \(m_j=n_{\pi (j)}\) and there are a unit-modulus scalar \(\xi _j\) and an invertible matrix \(Y_j\) such that, for every physical index \(i\),

Each \(B_k\) occurs for exactly one \(j\). This is [ 191 , Theorem 3.4, lines 613–623 ] .

This theorem matches the bases of periodic tensors, leaving the multiplicity matrices unconstrained. The periods \(m_j\) and \(n_k\) are derived from the transfer maps ( [ 191 , lines 257–258 ] ).

Under the hypotheses of Theorem 22.2.3.62, suppose in addition that positive integers \(m_j\) and \(n_k\) are given such that each \(A_j\) is spectrally \(m_j\)-periodic and each \(B_k\) is spectrally \(n_k\)-periodic (Definition 22.1.1). Then the conclusions of Theorem 22.2.3.62 hold for these specified periods.

Under the hypotheses of Theorem 22.2.2.1, suppose in addition that positive integers \(m_j\) and \(n_k\) are given such that each \(A_j\) is spectrally \(m_j\)-periodic and each \(B_k\) is spectrally \(n_k\)-periodic (Definition 22.1.1). Then the conclusions of Theorem 22.2.2.1 hold for these specified periods.

Let \(A_k\) be left-canonical periodic blocks and let every \(\mu _k\) be nonzero. Set \(T^i=\bigoplus _k\mu _k A_k^i\). There are phases \(\xi _{j,q}\) in Definition 22.2.3.67 such that \(P\) has the same total bond dimension as \(T\), its representative blocks are periodic and pairwise non-repeated, and \(|\mu '_{j,q}|=|\mu _{e(j,q)}|\) for every original copy. There is a matrix \(Y\) from the grouped bond space to the original bond space with \(YY^\dagger =I\) and \(Y^\dagger Y=I\), satisfying \(T^i=YP^iY^\dagger \) for every physical index \(i\). In particular, \(V^{(N)}(T)_\sigma =V^{(N)}(P)_\sigma \) at every positive length. The same \(Y\) transports multiplicity scalars: for all scalars \(c_{j,q}\), with \(Z_c=\bigoplus _j\operatorname {diag}(c_{j,0},\ldots ,c_{j,n_j-1})\otimes \mathbb {1}\),

This gives the grouping of [ 191 , Section 2.1 and the paragraph after Theorem 3.4 ] in a unitary gauge.

Let \(\mathcal F\) and \(\mathcal G\) be completely positive maps on \(M_D(\mathbb {C})\), and suppose \(\mathcal F\) is injective. Then \(\operatorname {rank}C_{\mathcal G}\le \operatorname {rank}C_{\mathcal F\circ \mathcal G}\). Consequently \(\operatorname {rank}C_{\mathcal F}\le \operatorname {rank}C_{\mathcal F^p}\) for every \(p{\gt}0\).

Let \(\Phi :M_D(\mathbb {C})\to M_D(\mathbb {C})\) be an injective positive linear map. If \(A\ge 0\), then \(\operatorname {rank}A\le \operatorname {rank}\Phi (A)\).

If \(A\) of bond dimension \(D\) has an irreducible-form presentation \(\bigoplus _k\mu _kA_k\) with blocks of bond dimensions \(D_k\), then \(\sum _kD_k=D\). In particular this holds for the blocked presentation of Definition 22.4.51.

Let \(A\) be a tensor and let \(p\geq 0\). If its blocked tensor \(A^{(p)}\) is irreducible, then \(A\) is irreducible. In particular, this holds if \(\mathcal E_A^p=\mathcal E_B\) for an irreducible tensor \(B\), even when \(A\) and \(B\) have different physical dimensions. If moreover \(A\) is periodic of period \(m\), then \(\gcd (m,p)=1\). These are necessary conditions for the converse construction in [ 191 , Theorem 4.1 ] .

Let \(e\) be a bijection between the blocks of \(\bigoplus _k\mu _kA_k\) and of \(\bigoplus _t\mu '_tB_t\) with \(D_k=D'_{e(k)}\), and suppose \(\mu _kA_k^i=\mu '_{e(k)}X_{e(k)}B_{e(k)}^iX_{e(k)}^{-1}\) for invertible \(X_t\). Then there are matrices \(Y\) and \(Y'\) with \(YY'=\mathbb {1}\), \(Y'Y=\mathbb {1}\), and

If every \(X_t\) is unitary, then \(Y'=Y^\dagger \). This is the matrix \(Y=\bigoplus _jT_j\otimes Y_j\) of [ 191 , line 690 ] . The matrices \(Y\) and \(Y'\) do not depend on the weights: for all scalars \(c_k\),

Let

where \(R_j=\operatorname {diag}(\mu _{j,1},\ldots ,\mu _{j,r_j})\) and \(S_k=\operatorname {diag}(\nu _{k,1},\ldots ,\nu _{k,s_k})\) have nonzero complex diagonal entries. Suppose that the finite families \(\{ A_j\} _{j\in J}\) and \(\{ B_k\} _{k\in K}\) are pairwise non-repeated families of left-canonical periodic tensors. If, for every positive integer \(N\), there is a scalar \(c_N\ne 0\) such that

Then every \(A_j\) has an index \(k\in K\) for which

and, conversely, every \(B_k\) has such a partner in the first family. Moreover, whenever the overlap between \(A_j\) and \(B_k\) does not tend to zero, the two blocks have equal bond dimension and there are an invertible matrix \(Y\) and a scalar \(\zeta \) with \(|\zeta |=1\) such that, for every physical index \(i\),

Normalization. In [ 191 , lines 313–332 ] , the trace-preserving normalization is obtained via a block-diagonal similarity from irreducible transfer maps of spectral radius one. Here, the basis tensors are taken to be normalized in advance; Theorem 22.2.3.59 transfers the result to unnormalized spectrally periodic blocks.

Let \(A\) be periodic of period \(m\), let \(p{\gt}0\), and put \(r=\gcd (m,p)\). Suppose \(P_u\) are orthogonal projections summing to \(\mathbb {1}_D\) with \(P_uA^i=A^iP_{u+1}\). Write \(F=\mathcal{E}_A^\dagger \) and fix \(\alpha {\lt}r\). Then \(\sum _{j=0}^{r-1}F^j(Q_\alpha )=\mathbb {1}_D\). If \(F^p(X)=X\) and \(Q_\alpha X=X\), there is \(c\in \mathbb {C}\) such that \(X=cQ_\alpha \).

Let \(G\) be an abelian group, \(m{\gt}0\), \(p\ge 0\), \(r=\gcd (m,p)\), and \(q=m/r\). Suppose \(c_\alpha \in G\) satisfies \(c_\alpha ^q=1\) for every \(0\le \alpha {\lt}r\). There are \(b_u\in G\), indexed modulo \(m\), such that, writing \(u=\alpha _u+pk_u\) for the coordinates of Theorem 22.4.5,

For \(G=U(1)\), these are the local phases in [ 191 , proof of Theorem 4.1 ] , with the wraparound of \(k_u\) handled as in  [ 179 ] .

Let \(A\) be left canonical and \(m{\gt}0\). Suppose the Hermitian projections \(P_u\) satisfy \(P_uP_v=\delta _{uv}P_u\), \(\sum _uP_u=\mathbb {1}\), and \(P_uA^i=A^iP_{u+1}\). Let \(p\ge 0\), \(r=\gcd (m,p)\), and \(c_\alpha ^{m/r}=1\). There is a left-canonical tensor \(A'\) such that, for every word \(w\) of length \(p\),

This is the rephasing identity in [ 191 , proof of Theorem 4.1 ] .

If orthogonal projections \(P_u\) satisfy \(\sum _u P_u=\mathbb {1}\), then \(P_uP_v=\delta _{uv}P_u\).

Let \(A\) and \(B\) be periodic tensors. Then at least one of the following alternatives holds:

1. \(\langle V^{(N)}(A) | V^{(N)}(B) \rangle \to 0\) as \(N\to \infty \);

2. \(A\) and \(B\) are related by the repeated-block relation, i.e. they differ by a unit-modulus phase and an invertible gauge.

Let \(A\) and \(B\) be periodic tensors with the same period \(m\). Let \(\bar A\sim \bigoplus _u A_u\) and \(\bar B\sim \bigoplus _v B_v\) be unit-weight cyclic sector decompositions whose sectors are left-canonical, with specified cyclic projectors \(P_u,Q_v\) and corner identifications satisfying \(A_u^\alpha =P_u\bar A^\alpha P_u\) and \(B_v^\alpha =Q_v\bar B^\alpha Q_v\). Assume every sector \(A_u\) has nonzero virtual dimension. If there are sectors \(u_0,v_0\) with \(d_{u_0}=e_{v_0}\) and \(A_{u_0}\sim _{\mathrm{gp}}B_{v_0}\) after identifying their virtual spaces, then \(A\) and \(B\) are repeated blocks.

If two periodic tensors have different bond dimensions, then \(\langle V^{(N)}(A) | V^{(N)}(B) \rangle \to 0\) as \(N\to \infty \).

If \(A\) and \(B\) have periods \(m_A\ne m_B\), then \(\lim _{N\to \infty }O_{AB}(N)=0\).

Suppose \(A\) and \(B\) have the same total bond dimension and the same period \(m\). Let \(A_u\) and \(B_v\) be period-blocked sector tensors of virtual dimensions \(d_u\) and \(e_v\), indexed by \(u,v\in \mathbb {Z}_m\), such that

with unit weights, and the two decompositions are cyclic sector decompositions. Assume each sector is left-canonical:

Assume \(d_u\ne 0\) for every \(u\) and \(e_v\ne 0\) for every \(v\), and whenever \(d_u=e_v\), the corresponding sector tensors are not gauge-phase equivalent after identifying their virtual spaces:

Then \(\lim _{N\to \infty }O_{AB}(N)=0\).

Let \(A_k\) be a finite family of left-canonical periodic tensors, with bond dimensions \(D_k\) and periods \(m_k\). Partition the family by the relation \(V^{(N)}(A_l)_\sigma =\zeta ^N V^{(N)}(A_k)_\sigma \) at every positive length for some \(\zeta \ne 0\). Choose one representative \(\rho _j\) in each class and enumerate its members as \(e(j,q)\). Distinct representatives are not repeated. Each member has \(D_{e(j,q)}=D_{\rho _j}\) and \(m_{e(j,q)}=m_{\rho _j}\), and admits a unitary matrix \(U_{j,q}\) and a scalar \(\xi _{j,q}\) of modulus one such that

More generally, any two repeated left-canonical periodic blocks admit such a unitary similarity. These are the blockwise identifications in [ 191 , Section 2.1 and the paragraph following Theorem 3.4 ] .

If \(A\) is a normalized periodic tensor, then \(d\geq 1\).

Every tensor \(A\) admits \(r\in \mathbb {N}\) and, for \(k=1,\ldots ,r\), left-canonical periodic tensors \(B_k\) of positive bond dimensions \(D_k\) and weights \(\mu _k{\gt}0\), with \(\sum _{k=1}^rD_k\le D\), such that \(A\) has the same matrix product vectors at all positive lengths as \(\bigoplus _{k=1}^r\mu _k B_k\). No physical sites are blocked [ 191 , lines 238–271 and 313–320 ] .

If \(A\) is periodic of period \(m\) and \(p\ge 0\), there is a family of nonzero cyclic projections \(P_u\) with \(\sum _uP_u=\mathbb {1}_D\) and \(P_uA^i=A^iP_{u+1}\) for which the dimensions \(D_\alpha \), isometries \(V_\alpha \), and left-canonical tensors \(C_\alpha \) of Theorem 22.4.7 exist, with the same identities and the same MPV at every length, including zero. If \(p{\gt}0\), every resulting block satisfies \(\operatorname{spec}_{\mathrm{per}}(\mathcal{E}_{C_\alpha })=\{ z\in \mathbb {C}:z^{m/\gcd (m,p)}=1\} \) and is periodic of period \(m/\gcd (m,p)\). This is the prescribed-blocking conclusion of [ 191 , Lemma 6 ] .

Let \(m{\gt}0\), let \(z\in \mathbb {C}\) be nonzero, and choose \(\xi \in \mathbb {C}\) with \(\xi ^m=z\). For each sector \(v=0,\ldots ,m-1\), let \(\{ A_v^i\} _i\) and \(\{ B_v^i\} _i\) be families of complex matrices with the same rectangular size. Suppose that for every choice of letters \(\sigma =(\sigma _v)_v\),

Suppose also that each sector has a reference letter \(i_v\) and a matrix entry \((r_v,c_v)\) with \((B_v^{i_v})_{r_v,c_v}\ne 0\). Then there are scalars \(\kappa _v\in \mathbb {C}\) such that

for every sector \(v\) and every letter \(i\).

This is the normalization in [ 191 , Appendix A, lines 1072–1080 ] : after Theorem 22.2.3.1 has extracted scalars independent of the sector coordinates, choosing a common \(m\)-th root of their product leaves a product-one family of sector scalars.

Let \(z\in \mathbb {C}\) be nonzero. For each sector \(v=0,\ldots ,m-1\), let \(\{ A_v^i\} _i\) and \(\{ B_v^i\} _i\) be families of complex matrices with the same rectangular size. Suppose that for every choice of letters \(\sigma =(\sigma _v)_v\),

Suppose also that for each \(v\) there is a reference letter \(i_v\) and a matrix entry \((r_v,c_v)\) with \((B_v^{i_v})_{r_v,c_v}\ne 0\). Then there are scalars \(\kappa _v\in \mathbb {C}\) such that

for every sector \(v\) and every letter \(i\).

This is the scalar extraction used in [ 191 , Appendix A, lines 1072–1080 ] , after the contraction with the maps \(F_u\) and \(\Omega _u\) has produced the displayed product-tensor identity.

Let \(A\) and \(B\) be spectrally periodic blocks. If their matrix product vectors are proportional by a nonzero scalar at every positive length, then \(A\) and \(B\) are repeated blocks. In particular, this holds when \(V^{(N)}(B)_\sigma =\zeta ^N V^{(N)}(A)_\sigma \) for a fixed \(\zeta \ne 0\). Left-canonical periodic blocks satisfy the spectral-radius-one hypothesis automatically. This is the single-block case of [ 191 , Theorem 3.4 ] .

Let \(A\) and \(B\) have the same one-site physical and bond dimensions, and let \(p{\gt}0\). If \(\mathcal E_B=\mathcal E_{A^{[p]}}\), then \(B\) is \(p\)-refinable.

Let \(B\) have physical dimension \(d\), let \(p{\gt}0\), and suppose \(\mathcal E_B=\mathcal F^p\), where \(\mathcal F\) has a Kraus representation with at most \(d\) operators. Then \(B\) is \(p\)-refinable.

Let \(B\) have physical dimension \(d\) and an invertible transfer map, and let \(p{\gt}0\). If \(\mathcal E_B\) is \(p\)-divisible as a channel, then \(B\) is \(p\)-refinable.

Let \(B\) have physical dimension \(d\), and let \(A\) have the same bond dimension but any finite physical dimension. Suppose \(p{\gt}0\) and \(\mathcal E_B=(\mathcal E_A)^p\). If the span of products of \(p-1\) letters of \(A\) contains an invertible matrix, then \(\operatorname {rank}_{\mathrm{Choi}}\mathcal E_A\le \operatorname {rank}_{\mathrm{Choi}}\mathcal E_B\le d\), and \(B\) is \(p\)-refinable.

Let the physical and bond dimensions both be one, and take \(B=(4)\). The tensor \(B\) is literally the period-one normalized block \((1)\) with positive weight \(4\). It is \(2\)-refinable with root \(A=(2)\), but its transfer map \(\mathcal{E}_B(x)=16x\) is not \(2\)-divisible as a channel. This contradicts the forward implication of the printed [ 191 , Theorem 4.1 ] ; see [ 171 ] .

Let \(A\) be any tensor of bond dimension \(D\). Suppose that, for \(p{\gt}0\), \(A^{[p]}\) has the same positive-length matrix product vectors as a periodic sector decomposition \(Q\) with a pairwise non-repeated basis and unit-modulus multiplicities. Then \(A\) has the same positive-length matrix product vectors as a unit-weight sum \(\bigoplus _k C_k\) of left-canonical periodic tensors. Their total bond dimension equals that of \(Q\) and is at most \(D\). This is the normalization step of [ 191 , Theorem 4.1, lines 750–756 ] for an arbitrary input tensor.

Let \(A\) and \(B\) be periodic tensors of periods \(m\) and \(n\) whose block maps are trace preserving. If \(A\) and \(B\) are repeated blocks, then \(m=n\).

Let \(S\) be the span of a finite family of square matrices, and let \(S^q\) denote the span of products of exactly \(q\) members, with \(S^0=\mathbb {C}I\). If \(p{\gt}0\) and \(S^{p-1}\) contains an invertible matrix, then \(\dim S\le \dim S^p\). In particular, the inequality holds whenever \(S\) contains an invertible matrix.

If \(uu^\dagger =\mathbb {1}\) and \(A\) is periodic with period \(m\), then \(A_u\) is periodic with the same period.

Let \(A\) and \(B\) each have the same length-\(N\) matrix-product vector, for every \(N\geq 0\), as a scalar-weight direct sum formed from a finite, pairwise non-repeated family of left-canonical periodic tensors \(\{ A_j\} _{j\in J}\) and \(\{ B_k\} _{k\in K}\), with a positive real scalar weight attached to each block. Suppose that, for every positive integer \(N\), there is a scalar \(c_N\ne 0\) such that

Then every block in either family has a block in the other family whose mixed overlap does not tend to zero. Every such non-decaying pair has equal bond dimension and differs by an invertible similarity and a unit-modulus scalar.

This specializes Theorem 22.2.3.58 to block-diagonal tensors with a single scalar weight per block.

Suppose that, for every \(j\in J\), there is an invertible matrix \(X_j\) such that

Then the two multiplicity-bearing tensors are related by the pure similarity

Suppose \(P_u^\dagger =P_u\), \(P_uP_v=\delta _{uv}P_u\), \(\sum _uP_u=\mathbb {1}\), and \(|c_u|=1\). If \(A\) is left canonical, then \(A'^i=\sum _uc_uP_uA^i\) is left canonical as well. Thus the rephasing in the proof of [ 191 , Theorem 4.1 ] keeps a left-canonical block trace preserving.

Suppose \(m\geq 1\), \(P_uP_v=\delta _{uv}P_u\), \(\sum _u P_u=\mathbb {1}\), and \(P_uA^i=A^iP_{u+1}\), with sector indices read modulo \(m\). For a word \(w=(i_1,\ldots ,i_n)\), write \(A^w=A^{i_1}\cdots A^{i_n}\). Then

The empty word has product \(\mathbb {1}\). The case \(n=p\) is the first equality of [ 191 , eq. Aprime-is-cPA ] .

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

where the diagonal entries of every \(R_j\) and \(S_k\) are nonzero. Suppose that the two representative families are pairwise non-repeated. Assign positive integers \(m_j\) and \(n_k\) to their representatives, and suppose that every transfer map is irreducible, has spectral radius one, and has peripheral spectrum respectively \(\{ z\in \mathbb {C}:z^{m_j}=1\} \) or \(\{ z\in \mathbb {C}:z^{n_k}=1\} \). If, for every positive \(N\), there is a scalar \(c_N\ne 0\) such that

then every representative in either family has a representative in the other family whose mixed overlap does not tend to zero. Every such pair has equal bond dimension and differs by an invertible similarity and a unit-modulus scalar.

Let \(m{\gt}0\), \(p\geq 0\), and \(r=\gcd (m,p)\). Every residue \(u\) modulo \(m\) has a unique expression \(u=\alpha +pk\pmod m\), with \(0\leq \alpha {\lt}r\) and \(0\leq k{\lt}m/r\). Adding \(p\) preserves \(\alpha \) and adds one to \(k\) modulo \(m/r\). These are the coordinates used in [ 191 , proof of Theorem 4.1 ] .

Suppose \(P_u\) are nonzero orthogonal projections with \(\sum _uP_u=\mathbb {1}_D\), \(A\) is left canonical, and \(P_uA^i=A^iP_{u+1}\). There are positive integers \(D_\alpha \) with \(\sum _\alpha D_\alpha =D\), isometries \(V_\alpha :\mathbb {C}^{D_\alpha }\to \mathbb {C}^D\), and left-canonical tensors \(C_\alpha \) such that

The tensors \(A^{[p]}\) and \(\bigoplus _\alpha C_\alpha \) have the same MPV at every length, including zero. The adjoint transfer map satisfies \(\mathcal{E}_{A^{[p]}}^\dagger (P_u)=P_{u+p}\), with indices taken modulo \(m\).

Decompose \(\mathbb C^7\) into five sectors of dimensions \((1,2,2,1,1)\). Let \(\mathcal F\) discard all off-diagonal sector blocks and send the diagonal block in sector \(i\) to its trace times the normalized identity in sector \(i+1\), with indices modulo five. Both \(\mathcal F\) and \(\mathcal F^2\) are irreducible channels. Their Kraus ranks are ten and nine, respectively. A minimal nine-letter tensor \(B\) for \(\mathcal F^2\) is trace preserving and periodic, and is a one-block irreducible form with weight one. Thus \(\mathcal E_B\) is \(2\)-divisible, but the supplied root \(\mathcal F\) has no Kraus representation on \(B\)’s nine-letter physical alphabet.

Let \(p{\gt}0\), let \(B_k\) be left-canonical periodic tensors on the blocked physical alphabet, and let \(|\nu _k|=1\). If a tensor \(A\) satisfies \(V^{(N)}(A^{[p]})=V^{(N)}\bigl(\bigoplus _k\nu _kB_k\bigr)\) for every \(N{\gt}0\), then there is a left-canonical tensor \(R\) on the original alphabet with \(R^{[p]}=\bigoplus _k\nu _kB_k\).

Let \(T^i=\bigoplus _k\mu _k A_k^i\), where the \(A_k\) are left-canonical periodic blocks and every \(\mu _k\) is nonzero. Let \(Q\) be a decomposition with pairwise non-repeated left-canonical periodic basis blocks and unit-modulus multiplicities. If \(V^{(N)}(T)=V^{(N)}(Q)\) for every positive \(N\), then \(|\mu _k|=1\) for every \(k\). This is the normalization consequence needed in the corrected forward direction of [ 191 , Theorem 4.1 ] ; the trace-preservation correction is recorded in [ 171 ] .

If \(A\) is left canonical and \((A^{[p]})^I=UC^IU^\dagger \) for a unitary \(U\), then \(C\) has a left-canonical root \(B\) at the original physical dimension: \(B^{[p]}=C\). This is the final conjugation in the proof of [ 191 , Theorem 4.1 ] .

Let \(A\) be left-canonical periodic of period \(m\) and let \(p{\gt}0\). Put \(r=\gcd (m,p)\). The prescribed orbit blocks \(C_a\), of period \(m/r\), admit a unitary \(U\) such that, with \(Q_a\) the step-orbit projections of Definition 22.4.6 and \(J_a\) the coordinate inclusions,

For every choice of \(c_a\) with \(c_a^{m/r}=1\), there is a left-canonical tensor \(A'\) with the same physical and bond dimensions as \(A\) such that

The unitary and orbit blocks are chosen independently of the phases.

Let \(V_a:\mathbb {C}^{D_a}\to \mathbb {C}^D\) satisfy \(\sum _a V_aV_a^\dagger =\mathbb {1}\) and \(\sum _a D_a=D\). If \(A^iV_a=V_aB_a^i\) for every \(a,i\), then there is a unitary \(U:\bigoplus _a\mathbb {C}^{D_a}\to \mathbb {C}^D\) with \(UJ_a=V_a\) and \(A^i=U(\bigoplus _a B_a^i)U^\dagger \), where \(J_a\) is the block inclusion.

If \(A_k\) and \(\bigoplus _{a=0}^{n_k-1}B_{k,a}\) have equal matrix product vectors at every length, then \(\bigoplus _k\mu _k A_k\) and \(\bigoplus _{k,a}\mu _k B_{k,a}\) do as well, for arbitrary complex weights \(\mu _k\).

If the blocked-to-root channel-root hypothesis holds, then the tensor-level blocked-to-root reconstruction hypothesis holds as well.

If the blocked equal-case \(Z\)-gauge extraction hypothesis and the construction of \(\widetilde A\) from that \(Z\)-gauge both hold, then the combined equal-case construction holds.

Let \(A\) and \(B\) be periodic tensors with the same matrix product vectors at every positive length. Then their bond dimensions agree, and after identifying the bond spaces, \(A\) and \(B\) are repeated blocks.

This is a normalized form of the state-vector conclusion in [ 228 , Theorem 5 ] . Let \(A\) be an irreducible left-canonical tensor whose peripheral transfer spectrum is the set of \(p\)-th roots of unity. If \(p\mid N\), there are orthogonal cyclic projections \((Q_u)_{u\in \mathbb Z/p\mathbb Z}\) such that \(\sum _uQ_u=\mathbb {1}\), \(Q_uA^i=A^iQ_{u+1}\), and \(Q_{u+N}=Q_u\). For every ring length \(N\geq 0\), its coefficients satisfy the boundary-aware decomposition

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

Each \(B_u\) is a \(p\)-periodic MPS of bond dimension \(D\) with local matrices given by (1). At \(N=0\), (6) reduces to the boundary traces \(\operatorname{tr}(Q_u)\) from (4). If \(p\nmid N\), then \(|V^{(N)}(A)\rangle =0\).

Let \(A\) have positive bond dimension and let \(p_0,p\ge 1\) with \(p_0\mid p\). If \(\mathcal{E}_{A^{[p_0]}}\) is primitive, then \(\mathcal{E}_{A^{[p]}}\) is primitive.

Under the periodic projective-rigidity hypothesis (Definition 22.3.4), if \(D{\gt}0\), then the virtual gauge family defines a projective representation of \(G\) on the bond space whose factor system is a \(2\)-cocycle.

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 spectrally periodic tensors of periods \(m\) and \(n\). If \(A\) and \(B\) are repeated blocks, then \(m=n\).

If \(A\) and \(B\) are repeated blocks, then their transfer maps have the same unit-circle eigenvalues.

Let \(A\) and \(B\) be repeated blocks. Then there is a scalar \(\xi \) with \(|\xi |=1\) such that \(\xi A\) and \(B\) generate the same vector at every positive length,

The phase belongs to the matched pair \(A,B\); distinct matched pairs of two bases of periodic tensors may require distinct phases, so no single phase is asserted for the direct-sum tensors ( [ 191 , lines 286–305, 625–627 ] ).

Let \(A\) and \(B\) be repeated blocks. Then there is a scalar \(\zeta \) with \(|\zeta |=1\) such that, for every length \(N\),

The proportionality scalar therefore depends on the length. Already the one-dimensional blocks with a single physical index, \(B^0=1\) and \(A^0=\zeta \), satisfy \(|V^{(N)}(A)\rangle =\zeta ^{N}|V^{(N)}(B)\rangle \), so no scalar independent of the length works unless \(\zeta =1\).

Let \(m,n{\gt}0\). If \(\{ z\in \mathbb {C}:z^m=1\} =\{ z\in \mathbb {C}:z^n=1\} \), then \(m=n\).

The rotated block-diagonal tensor satisfies

If two tensors have the same MPV coefficients at every positive length, then so do their blocked tensors after any positive blocking length.

If \(\mathcal{V}(A)=\mathcal{V}(B)\), then \(\mathcal{V}(A_u)=\mathcal{V}(B_u)\).

Let \(A\) and \(B\) be spectrally periodic tensors of periods \(m\) and \(n\). If their transfer maps have the same peripheral spectrum, then \(m=n\).

If \(A\) is spectrally periodic, then its bond dimension satisfies \(D\geq 1\).

Let \(A\) be spectrally \(m\)-periodic. There is a positive definite matrix \(\sigma \) such that

is a left-canonical \(m\)-periodic tensor. The similarity is pure, so it leaves every matrix-product vector unchanged and introduces no scalar weight into the normalization.

Let \(A\) be an MPS tensor whose transfer map is irreducible with spectral radius one ( [ 191 , lines 248–261 ] ). Then there exists a positive integer \(m\) such that \(A\) is spectrally \(m\)-periodic, so that the peripheral eigenvalues of \(\mathcal{E}_A\) are precisely the \(m\)-th roots of unity ( [ 191 , lines 257–258 ] ).

Suppose every representative \(A_j\) is spectrally \(m_j\)-periodic. There are left-canonical \(m_j\)-periodic representatives \(B_j\), each purely similar to \(A_j\), such that the two full multiplicity-bearing tensors are purely similar. Every multiplicity matrix \(R_j\) is unchanged.

Let \(B\) be an MPS tensor in irreducible form II and let \(p \ge 1\). Assume the forward construction of \(\widetilde A\) for refinability \(\Rightarrow \) divisibility and the reverse transfer-map construction for divisibility \(\Rightarrow \) refinability. Then \(B\) is \(p\)-refinable iff its transfer map \(\mathcal{E}_B\) is \(p\)-divisible.

Let \(B\) be an MPS tensor in irreducible form II. Assume that every refinement witness gives a tensor \(\widetilde A\) with \(\sum _i (\widetilde A^i)^\dagger \widetilde A^i = \mathbb {1}\) and \(\mathcal{E}_B = \mathcal{E}_{\widetilde A^{[p]}}\). Then \(p\)-refinability of \(B\) implies \(p\)-divisibility of \(\mathcal{E}_B\).

Assume the blocked equal-case construction. Then for every tensor \(B\) in irreducible form II, \(p\)-refinability of \(B\) implies \(p\)-divisibility of \(\mathcal{E}_B\).

Let \(B=\bigoplus _k\mu _kB_k\), where the \(B_k\) are left-canonical periodic tensors and \(|\mu _k|=1\), and let \(p{\gt}0\). If \(B\) is \(p\)-refinable, then \(\mathcal{E}_B\) is \(p\)-divisible. This is the forward implication of [ 191 , Theorem 4.1 ] with the trace preservation of \(\mathcal{E}_B\) made explicit; see Remark 22.4.30 and [ 171 ] .

Let \(B\) be an MPS tensor in irreducible form II and let \(p \ge 1\). Assume the reverse transfer-map root hypothesis: there is a tensor \(A\) with \(\mathcal{E}_B = \mathcal{E}_{A^{[p]}}\) whenever \(\mathcal{E}_B\) is \(p\)-divisible. Then \(p\)-divisibility of \(\mathcal{E}_B\) implies \(p\)-refinability of \(B\).

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

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

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).

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

If \(uu^\dagger =\mathbb {1}\), then the transfer map is invariant under physical-index rotation: \(\mathcal{E}_{A_u}=\mathcal{E}_A\).

If \(|c|=1\), then the tensors \(A\) and \(cA\) have the same transfer map.

If \(A\) and \(B\) are \(\mathbb {Z}_m\)-gauge equivalent, then their \(m\)-blocked tensors \(A^{[m]}\) and \(B^{[m]}\) are gauge equivalent in the ordinary sense.

Let \((\mu _i)_{i\in I}\) and \((\nu _i)_{i\in I}\) be two finite lists of nonzero multiplicity entries. If \(\mu _i^m=\nu _i^m\) for every \(i\in I\), then there exists a diagonal matrix \(Z\) such that

Assume the periodic equal-case Fundamental Theorem at the blocked physical dimension. Let \(C\) and \(A^{[p]}\) be MPS tensors of the same bond dimension, both in irreducible form II, and suppose that they have the same MPV family. Then there exists an integer \(m \ge 1\) and a \(\mathbb {Z}_m\)-gauge equivalence between \(C\) and \(A^{[p]}\).