22 Periodic MPS: irreducible form and the periodic Fundamental Theorem
This chapter develops the periodic equal-case Fundamental Theorem of MPS (Theorem 22.2.2.1) following [ DlCCSPG17 ] , and gives its two applications: the symmetry corollary lifting a physical on-site symmetry of a tensor in irreducible form II to a virtual \(Z\)-gauge, and the equivalence between \(p\)-refinability of the matrix-product-vector family and \(p\)-divisibility of the transfer map.
22.1 Periodic irreducible form and physical-index rotation
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 ] .
If \(A\) is spectrally periodic, then its bond dimension satisfies \(D\geq 1\).
In bond dimension zero the matrix space and its endomorphism algebra are zero-dimensional, so the transfer map has spectral radius zero. This contradicts the spectral-radius-one hypothesis.
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.
Conjugation by \(X\mapsto CXC^\dagger \) gives a linear equivalence of the matrix space. Transporting eigenvectors through this equivalence preserves every eigenvalue and hence the unit-circle eigenvalue set.
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” ] ).
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
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
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
Repeat \(X_j\) on every copy indexed by a diagonal entry of \(R_j\), and take their block-diagonal direct sum. On the \((j,q)\) block, conjugation by this matrix sends \(\mu _{j,q}A_j^i\) to \(\mu _{j,q}B_j^i\).
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.
Apply the positive Perron gauge separately to every representative and assemble these gauges with Theorem 22.1.7. Since every spectral radius is already one, no representative is rescaled.
If \(A\) is periodic, then its bond dimension satisfies \(D\geq 1\).
The prescribed peripheral spectrum contains the eigenvalue \(1\). In bond dimension zero the matrix space is zero-dimensional and admits no eigenvector, which is a contradiction.
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.
Let \(A^i=\bigoplus _j\mu _j A_j^i\) be a normal canonical block family with nonzero complex weights. Replacing \(\mu _j\) by \(|\mu _j|\) and \(A_j\) by \((\mu _j/|\mu _j|)A_j\) gives the positive-weight irreducible form convention and does not change the MPV family. The corresponding irreducible-form witness has all periods equal to \(1\). This is the period-one part of the statement in [ DlCCSPG17 , Section II ] that normal tensors are periodic tensors with period \(1\).
The converse containment is not part of the implication above. Irreducible form permits periodic blocks with period larger than \(1\); for instance, the antiferromagnetic example has period \(2\).
Physical-index rotation and its MPV expansion are recorded in Section G.1. The compatibility results below supply the physical action used in the periodic analysis of [ DlCCSPG17 ] .
If \(uu^\dagger =\mathbb {1}\), then the transfer map is invariant under physical-index rotation: \(\mathcal{E}_{A_u}=\mathcal{E}_A\).
This is the unitary freedom of Kraus representations from [ Wol12 , Theorem 2.1(4) ] .
If \(uu^\dagger =\mathbb {1}\) and \(A\) is left-canonical, then \(A_u\) is left-canonical.
Expand \(\sum _i(B^i)^\dagger B^i\), where \(B^i=\sum _j u_{ij}A^j\), swap sums, apply orthogonality \(u^\dagger u=\mathbb {1}\), and collapse to \(\sum _j(A^j)^\dagger A^j=\mathbb {1}\).
If \(uu^\dagger =\mathbb {1}\) and \(A\) is irreducible, then \(A_u\) is irreducible.
Contrapositive: any nontrivial invariant projection \(P\) for the rotated tensor yields \((\mathbb {1}-P)A^kP=0\) for all \(k\) via the orthogonality of \(u\), contradicting irreducibility of \(A\).
If \(uu^\dagger =\mathbb {1}\) and \(A\) is periodic with period \(m\), then \(A_u\) is periodic with the same period.
Assembles transfer-map invariance, left-canonical preservation, and irreducibility preservation.
If \(\mathcal{V}(A)=\mathcal{V}(B)\), then \(\mathcal{V}(A_u)=\mathcal{V}(B_u)\).
Induction on the word length with a strengthened hypothesis carrying an arbitrary prefix matrix, reducing each step to the hypothesis via word concatenation.
The rotated block-diagonal tensor satisfies
Pointwise matrix-entry computation using linearity of the block-diagonal embedding and reindexing.
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.
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.
Apply the assumed periodic equal-case theorem to \(A\) and \(B\). Its \(Z\)-gauge conclusion supplies \(m\), \(Y\), and \(Z\) with the three displayed identities by Definition 22.1.1.7.
22.1.1 The \(Z\)-gauge degree of freedom
Given complex multiplicity entries \(\mu ,\nu \), define the entrywise \(Z\)-gauge ratio by \(z(\mu ,\nu ):=\mu /\nu \).
If \(\mu ^m=\nu ^m\) and \(\nu \neq 0\), then \(z(\mu ,\nu )^m=1\).
If \(\nu \neq 0\), then \(z(\mu ,\nu )\nu =\mu \).
For matched multiplicity entries \((\mu _i)_i\) and \((\nu _i)_i\) on a finite index type, define the diagonal matrix \(Z:=\operatorname{diag}\! (z(\mu _i,\nu _i))_i\).
If \(\mu _i^m=\nu _i^m\) and \(\nu _i\neq 0\) for all \(i\), then \(Z^m=\mathbb {1}\).
Under \(\nu _i\neq 0\) for all \(i\), one has \(Z\operatorname{diag}(\nu )=\operatorname{diag}(\mu )\).
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:
\(Z\) commutes with every tensor matrix: \([A^i,Z]=0\) for all physical indices \(i\), and
\(Z^m=\mathbb {1}_D\).
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.
When \(m=1\), the condition \(Z^1=\mathbb {1}\) forces \(Z=\mathbb {1}\), so \(Z\)-gauge equivalence reduces to ordinary gauge equivalence (Definition 2.3.1). The \(Z\)-gauge is therefore a genuine new degree of freedom that appears only for \(m{\gt}1\).
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.
Write the \(\mathbb {Z}_m\)-gauge relation as \(ZA^i=YB^iY^{-1}\) with \(Z^m=\mathbb {1}\) and \([A^i,Z]=0\). For a blocked letter, commute one factor of \(Z\) past each physical letter in the associated length-\(m\) word. The total prefactor becomes \(Z^m=\mathbb {1}\), so the entire blocked word is related by the same ordinary gauge \(Y\).
If \(A\) and \(B\) are \(\mathbb {Z}_m\)-gauge equivalent, then the blocked tensors \(A^{[m]}\) and \(B^{[m]}\) have the same MPV family.
Apply Theorem 22.1.1.9 and then the gauge invariance of MPV families.
22.2 Periodic Fundamental Theorem
The theorems in Chapter 10 all require that each block has a primitive transfer map, i.e. peripheral spectrum exactly \(\{ 1\} \). This is achieved by blocking the original tensor by a common period. [ DlCCSPG17 ] develops a framework that avoids this blocking step, giving a fundamental theorem that applies directly to periodic tensors.
22.2.1 Common-period blocking lemmas
Given a finite family of blocking periods \((p_i)\) indexed by a finite set, their least common multiple serves as the shared blocking period.
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.
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.
Each \(p_i\) divides \(\operatorname{lcm}(p_i)\). Apply the divisibility monotonicity theorem for primitive blocked transfer maps to each family member separately.
22.2.2 Statement of the periodic Fundamental Theorem
The following is the equal-case theorem of [ DlCCSPG17 , lines 643–656 ] . The statements recorded here are conditional components of that theorem: the final comparison still assumes periodic-overlap and power-equality hypotheses, so the literature theorem has not yet been obtained here as a single unconditional result.
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:
an invertible matrix \(Y_j\) of size \(D_{\pi (j)}\times D_j\), and
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 \((\mu _i)_{i\in I}\) 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
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 \(Z\, \operatorname{diag}(\nu )=\operatorname{diag}(\mu )\).
Apply Theorem 22.2.2.2 to obtain \(Z\), and apply the Newton–Girard power-sum identity to recover the multiplicity multiset.
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.
Apply the conditional block-matching theorem to the two bases extracted from the irreducible-form representations.
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.
22.2.3 Periodic overlap dichotomy
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.
For each sector choose the coordinate functional at the nonzero reference entry and normalize it so that it takes the value \(1\) on \(B_v^{i_v}\). Applying the tensor product of these functionals to the reference tensor identity gives \(\prod _v\kappa _v=z\). To prove the proportionality in sector \(v\), replace only the \(v\)-th functional by the coordinate functional at an arbitrary entry \((r,c)\) and apply the tensor identity with \(\sigma _v=i\) and all other letters fixed at their references. The product over the other sectors is nonzero because its product with \(\kappa _v\) is \(z\), so cancellation gives \((A_v^i)_{r,c}=(\kappa _v B_v^i)_{r,c}\) for every entry.
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.
Apply Theorem 22.2.3.1 to obtain scalars \(\mu _v\) with \(\prod _v\mu _v=z\) and \(A_v^i=\mu _vB_v^i\). Since \(m{\gt}0\), the identities \(\xi ^m=z\) and \(z\ne 0\) imply \(\xi \ne 0\). Put \(\kappa _v=\mu _v/\xi \). Then \(\prod _v\kappa _v=(\prod _v\mu _v)/\xi ^m=z/z=1\), and \(A_v^i=(\kappa _v\xi )B_v^i\).
Let \(A\) have positive bond dimension, and let \(c\in \mathbb {C}\). If both \(A\) and the scalar multiple \(cA\) are left-canonical, then \(|c|=1\).
The two left-canonical equations give
Substituting the first equation into the second gives \((\bar c c)\mathbb {1}=\mathbb {1}\). Since the bond dimension is positive, comparison of any diagonal entry gives \(\bar c c=1\), hence \(|c|=1\).
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.
By Theorem 22.2.3.3, applied to \(B\) and the scalar \(\kappa \xi \), one has \(|\kappa \xi |=1\). Since \(|\xi |=1\), the multiplicativity of the complex norm gives \(|\kappa |=1\).
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.
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.
Put \(K_i=(A^i)^\dagger \) and let \(\Phi _K\) be the Kraus map of the family \(K\). Since \(A\) is left-canonical, \(\Phi _K\) is unital. The projection identities and the shift relation give
because both sides are equal to \(P_k\). The equality case in the Kadison–Schwarz inequality therefore gives \(P_{k+1}K_i^\dagger =K_i^\dagger \Phi _K(P_{k+1})\). Substituting \(K_i^\dagger =A^i\) and \(\Phi _K(P_{k+1})=P_k\) gives \(P_{k+1}A^i=A^iP_k\).
Insert \(\sum _u P_u=\mathbb {1}\) on the right of \(A^i\), then use Theorem 22.2.3.6 and \(P_u^2=P_u\) term by term.
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.
Induct on the length of \(w\), using Theorem 22.2.3.6 for the first letter and the induction hypothesis for the remaining word.
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 \(\ell =mN_0\).
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 \(m\).
Induct on the word \(\mathbf i\). For the empty word the identity is \(P_uF_u^{\mathbf j}=F_u^{\mathbf j}\), which follows from the idempotent part of the orthogonal-projection hypothesis. For a nonempty word, remove the first letter and apply the induction hypothesis at the next cyclic sector.
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 \(c_u\).
Induct on \(k\). The base case \(k=1\) is the hypothesis. For the inductive step split \(W\) into its first length-\(m\) block and the remaining length-\(km\) word, apply the concatenation law, and use that the junction shift \(m\cdot 1=0\) returns to the base sector before applying the block identity and the induction hypothesis.
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.
Expand each \(A_k^{\sigma _k}X_k =\sum _{\mathbf j}(\Omega _{k+1}(X_k))_{\mathbf j} A_k^{\sigma _k}F_{k+1}^{\mathbf j}\) using the right inverse, apply the noncommutative distributivity identity
and pull the scalar coefficients \(\prod _k(\Omega _{k+1}(X_k))_{\rho _k}\) out of each ordered product.
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.
Apply the grading at sector \(-(k+1)\) and simplify the cyclic index.
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.
Induct on the word. The head junction \(P_uA^iP_{u+1}=P_uA^i\) collapses by the shift and idempotency; the induction hypothesis handles the tail, and the accumulated offset realigns.
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.
A blocked letter is a word of length exactly \(m\); the accumulated offset \(m\cdot 1=0\) vanishes, so the telescoping gives the diagonal corner.
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.
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 \(W\).
The decoding bijection identifies a word-indexing function with its ordered list of letters, and the iterated blocking bijection concatenates the decoded length-\(m\) blocks in order.
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 \(\varphi \).
Induct on \(W\). The head blocked letter collapses by the blocked-letter diagonal corner to \(\varphi _k(\bar A_k^I)\), and multiplicativity of \(\varphi \) reassembles the letters:
The empty word is excluded so the induction stays inside the corner.
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, \(P_kYP_k=Y\),
Set \(\hat\Omega _k(Y)=\Omega _k(\varphi _k^{-1}(Y))\) for corner-supported \(Y\). Applying \(\varphi _k\) to the sector recovery and using the corner product of a flattened word,
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 \(X_k\),
Apply the right-inverse contraction with the ambient corner right inverse in place of the abstract right inverse.
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 \(P_{k+1}A^i=A^iP_k\).
Choose the projection family contained in the cyclic sector decomposition. It satisfies
Apply Theorem 22.2.3.6.
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
Choose the projection family contained in the cyclic sector decomposition. It satisfies
Apply Theorem 22.2.3.7.
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.
This is the cyclic-sector existence part of Lemma bdcf, translated into the inverse indexing convention of Definition 22.2.3.5.
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 ] .
Let \(U\) diagonalize \(P_k\), so that
Write \(B^i=U^\dagger (A^{(m)})^iU\), and let \(B_{kk}^i\) be the corner block on the support of \(P_k\). The sector tensor is \(C_k^i=V_k^\dagger B_{kk}^iV_k\), where \(V_k\) identifies the support with \(\mathbb {C}^{\dim C_k}\). The linear corner isomorphism is the inverse transport
Hence
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 \(\varphi _k(C_k^i)=P_k(A^{(m)})^iP_k\).
Apply Theorem 8.11.1 to the original tensor \(A\) with its period-\(m\) peripheral-spectrum hypotheses. The resulting decomposition is the decomposition of \(A^{(m)}\), and the isometries \(V_k\) in its conclusion are the required support isometries.
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 \(N{\gt}0\), \(O_{A_{u+1},B_{v+1}}(N)=O_{A_u,B_v}(N)\).
Choose the projection families \(P_u\) and \(Q_v\) from the two cyclic sector decompositions. The single-site relations \(P_{u+1}A^i=A^iP_u\) and \(Q_{v+1}B^i=B^iQ_v\) give the following trace-sum identity, where \(T\) is the one-site rotation of physical words:
The first and last equalities in (1) are the cyclic-sector trace formula after reindexing blocked words as physical words of length \(Nm\).
Let \(A\) be periodic of period \(m\). Let \(P_0,\ldots ,P_{m-1}\) be orthogonal projections satisfying
If
then \(U=0\).
Since the peripheral spectrum of \(\mathcal{E}_A\) consists of \(m\)-th roots of unity, the corresponding lemma in the companion quantum-channel volume [ LTC26 , “Powering collapses peripheral eigenvalues” ] gives \(\operatorname{spec}_{\mathrm{per}}(\mathcal{E}_A^m)=\{ 1\} \), hence \(\zeta =1\). The cyclic trace gives \(\operatorname{tr}U=\operatorname{tr}(P_uUP_v)=\operatorname{tr}(P_vP_uU)=0\).
Put \(W=\sum _{t=0}^{m-1}\mathcal{E}_A^t(U)\). The equation \(\mathcal{E}_A^m(U)=U\) gives \(\mathcal{E}_A(W)=W\), and trace preservation gives \(\operatorname{tr}W=0\). Irreducibility of \(A\) therefore gives \(W=0\). The shift relation \(P_{k+1}A^i=A^iP_k\) moves \(\mathcal{E}_A^t(U)\) into the \((u+t,v+t)\) corner, so only the \(t=0\) term contributes after multiplying by \(P_u\) and \(P_v\). Thus \(U=P_uWP_v=0\).
If \(A\) and \(B\) have periods \(m_A\ne m_B\), then \(\lim _{N\to \infty }O_{AB}(N)=0\).
If \(D_A\ne D_B\), use Theorem 22.2.3.40. Assume \(D_A=D_B\) and \(A\sim _{\mathrm{gp}}B\), say \(B^i=\zeta XA^iX^{-1}\) with \(X\in \mathrm{GL}_D(\mathbb {C})\) and \(\zeta \ne 0\). The transfer maps satisfy
With \(\mathcal{E}_A(\rho _A)=\rho _A\), put \(\sigma =X\rho _AX^*\). Then \(\mathcal{E}_B(\sigma )=|\zeta |^2\sigma \), while \(\mathcal{E}_B(\rho _B)=\rho _B\) for a nonzero positive fixed point \(\rho _B\). Perron–Frobenius eigenvalue uniqueness [ Wol12 , Theorem 6.3 ] gives \(|\zeta |^2=1\). Hence \(\mathcal{E}_B\) is conjugate to \(\mathcal{E}_A\), so \(\operatorname{spec}_{\mathrm{per}}(\mathcal{E}_A)=\operatorname{spec}_{\mathrm{per}}(\mathcal{E}_B)\). Since
equality of the two finite sets in (2) gives \(m_A=m_B\), a contradiction. Thus \(A\not\sim _{\mathrm{gp}}B\), and the irreducible trace-preserving overlap dichotomy gives \(\lim _{N\to \infty }O_{AB}(N)=0\).
For a periodic tensor equipped with a cyclic sector decomposition, each nonzero blocked sector tensor is normal.
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 \(\lim _{N\to \infty }O_{AB}(N)=0\).
Write \(\bar A\sim \bigoplus _{u\in \mathbb {Z}_m}A_u\) and \(\bar B\sim \bigoplus _{v\in \mathbb {Z}_m}B_v\). For every \(N\),
For each sector pair \((u,v)\),
The finite double sum (3) therefore gives
If \(A\sim _{\mathrm{gp}}B\), then \(\bar A\sim _{\mathrm{gp}}\bar B\), so for some \(X\in \mathrm{GL}_D(\mathbb {C})\) and \(\zeta \ne 0\), \((\bar B)^\alpha =\zeta X(\bar A)^\alpha X^{-1}\). Thus \(V^{(N)}(\bar B)_\sigma =\zeta ^N V^{(N)}(\bar A)_\sigma \), and
Since \(\lim _{N\to \infty }O_{\bar A\bar A}(N) =\lim _{N\to \infty }O_{\bar B\bar B}(N)=m{\gt}0\),
hence \(|\zeta |=1\). Therefore
contradicting \(\lim _{N\to \infty }O_{\bar A\bar B}(N)=0\). Thus \(A\not\sim _{\mathrm{gp}}B\), and the irreducible trace-preserving overlap dichotomy gives \(\lim _{N\to \infty }O_{AB}(N)=0\).
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 \(G\) 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.
Set \(\varphi _k=(\kappa _0\cdots \kappa _{k-1})^{-1}\), with the empty product equal to \(1\). For \(k{\lt}n\),
Separating the final factor in the product-one hypothesis gives \((\kappa _0\cdots \kappa _{n-1})\kappa _n=1\), which gives
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 \(m\ge 1\). 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 \(U=\sum _vU_v\).
The corner relation \(U_v=P_{v-q}U_vQ_v\) gives \(P_uU_{u+q}Q_{u+q}=U_{u+q}\); reindexing the sum by \(v=u+q\) turns \(\sum _uP_uU_{u+q}Q_{u+q}\) into \(\sum _vU_v\).
Suppose \(B^i=\zeta XA^iX^{-1}\) and \(|\zeta |=1\), with \(X\) invertible. Then \(A\) and \(B\) satisfy the repeated-block relation \(A^i=\zeta ^{-1}X^{-1}B^iX\).
Since \(|\zeta |=1\), the scalar is nonzero. Multiplication by \(X^{-1}\) on the left and by \(X\) on the right gives \(X^{-1}B^iX=\zeta A^i\); multiplication by \(\zeta ^{-1}\) proves the claim.
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.
The corner relation \(U_v=P_{v-q}U_vQ_v\) gives \(P_uU_{u+q}Q_{u+q}=U_{u+q}\), so \(U=\sum _vU_v\) after reindexing. Distinct projections are orthogonal, \(Q_vQ_w=0\) and \(P_{v-q}P_{w-q}=0\) for \(v\ne w\), which kills the cross terms in \(UU^\dagger \) and \(U^\dagger U\); the diagonal terms are \(U_vU_v^\dagger =P_{v-q}\) and \(U_v^\dagger U_v=Q_v\), so
The off-diagonal structure of \(B\) makes \(Q_vB^iQ_w\) vanish unless \(w=v+1\), so \(UB^iU^\dagger =\sum _vU_vQ_vB^iQ_{v+1}U_{v+1}^\dagger \). Summing the per-site proportionality over \(u\) and reindexing by \(v=u+q\) gives
The last equality is the off-diagonal structure of \(A\). The unit-modulus scalar and the unitary \(U\) exhibit \(A\) and \(B\) as repeated blocks.
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 \(i\).
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 \(N\to \infty \).
Let \(A\) and \(B\) be periodic tensors. Then at least one of the following alternatives holds:
\(\langle V^{(N)}(A) | V^{(N)}(B) \rangle \to 0\) as \(N\to \infty \);
\(A\) and \(B\) are related by the repeated-block relation, i.e. they differ by a unit-modulus phase and an invertible gauge.
If \(m_A\ne m_B\), then \(\lim _{N\to \infty }O_{AB}(N)=0\) by Theorem 22.2.3.29. Assume \(m_A=m_B=m\). If \(D_A\ne D_B\), then \(\lim _{N\to \infty }O_{AB}(N)=0\) by Theorem 22.2.3.40. Assume \(D_A=D_B\).
Choose period-blocked sector decompositions
If no pair \((u,v)\) satisfies \(d_u=e_v\) and \(A_u\sim _{\mathrm{gp}}B_v\), then Theorem 22.2.3.31 gives \(\lim _{N\to \infty }O_{AB}(N)=0\).
Otherwise there are \(u_0,v_0\in \mathbb Z_m\) such that \(d_{u_0}=e_{v_0}\) and \(A_{u_0}\sim _{\mathrm{gp}}B_{v_0}\). Theorem 22.2.3.39 then gives repeated-block equivalence between \(A\) and \(B\).
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 \(N\).
The Gram matrix converges to the diagonal matrix whose \(j\)-th diagonal entry is the positive period of \(A_j\). Its determinant therefore has a nonzero limit, so the Gram matrix is invertible for every sufficiently large \(N\).
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.
Theorem 22.2.2.1 and the equal-MPV corollary of Section 10.4 both characterize the gauge freedom between two tensors with equal MPV families, but they do so at different levels:
Blocking approach (Section 10.4, following [ CPGSV16 ] ): for a common multiple \(p\) of the periods,
\begin{align} A^{[p]} & \sim \bigoplus _a C_a, \notag \\ B^{[p]} & \sim \bigoplus _b D_b, \notag \end{align}with all \(C_a,D_b\) normal. The Fundamental Theorem gives, after a permutation of the normal summands,
\begin{align} D_{\pi (a)}^\alpha & = X_a C_a^\alpha X_a^{-1}. \notag \end{align}This is a gauge statement for \(A^{[p]}\) and \(B^{[p]}\); the cyclic phases have been absorbed into the period-\(p\) words.
Periodic approach (Theorem 22.2.2.1, following [ DlCCSPG17 ] ): keep the cyclic sectors
\begin{align} A^i & =\sum _{u\in \mathbb Z_m}P_uA^iP_{u+1}, \notag \\ B^i & =\sum _{v\in \mathbb Z_m}Q_vB^iQ_{v+1}. \notag \end{align}Sector matching gives phases \(\phi _v\), a shift \(q\), and unitaries \(U_v\) such that
\begin{align} P_uA^iP_{u+1} & = e^{i\xi }e^{i\phi _{u+q}} U_{u+q}Q_{u+q}B^iQ_{u+q+1}U_{u+q+1}^{\dagger } e^{-i\phi _{u+q+1}}. \notag \end{align}In the equal-MPV theorem the proportional phase is absorbed, so one obtains \(ZA^i=UB^iU^{-1}\) with \(Z^m=\mathbb {1}\); the trace only sees powers of the corresponding \(m\)-th roots of unity.
Chapters 8 and 10 follow only the blocking argument: remove periodicity by blocking, then apply the Fundamental Theorem. The periodic extension treats periodic blocks directly.
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.
By Theorem 22.2.3.41, either the overlap \(\langle V^{(N)}(A) | V^{(N)}(B) \rangle \) tends to \(0\) or the bond dimensions agree and \(A\) and \(B\) are repeated blocks. In the decay case, equality of the MPV families gives, for every \(N{\gt}0\),
Along multiples of the period \(m_a\), (5) would imply
contradicting \(m_a{\gt}0\).
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.
Invert the pure gauge and take the repeated-block phase to be one.
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.
Cyclicity of the trace makes every closed-chain coefficient, and hence every mixed overlap, invariant under the two similarities. Thus decay and non-decay are unchanged. For a non-decaying pair, compose its primed repeated-block relation with the two pure similarities.
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.
Fix \(j_0\in J\) and suppose that its overlap with every \(B_k\) tends to zero. Let \(T\subseteq J\) be the set of all indices with this property. For every \(j\notin T\), choose \(k(j)\) for which the mixed overlap does not decay. The periodic overlap dichotomy identifies \(A_j\) with \(B_{k(j)}\) up to a gauge transformation and a phase; consequently their length-\(N\) vectors are scalar multiples of one another.
Choose a common positive multiple \(p\) of all periods. Along the lengths \(pN\), define the coproduct-indexed family \(w_N\colon T\amalg K\to (\mathbb C^d)^{\otimes pN}\) by \(w_N(\operatorname {inl}(j))=|V^{(pN)}(A_j)\rangle \) and \(w_N(\operatorname {inr}(k))=|V^{(pN)}(B_k)\rangle \). This family is eventually linearly independent. Indeed, distinct vectors within either summand have decaying overlaps by non-repetition and the overlap dichotomy, while the mixed overlaps vanish by the definition of \(T\).
Expand the proportionality relation at length \(pN\) using the grouped coefficients
Replace each term indexed by \(j\notin T\) by its scalar multiple of the corresponding \(B_{k(j)}\) term, and collect terms with the same value of \(k(j)\). Linear independence forces \(\operatorname {tr}(R_{j_0}^{pN})=0\) for every sufficiently large \(N\). Since each \(\mu _{j_0,q}^p\) is nonzero, geometric extrapolation gives the same identity at exponent zero. This says \(r_{j_0}=0\), contradicting the positive size of \(R_{j_0}\). Hence \(j_0\) has a non-decaying partner. Interchanging \(A\) and \(B\), inverting the nonzero scalars \(c_N\), and using conjugate symmetry of the mixed overlap gives the converse assertion. For any non-decaying pair, the periodic overlap dichotomy excludes its decay alternative and supplies the repeated-block relation.
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.
Normalize all representatives by their pure Perron gauges and assemble the gauges block diagonally. The multiplicity matrices and their grouped coefficients remain unchanged. Pure similarity preserves the total positive-length proportionality and pairwise non-repetition, so Theorem 22.2.3.48 applies to the normalized families. Finally apply Lemma 22.2.3.47.
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 ] .
Repeat the coefficient argument above with the singleton coefficients \(\mu _j^N\) and \(\nu _k^N\). Their nonvanishing at every length replaces the geometric-extrapolation step. Equality with the two scalar-weight direct sums transports the positive-length proportionality relation to the corresponding block-diagonal tensors.
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.
22.3 Physical symmetries on periodic MPS
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 \(i\).
The twisted tensor \(\widetilde{A}_g\) coincides with the physical-index rotation \(A_{U(g)}\) of Definition G.1.1.1. By Theorem 22.1.19 the rotated tensor remains in irreducible form II, and the on-site symmetry hypothesis supplies the equality \(\mathcal{V}(A) = \mathcal{V}(A_{U(g)})\). Apply the periodic equal-case Fundamental Theorem (Theorem 22.2.2.1) to obtain the asserted \(\mathbb {Z}_{m_g}\)-gauge equivalence.
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
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
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.
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 \(\widetilde{A}_g\).
Unpack the rigidity hypothesis to extract the family \((Y, u)\) together with the pointwise \(Z\)-gauge intertwiners. Set \(X(g) := Y(g^{-1})\) and \(\omega (g, h) := u(h^{-1}, g^{-1})\). The projective multiplication law \(X(g)X(h) = \omega (g,h)X(gh)\) is then exactly the \((h^{-1}, g^{-1})\)-instance of the factor-system identity, using \((gh)^{-1} = h^{-1}g^{-1}\). The intertwining relation between \(A\) and \(\widetilde{A}_g\) transports unchanged.
If the bond dimension \(D\) is positive, the factor system \(\omega \) produced by Theorem 22.3.3 satisfies the multiplicative \(2\)-cocycle identity \(\omega (g,h)\omega (gh,k) = \omega (g,hk)\omega (h,k)\).
Associativity of matrix multiplication applied to \(X(g)X(h)X(k)\), combined with the projective multiplication law, gives
Since \((gh)k = g(hk)\), both sides of (6) are scalar multiples of the same matrix \(X((gh)k)\). The matrix lies in \(\mathrm{GL}_D(\mathbb {C})\) and is therefore invertible, but the scalar cancellation leading from an equality of scalar multiples of a matrix to equality of the scalar coefficients additionally requires \(D{\gt}0\) so that the underlying index set is inhabited. Cancelling then yields \(\omega (g,h)\omega (gh,k) = \omega (g,hk)\omega (h,k)\) in \(\mathbb {C}^{\times }\), hence in \(\mathrm{Units}(\mathbb {C})\).
In the one-dimensional SPT literature, a \(U(1)\)-valued factor system of a symmetric MPS gives a cohomology class in \(H^{2}(G,U(1))\) (cf. Chen–Gu–Wen and Pérez-García et al.). This interpretation is external to [ DlCCSPG17 ] : Section 4.2 proves the \(Z\)-gauge symmetry corollary and leaves its consequences to later work. The result established here is only the 2-cocycle identity of Theorem 22.3.4, under the conditional rigidity hypothesis of Definition 22.3.2.
22.4 Refinement and divisibility
The next theorem relates refinement of a matrix-product-vector family to divisibility of its transfer map, following [ DlCCSPG17 , Theorem 4.1 ] .
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 \(\tau \in \{ 0,\ldots ,d^p-1\} ^N\).
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 \(A^{[p]}\).
Choose \((A,W)\) from the refinement data. The definition of \(C\) gives \(\mathcal{E}_C = \mathcal{E}_B\) and equality of MPV families between \(C\) and \(A^{[p]}\). To see that \(C\) is still in irreducible form II, transport each periodic block \(B_j\) of \(B\) through the same physical isometry \(W\). The left-canonical normalization is preserved because \(W^\dagger W = \mathbb {1}\), while irreducibility and the peripheral spectrum are preserved because the transfer map of each transported block agrees with that of \(B_j\). Reassembling these transported blocks with the same weights gives an irreducible-form decomposition of \(C\).
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 \(A^{[p]}\).
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 \(A^{[p]}\).
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 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 \(\mathcal{E}_C = \mathcal{E}_{\widetilde A^{[p]}}\).
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.
Apply the \(Z\)-gauge extraction hypothesis to obtain a blocked \(\mathbb {Z}_m\)-gauge witness between \(C\) and \(A^{[p]}\), and then apply the construction of \(\widetilde A\) to that witness.
Assume the blocked equal-case construction. Then every \(p\)-refinable tensor \(B\) in irreducible form II admits a left-canonical tensor \(\widetilde A\) with \(\mathcal{E}_B = \mathcal{E}_{\widetilde A^{[p]}}\).
Apply Theorem 22.4.3 to the refinement witness of \(B\). This produces a blocked tensor \(C\) in irreducible form II with \(\mathcal{E}_C = \mathcal{E}_B\) and the same MPV family as \(A^{[p]}\). The blocked equal-case construction then gives a left-canonical tensor \(\widetilde A\) with \(\mathcal{E}_C = \mathcal{E}_{\widetilde A^{[p]}}\). Combining the two transfer-map identities yields \(\mathcal{E}_B = \mathcal{E}_{\widetilde A^{[p]}}\).
Assume the blocked equal-case construction. Then for every tensor \(B\) in irreducible form II, \(p\)-refinability of \(B\) implies \(p\)-divisibility of \(\mathcal{E}_B\).
Combine Theorem 22.4.9 with the witness-level forward implication: once \(\mathcal{E}_B\) is identified with \(\mathcal{E}_{\widetilde A^{[p]}}\) for a left-canonical \(\widetilde A\), the blocking identity gives \(\mathcal{E}_B = (\mathcal{E}_{\widetilde A})^p\).
The following conditional statements separate the refinability–divisibility equivalence of [ DlCCSPG17 , Theorem 4.1 ] from the additional constructions used below.
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.
Combine the forward implication (Theorem 22.4.12) with the reverse implication (Theorem 22.4.15) into a single biconditional. The forward direction assumes the construction of \(\widetilde A\); the reverse direction assumes the transfer-map construction. Together they give the full equivalence under those hypotheses.
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 \(\mathcal{E}_B\).
The construction supplies \(\widetilde A\) with \(\sum _i (\widetilde A^i)^\dagger \widetilde A^i = \mathbb {1}\) and \(\mathcal{E}_B = \mathcal{E}_{\widetilde A^{[p]}}\). The left-canonical property makes \(\mathcal{E}_{\widetilde A}\) a CPTP channel, and the blocking identity \(\mathcal{E}_{\widetilde A^{[p]}} = (\mathcal{E}_{\widetilde A})^p\) then identifies \(\mathcal{E}_B\) with the \(p\)-th power of the channel \(\mathcal{E}_{\widetilde A}\).
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.
If the blocked-to-root channel-root hypothesis holds, then the tensor-level blocked-to-root reconstruction hypothesis holds as well.
Choose a tensor \(A'\) whose matrix family realizes the root channel with at most \(d\) nonzero terms. Since this channel is trace-preserving, the matrices of \(A'\) satisfy \(\sum _i (A'^i)^\dagger A'^i = \mathbb {1}\). The identity \(\mathcal{E}_C = (\mathcal{E}_{A'})^p\) is then equivalent to \(\mathcal{E}_C = \mathcal{E}_{A'^{[p]}}\) by the blocking formula.
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 \(B\).
From the reverse transfer-map root hypothesis, extract \(A\) with \(\mathcal{E}_B = \mathcal{E}_{A^{[p]}}\). This matches two finite Kraus families of the same completely positive map: the blocked family \(A^{[p]}\) with \(d^p\) operators and the original family \(B\) with \(d\) operators. The cardinality comparison \(d \le d^p\) holds whenever \(p \ge 1\), so [ Wol12 , Theorem 2.1(4) ] supplies an isometry \(V \in M_{d^p \times d}(\mathbb {C})\) with \(V^\dagger V = \mathbb {1}\) and \((A^{[p]})^\alpha = \sum _j V_{\alpha ,j}B^j\). Expanding the coefficient of \(A^{[p]}\) at the word \(\tau \) by linearity of matrix multiplication and of the trace, the contributions reorganise into the \(V\)-weighted sum over length-\(N\) words of \(B\) required by Definition 22.4.2, exhibiting \(V\) as the isometry \(W\) in the statement of \(p\)-refinability.
The original motivation for Theorem 22.4.11 is a continuum-limit construction in the spirit of [ DlCCSPG17 ] : refinability along arbitrarily large \(p\) is equivalent to infinite divisibility of the transfer map, which selects out continuum limits of translationally invariant MPS. The further consequences are explored elsewhere.