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 moreover the periodic equal-case Fundamental Theorem of MPS (Theorem 22.2.2.1) as an explicit hypothesis on the pair \((d, D)\), expressed by the hypothesis  \(d\, D\). 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 \(i\),

Thus \(A\) and \(B\) are related by a genuine \(\mathbb Z_m\) virtual gauge, not merely by a scalar root of unity.

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 a tensor \(A\) and a blocking length \(L\), the \(L\)-blocked tensor is the tensor with physical index set \(\{ 0,\ldots ,d{-}1\} ^L\) (hence physical dimension \(d^L\)) and the same bond dimension \(D\), whose matrices are indexed by words \((i_1, \ldots , i_L) \in \{ 0,\ldots ,d{-}1\} ^L\):

Blocking coarse-grains \(L\) neighbouring sites into one tensor: the inner virtual bonds are contracted, the \(L\) physical legs merge into a single composite index, and the two outer bonds remain as the bond of \(A^{[L]}\).

Given a finite family of tensors \((B_i)\) with designated periods \((p_i)\), block each \(B_i\) to the shared period \(\operatorname{lcm}(p_i)\), so every block lives in the same physical space.

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 \(P_k\). We shall sometimes keep these projections and corner isomorphisms as specified witnesses; forgetting the witnesses gives the displayed decomposition. This is the inverse cyclic indexing of the off-diagonal convention in [ DlCCSPG17 , 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 above 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, (5) is represented by

where the black node denotes the tensor \(A\), the upper leg is the physical index \(i\), and the two red side nodes denote the gauge matrices acting on the virtual legs.

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 [ DlCCSPG17 , lines 266–271 ] . Following [ DlCCSPG17 , 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 8.9.2), which requires the stronger condition that every block is \(1\)-periodic.

Given a finite family of blocking periods \((p_i)\) indexed by a finite set, their least common multiple serves as the shared blocking period.

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

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

The 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.

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 ] .

The predicate called on-site symmetry requires \(\mathcal{V}(A)=\mathcal{V}(\widetilde A_g)\) for every \(g\in G\), where \(U:G\to \mathrm{GL}_d(\mathbb {C})\) acts on the physical index. This is equality of periodic MPV families and is the hypothesis used to construct virtual local gauges. It is not, by itself, the source notion that every reduced density operator of an infinite physical state is invariant under \(U(g)^{\otimes N}\).

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 [ DlCCSPG17 , 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 [ DlCCSPG17 , 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

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. §11.1); in the genuinely 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

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

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 hypothesis is the transfer-map reading of the construction in [ DlCCSPG17 , lines 765–810 ] : for each periodic block \(A_j\), the scalars \(c_{j,\alpha }\) define matrices \(A'_j\), and then \(\widetilde A = U^\dagger A'U\). This source description is the intended irreducible-form subcase of the hypothesis; the definition above keeps the construction as a conditional input and does not assert these block scalars \(c_{j,\alpha }\) and matrices \(A'_j\) for arbitrary \(A\). The cited source moreover obtains the blocked-tensor identity \(\widetilde A^{[p]} = C\), whereas the formal hypothesis above records only its channel-level consequence \(\mathcal{E}_C = \mathcal{E}_{\widetilde A^{[p]}}\) (equality of \(p\)-blocked transfer maps). This conditional input is therefore the transfer-map weakening of the source construction, not the full blocked tensor equality.

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 \(M \in M_{d}(\mathbb {C})\) and an MPS tensor \(A\), the physical-index rotation is the tensor \((A_M)^i := \sum _j M_{ij}\, A^j\). 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, we write \(\mathcal{V}^+(A) = \mathcal{V}^+(B)\) if \(|V^{(N)}(A)\rangle = |V^{(N)}(B)\rangle \) for every \(N {\gt} 0\).

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 [ DlCCSPG17 , 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 \(i\). The source equal-case theorem uses the finer block-multiplicity diagonal gauges \(Z_j\) from Theorem 22.2.2.1; those multiplicity-space matrices are not part of this global relation.

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 \(A^i=\zeta ^{-1}X^{-1}B^iX\).

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 \(i\),

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 \(U=0\).

From a sector right inverse \(\Omega \) recovering every sector word product \(\bar A_k^{\, W}\) over blocked words \(W\in [d^m]^L\) with \(L{\gt}0\) ( [ DlCCSPG17 , 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: \(\sum _{\mathbf j}(\Omega _{k+1}(X_k))_{\mathbf j}\, F_{k+1}^{\mathbf j}=X_k\) ( [ DlCCSPG17 , 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 [ DlCCSPG17 , 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 [ DlCCSPG17 , 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 \(m\ge 1\), 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 [ DlCCSPG17 , 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 [ DlCCSPG17 , 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.

Feeding the ambient corner right inverse into the right-inverse contraction yields the contracted product-tensor identity [ DlCCSPG17 , 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.

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 \(N{\gt}0\), \(O_{A_{u+1},B_{v+1}}(N)=O_{A_u,B_v}(N)\).

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 (4) 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 (4) 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 10.3.1. Hence the two assembled tensors are gauge equivalent.

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 \(z(\mu ,\nu )^m=1\).

If each tensor \(B_i\) has a primitive transfer map after blocking by its own positive period \(p_i\), then blocking the entire family to the common LCM period preserves primitivity of each member’s transfer map.

Assume the periodic projective-rigidity hypothesis (Definition 22.3.2) 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 22.3.3 satisfies the multiplicative \(2\)-cocycle identity \(\omega (g,h)\omega (gh,k) = \omega (g,hk)\omega (h,k)\).

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 [ DlCCSPG17 , 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 [ DlCCSPG17 , 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 [ DlCCSPG17 , 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 \(V^{(N)}(B)_\sigma = \zeta ^N V^{(N)}(A)_\sigma \).

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.

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

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

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 [ DlCCSPG17 , Appendix A, lines 1082–1084 ] , after the sectorwise proportionality relation has been obtained.

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 [ DlCCSPG17 , eq. Aoffdiag ] ; the compressed blocks of [ DlCCSPG17 , 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 \(A^{[p]}\).

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

satisfies \(\ker \Phi _N=\{ 0\} \).

Scope restriction. The source consequence following the periodic-overlap dichotomy [ DlCCSPG17 , lines 589–609 ] states eventual independence at every sufficiently large length and also states that, for an ambient tensor \(A\) in irreducible form having \(\{ A_j\} _{j\in J}\) as its basis of periodic tensors, the non-zero vectors \(|V^{(N)}(A_j)\rangle \) span the vector \(|V^{(N)}(A)\rangle \); this spanning assertion is what the paper cites as justification for the phrase “basis of periodic vectors”. The theorem recorded here proves independence only along the common-multiple subsequence \(pN\) and omits the spanning assertion. The corresponding paper-gap note, [ con26s ] , Section “Scope restriction: independence without spanning”, records this restriction and its elimination plan.

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

Let \(A\) and \(B\) be in irreducible form, and suppose their bases of periodic tensors contain no repeated pair of distinct basis elements. If the periodic-overlap hypothesis holds for the two bases, then the two bases have the same cardinality and are matched by a bijection whose matched elements are repeated blocks.

Let \(A\) and \(B\) be in irreducible form, and suppose their bases of periodic tensors contain no repeated pair of distinct basis elements. Assume the periodic-overlap hypothesis and the corresponding power equality for the scalar multiplicity entries. Then the bases are matched by a bijection, and each matched pair has a scalar diagonal \(Z_j\in M_{1}(\mathbb {C})\) with

The two scalar multiplicity entries also determine the same singleton multiset.

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. Write

where \(\{ A_j\} _{j \in J}\) and \(\{ B_k\} _{k \in K}\) are bases of periodic tensors, \(R_j\) and \(S_k\) are diagonal multiplicity matrices of sizes \(r_j\times r_j\) and \(s_k\times s_k\), and \(A_j^i\) and \(B_k^i\) act on bond spaces of dimensions \(D_j\) and \(D_k\). Suppose that \(\mathcal{V}(A)=\mathcal{V}(B)\) (equal MPV families for all lengths \(N\)).

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 exists:

1. an invertible matrix \(Y_j\) of size \(D_{\pi (j)}\times D_j\), and

2. a diagonal unitary matrix \(Z_j\) of size \(r_j\times r_j\) satisfying \(Z_j^{m_j}=\mathbb {1}_{r_j}\),

such that \(B_{\pi (j)}^i=Y_jA_j^iY_j^{-1}\) for every physical index \(i\), and, after reordering the diagonal entries of the multiplicity matrix \(S_{\pi (j)}\), one has \(Z_jR_j=S_{\pi (j)}\). Equivalently, setting \(Z=\bigoplus _j Z_j\otimes \mathbb {1}_{D_j}\) gives an invertible matrix \(Y\) such that \(ZA^i=YB^iY^{-1}\) for all \(i\), with \([A^i,Z]=0\) and \(\mathcal{V}(A)=\mathcal{V}(ZA)\).

Remark. The proof is conditional on the periodic-overlap dichotomy of [ DlCCSPG17 , lines 589–609 ] , which is left here as a separate hypothesis.

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. Suppose that, for every positive \(N\), the vectors \(|V^{(N)}(A)\rangle \) and \(|V^{(N)}(B)\rangle \) are proportional by a nonzero scalar. Then there is a bijection \(\pi \colon J\to K\) such that, for every \(j\in J\), the blocks \(A_j\) and \(B_{\pi (j)}\) have the same period 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 [ DlCCSPG17 , Theorem 3.4, lines 613–632 ] .

The theorem relates the two bases of periodic tensors but makes no assertion relating their multiplicity matrices.

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 boundary. The source initially assumes that the transfer maps of the basis tensors are irreducible and have spectral radius one, and then obtains the trace-preserving normalization by a block-diagonal similarity [ DlCCSPG17 , lines 313–332 ] . The present theorem starts with the normalized basis tensors. The multiplicity matrices are otherwise arbitrary diagonal matrices with nonzero complex entries, the displayed direct sums are literal, and proportionality is assumed only at positive lengths. Theorem 22.1.5 constructs each normalizing similarity, while Theorem 22.1.8 assembles them without changing the multiplicity matrices. Theorem 22.2.3.49 below applies the present result and returns its conclusion to the unnormalized blocks.

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 \(m{\gt}0\), 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 [ DlCCSPG17 , Appendix A, lines 1072–1080 ] : after the preceding translation step has made the scalar independent of the sector label, choosing a common \(m\)-th root of that scalar 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 [ DlCCSPG17 , Appendix A, lines 1072–1080 ] , after the contraction with the maps \(F_u\) and \(\Omega _u\) has produced the displayed product-tensor identity.

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.

Scope restriction. This formulation records one positive scalar per listed block and equality with the corresponding direct sum also at the empty word. Thus it is a specialization of the preceding multiplicity-bearing theorem and is not itself the unrestricted irreducible-form statement of [ DlCCSPG17 , Theorem 3.4 ] .

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

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.

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.

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.

Theorems 22.3.3 and 22.3.4 combine into a single statement: from the periodic projective-rigidity hypothesis (Definition 22.3.2), one obtains a projective representation of \(G\) on the bond space whose factor system is a genuine \(2\)-cocycle whenever \(D{\gt}0\).

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.

The rotated block-diagonal tensor satisfies

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

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.

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 \(B\) 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 \(B\).

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

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