Dependencies

Let \(P\) satisfy the auxiliary BNT sector hypotheses, with distinct basis tensors \((B_j)_j\). Then the direct sum containing one copy of each basis tensor is a renormalization fixed point if and only if every \(B_j\) has the square-root fixed-point form and \(\mathcal{E}_{j,j'}=0\) for \(j\ne j'\). This is the multiplicity-one, phase-one specialization of [ CPGSV16 , Theorem 3.11 and Corollary 3.12 ] . It concerns the direct sum of the basis representatives, not the repeated-copy tensor carrying the weights \(\mu _{j,q}\).

If the basis direct sum is a renormalization fixed point, there are also invertible matrices \(X_j\), trace-one positive diagonal matrices \(\rho _j=\operatorname{diag}(\lambda _{j,\alpha })\), and tensors \(U_j\) such that \(B_j^i=X_j\sqrt{\rho _j}\, U_j^iX_j^{-1}\), and the family \((U_j)_j\) satisfies the joint isometry condition, including the cross-block equations for \(j\ne j'\).

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

Let \(A=(A^i)_{i=0}^{d-1}\) have bond dimension \(D\). Its physical-isometry class appears as a limit of the renormalization flow if there are a physical dimension \(e\) and a tensor \(B=(B^a)_{a=0}^{e-1}\) of the same bond dimension \(D\) such that

Here \(\widehat{\mathcal{E}}_A\) and \(\widehat{\mathcal{E}}_B\) are the transfer matrices, and convergence is taken in the standard topology on the finite-dimensional matrix space \(M_{D^2}(\mathbb {C})\). The physical dimension is quantified because blocking changes it, whereas the bond dimension remains fixed. Transfer matrices represent the quotient by physical isometries described in [ CPGSV16 , lines 382–394 and 1205–1209 ] .

A tensor \(A\) with one physical component has a blocking witness if there are an invertible matrix \(X\) and a scalar \(\zeta \) with \(|\zeta |=1\) such that

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

The coefficient form of a basis of normal tensors: each block \(A_j\) is a CPSV normal tensor (Definition 8.6.1), the MPV of \(A\) is given at every positive length by explicit coefficients \(c_j^{(N)} \in \mathbb {C}\) with \(|V^{(N)}(A)\rangle = \sum _{j=1}^{g} c_j^{(N)} |V^{(N)}(A_j)\rangle \), and the block MPVs \(\{ |V^{(N)}(A_j)\rangle \} \) are eventually linearly independent. This is the literal coefficient-form definition of [ CPGSV16 , Section 2.3 ] . It differs from Definition 9.1.1 only in the chosen normality predicate. Theorem 9.2.5 proves the implication between the two forms.

The connected two-point function is

Let \(A\) have bond dimension \(D\). It is in the canonical form of [ CPGSV16 , Section 2.3 ] if there are positive integers \(D_1,\ldots ,D_r\), normal tensors \(A_k\) of bond dimension \(D_k\), nonzero weights \(\mu _k\in \mathbb {C}\), and, writing \(S=\sum _kD_k\), a matrix \(U\in \mathbb {C}^{S\times D}\) such that \(S\le D\) and, for every physical index \(i\),

Thus the complement of the retained direct sum consists literally of zero bond-space coordinates. In particular, \(r=0\) is possible only when \(A=0\).

The convention in line 246 of [ CPGSV16 ] is a separate normalization of these data, not part of canonical-form membership:

Every weight \(\mu _k\) is nonzero. This is the reading of [ CPGSV16 , Section 2.3 ] adopted here: the remark in line 219 there places the degenerate case in the block dimensions (\(D_k=0\)), and lines 224–225 choose each \(\mu _k\) so that the transfer map of \(\mu _kA_k\) has spectral radius one, which a vanishing weight cannot satisfy. The reading loses nothing, since if \(\mu _{k_0}=0\) then \(\mu _{k_0}A_{k_0}^i=0\) is the zero matrix of size \(D_{k_0}\) and

so \(A^i\) is the same coisometric image of the direct sum over the remaining indices, with the \(D_{k_0}\) coordinates of the omitted summand moved to the zero complement. No ordering or separation condition is imposed.

The basis-of-normal-tensors refinement is Definition 9.1.1; it is not part of canonical form itself.

The permutation matrix

defines an invertible virtual gauge that exchanges the \(1\) and \(\omega \) coordinates of the cube-phase tensor.

For an MPS tensor \(A\) on bond space \(\mathbb {C}^D\), define the associated MPO tensor \(M\) on bond space \(\mathbb {C}^{D^2}\) by \(M^{ij}=A^i\otimes \overline{A^j}\). This is the single-ancilla purification tensor of [ CPGSV16 ] .

Given an invertible virtual matrix \(X\), the induced gauge on the doubled bond space is \(X\otimes \overline X\).

Given a word \(w = (i_1, \ldots , i_L) \in \{ 0,\ldots ,d{-}1\} ^L\), the word evaluation is the matrix product

The empty word evaluates to the identity, \(A^\varnothing = \mathbb {1}_D\). In tensor-network notation,

in which each black node denotes the same local tensor \(A\), the virtual legs remain open, and the physical legs are labelled by the word \((i_1,\ldots ,i_L)\).

For a permutation \(\rho \) of a finite bond-index set, let \(\Pi _\rho \) be its permutation matrix. The corresponding element of the general linear group is denoted \(G_\rho \).

Two tensors \(A\) and \(B\) of the same bond dimension \(D\) are gauge equivalent if there exists an invertible matrix \(X \in \mathrm{GL}_D(\mathbb {C})\) such that, for every \(i \in \{ 0,\ldots ,d{-}1\} \),

In tensor-network notation, (5) is represented by

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

Two tensors \(A\) and \(B\) are gauge-phase equivalent if there exist \(X \in \mathrm{GL}_D(\mathbb {C})\) and \(\zeta \in \mathbb {C}\setminus \{ 0\} \) such that, for every physical index \(i\),

This is the source quantification in [ CPGSV16 , Theorem 3.10(iii) ] . For every \(N{\gt}2\), the fixed-chain condition of Definition 13.1.17 holds:

This condition records the ground-space spanning equation, but does not prove it for a concrete canonical-form tensor.

A tensor \(A\) is injective if its matrices span the full matrix algebra: \(\operatorname{span}_{\mathbb {C}}\{ A^i : i = 0,\ldots ,d{-}1\} = M_{D}(\mathbb {C})\).

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

Let \((A_j)_j\) be a basis of normal tensors. The family is locally orthogonal when, for every pair of distinct components,

This is the mixed-sector condition of [ CPGSV16 , Definition 3.5 ] .

Let \((A_j)_j\) be a basis of normal tensors for \(A\). This auxiliary condition requires virtual-insertion distance independence and \(\mathcal{E}_{j,j'}=0\) for every pair of distinct BNT components. It is not [ CPGSV16 , Definition 3.6 ] , which quantifies over physical observables on all disjoint regions.

A collection of scaling factors \((\mu _k)_{k=1}^r\) and block tensors \((A_k)_{k=1}^r\) satisfies the auxiliary block-family hypotheses if:

1. each \(A_k\) is injective,

2. each \(A_k\) satisfies the TP normalization \(\sum _i(A_k^i)^\dagger A_k^i=\mathbb {1}\),

3. the moduli are non-increasing: \(|\mu _1|\ge |\mu _2|\ge \cdots \ge |\mu _r|\),

4. all \(\mu _k\neq 0\),

5. every block bond dimension is positive: \(D_k\ge 1\) for every \(k\),

6. the self-overlap converges: \(O_{A_kA_k}(N)\to 1\) for each \(k\).

Only this one-sided normalization is assumed; no unital condition is assumed here. Condition (6) is a primitivity hypothesis: for a primitive channel ( [ Wol12 , Theorem 6.7(3) ] ), \(T^n(\rho )\to \rho _\infty \) for every initial state, which forces the self-overlap to converge to \(1\). See also  [ Wol12 , Theorem 6.8 ] for the completely-positive characterisation and Chapter 6 for the transfer-operator gap proof.

For every positive-definite right fixed point \(\rho ^R{\gt}0\), every pair of virtual bond matrices \(X,Y\), and all separations \(n,m\geq 1\), the virtual-insertion expression is independent of distance: \(C(X,Y;n)=C(X,Y;m)\).

Let \(\lambda _\alpha {\gt}0\) satisfy \(\sum _\alpha \lambda _\alpha =1\), and set \(\rho =\operatorname{diag}(\lambda _\alpha )\). A normal tensor \(A\) has the square-root fixed-point form when there are an invertible matrix \(X\) and a tensor \(U\) satisfying the unit pair-index isometry

such that

Local fix (square-root diagonal). The display at line 1278 uses the bare diagonal \(\Lambda \), while lines 1281–1283 impose a unit pair-index isometry and line 1300 constructs the reference tensor with coefficients \(\sqrt{\Lambda _\alpha }\). These equations force the repaired form \(A^i=X\sqrt{\rho }\, U^iX^{-1}\) used in this definition. The correction is documented in [ con26p ] .

Explicitly, the reference tensor is

Thus \(\rho =\operatorname{diag}(\lambda _\alpha )\) is the trace-one diagonal fixed point. This records the \(j=j'\) part of the joint isometry condition [ CPGSV16 , Appendix B, lines 1278, 1281–1283, 1300 ] .

This auxiliary one-block convention means \(\mathcal{E}_A^2=\mathcal{E}_A\). It is not local orthogonality in the sense of the source, which is the family of mixed-sector equations \(\mathcal{E}_{j,j'}=0\) for distinct BNT components.

A family \((A_j)_j\) of MPS tensors has positive-gap BNT zero correlation length for \(A\) when \((A_j)_j\) is a basis of normal tensors for \(A\) in the sense of Definition 9.1.2, its components are locally orthogonal, and the physical correlations of \(A\) are independent of separation whenever both complementary gaps are positive. Adjacent regions are not included.

For an observable \(O\) on a block of \(L\) physical spins, write

Place two observables \(O_1,O_2\) on nonempty finite blocks of a periodic chain, with positive complementary gap lengths \(n_1,n_2\). The tensor has positive-gap physical correlations independent of distance when the expectation is unchanged upon replacing these gaps by positive \(m_1,m_2\) satisfying \(n_1+n_2=m_1+m_2\):

This positive-gap restriction is the transfer-matrix form of [ CPGSV16 , Definition 3.3 and the correlation formula preceding Theorem 3.8 ] , restricted to positive complementary gaps. It excludes adjacent regions.

A family \((U_j)_j\) satisfies the joint isometry condition when

Each block satisfies the within-block pair-index orthonormality, and the cross-block sums between distinct blocks vanish. This is the isometry condition of [ CPGSV16 , Theorem 3.11 ] , split into its \(j=j'\) and \(j\ne j'\) cases.

This auxiliary one-block predicate is the conjunction of \(\mathcal{E}_A^2=\mathcal{E}_A\) and virtual-insertion distance independence. It is not the physical zero-correlation-length condition of Definition 18.2.6.

For MPS tensors \(A\) of bond dimension \(D_1\) and \(B\) of bond dimension \(D_2\), possibly with \(D_1\neq D_2\), the rectangular mixed transfer operator is the map \(F_{AB}^{\mathrm{rect}}: M_{D_1\times D_2}(\mathbb {C})\to M_{D_1\times D_2}(\mathbb {C})\) defined by

An MPO tensor is an MPDO if, for every \(N\geq 1\),

This is the global positivity condition of [ CPGSV16 , Section 4 ] .

For an MPO tensor \(K\), its two-site blocking \(K^{[2]}\) has physical indices \((i_0,i_1)\) and \((j_0,j_1)\) and matrices

This is the blocking used in [ CPGSV16 , Theorem 4.9, lines 851–856 ] .

The transfer map of \(A\) satisfies the idempotence criterion when \(\mathcal{E}_A\circ \mathcal{E}_A=\mathcal{E}_A\).

A tensor \(A\) is normal [ CPGSV21 ] if there exists a positive integer \(L_0\) such that \(A\) is \(L_0\)-block injective.

A tensor \(A\) of physical dimension \(d\) and bond dimension \(D\) is a normal tensor if, after the spectral-radius normalization of  [ CPGSV16 ] , (i) \(A\) admits no nontrivial invariant orthogonal projection and (ii) the associated CPM \(\mathcal{E}_A(X)=\sum _i A^iX(A^i)^\dagger \) has spectral radius \(r(\mathcal{E}_A)=1\) and peripheral spectrum \(\sigma _\partial (\mathcal{E}_A)=\{ 1\} \).

For an MPO tensor \(M\) with one physical component, the Appendix D blocking diagram holds up to virtual gauge if \(M^{[2]}\) and \(M\) are related by simultaneous similarity on the virtual indices. For a pure tensor \(A\), this condition is imposed on the doubled tensor

Both physical matrix algebras are one-dimensional, so every physical unitary-conjugation channel in the diagram is the identity. Similarity is symmetric and hence gives both directions of the diagram in [ CPGSV16 , Appendix D, equation RFP-gauge ] .

An MPS tensor \(A\) is a renormalization fixed point if there is an isometry \(V\colon \mathbb {C}^d\to \mathbb {C}^{d^2}\), with coefficients \(V_{(i_1,i_2),j}\) and \(V^\dagger V=\mathbb {1}\), such that, for all physical indices \(i_1,i_2\),

This is the relation \(AA=A\) in [ CPGSV16 , Definition 3.2 ] .

For an observable \(O\) on a block of \(L\) physical spins, define the linear map on the virtual matrix space by

This is the observable transfer map of [ CPGSV16 , lines 490–496 ] .

Let \(O_1,O_2\) act on blocks of \(L_1,L_2\) physical spins, separated around the periodic chain by complementary gaps of lengths \(n_1,n_2\). Their two-observable expectation is

The trace is the operator trace on the virtual matrix space, as in [ CPGSV16 , lines 490–496 ] .

For tensors \(A\) and \(B\) of possibly different bond dimensions \(D_1\) and \(D_2\), we write \(\mathcal{V}(A) = \mathcal{V}(B)\) if \(|V^{(N)}(A)\rangle = |V^{(N)}(B)\rangle \) for every word length \(N\), including the empty word. This algebraic relation is used for exact direct-sum identities.

Given a sector decomposition \(P\), these conditions define the normalized, left-canonical, aperiodic BNT surface used below. This is stronger than the abstract BNT of Definition 9.1.1 and the literal coefficient-form BNT of Definition 9.1.2: besides spanning and eventual linear independence, it fixes irreducible left-canonical representatives, their self-overlap normalization, gauge-phase separation, and the canonical-form weight normalization. It imposes no equal-modulus factorization or strict ordering on the raw weights. Explicitly, each basis block has positive bond dimension, is irreducible, and is left-canonical; the normalized self-overlap of every basis block converges to \(1\); the basis MPV family is eventually linearly independent; distinct same-dimension basis blocks are not gauge-phase equivalent after identifying equal bond dimensions; and the [ CPGSV16 , Section 2.3 ] canonical-form normalization holds,

The MPV expansion takes the raw two-layer form

matching [ CPGSV16 , Section 2.3 ] and [ CPGSV21 , Definition IV.2 ] . The canonical-form normalization admits every example of the source paper (in particular \(C \oplus D\), \(C \oplus (-C)\), \(C \oplus (1/2) C\), and \(C \oplus (-C) \oplus (1/2) C\)) and is not derivable from the per-block normality hypotheses alone.

A BNT family has zero correlation length when the associated matrix-product vectors have correlations independent of distance for all disjoint physical regions and the BNT components are locally orthogonal. This is [ CPGSV16 , Definition 3.6 ] .

Let \(O_1,O_2\) act on two disjoint contiguous regions of positive lengths \(L_1,L_2\geq 1\) in a periodic chain, with complementary gap lengths \(n_1,n_2\geq 0\). Physical correlations are independent of distance when, for every \(m_1,m_2\geq 0\) satisfying \(n_1+n_2=m_1+m_2\), one has

Thus either region may be translated without crossing the other. Zero gaps, corresponding to adjacent regions, are included. This is [ CPGSV16 , Definition 3.3 and lines 490–496 ] .

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.

Two tensors are unit gauge-phase equivalent when the gauge-phase equation holds with \(|\zeta |=1\).

Let \((B_k)_k\) be a family of tensors and let \(\bigoplus _k B_k\) be their direct sum on the total bond space. For every bond matrix \(X\) and every pair of blocks, the \((j,j')\) bond block of the transfer sum

equals \(\mathcal{E}_{j,j'}\) applied to the \((j,j')\) bond block of \(X\).

Let \((B_k)_k\) be a family of tensors. If every mixed transfer operator \(\mathcal{E}_{j,j'}\) is idempotent, then the block transfer sum \(\mathcal S_B\) is idempotent: \(\mathcal S_B(\mathcal S_B(X))=\mathcal S_B(X)\).

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

The matrix of \(G_\rho \) is \(\Pi _\rho \), and the matrix of its inverse is \(\Pi _{\rho ^{-1}}\).

For a permutation \(\rho \) and a square matrix \(M\), conjugation by the permutation matrix reindexes both coordinates by \(\rho \):

Equivalently, the right-hand side is the submatrix of \(M\) along \(\rho \times \rho \).

Let \(a\) be an element of a complex Banach algebra with \(a^2=a\) and \(\rho _{\operatorname{spec}}(a){\lt}1\). Then \(a=0\).

The positive-gap restriction is equivalent to the conjunction of the BNT relation, positive-gap physical distance independence, and BNT local orthogonality.

If a BNT family has zero correlation length for all disjoint physical regions, then it has positive-gap BNT zero correlation length.

Correlations independent of distance for all disjoint physical regions are independent of distance when both complementary gaps are positive.

Gauge-equivalent tensors have the square-root fixed-point form simultaneously.

If the direct sum \(\bigoplus _k B_k\) is a renormalization fixed point, with \(\dim _k\ge 1\) for all \(k\), then each block \(B_j\) is a renormalization fixed point.

For decompositions \(A^i=X_AD_AU^iX_A^{-1}\) and \(B^i=X_BD_BV^iX_B^{-1}\), the mixed transfer operator of \(A,B\) is the mixed transfer operator of the tensors \(U,V\) conjugated by the outer factors:

With decompositions \(A^i=X_AD_AU^iX_A^{-1}\) and \(B^i=X_BD_BV^iX_B^{-1}\) whose factors \(X_A,D_A,X_B,D_B\) are invertible, if \(\mathcal{E}_{A,B}=0\) then \(\mathcal{E}_{U,V}=0\).

For complex scalars \(c,e\) and tensors \(A,B\) of possibly different bond dimensions, \(\mathcal{E}_{cA,eB}=c\overline e\, \mathcal{E}_{A,B}\).

Let \(P\) be an auxiliary BNT sector decomposition in BNT canonical form, let \(A=\bigoplus _j A_j\), and fix a sector \(j\). For every \(u,v\in M_{D_j}(\mathbb {C})\), there are a positive integer \(L\) and observables \(O_1,O_2\) on \(L\) sites such that, for all \(n_1,n_2\in \mathbb {N}\),

If \(\mathcal{E}_{U,V}=0\), then, for all virtual indices,

A virtual similarity preserves the physical blocking relation and hence transfer idempotence. It also preserves every local MPS space, so the canonical parent interaction and all its translated local terms are unchanged.

Let \(U\) be a family of bond matrices satisfying the unit pair-index isometry condition

Then for every bond matrix \(Z\),

Let \(A=\bigoplus _k A_k\). Denote compression to the \((j,j)\) bond block by \(C_j\) and its trace-pairing adjoint by \(I_j\). Then

If \(\mathcal{E}_A\) and \(\mathcal{E}_j\) are the transfer maps of \(A\) and \(A_j\), respectively, then, for every \(n\geq 0\),

The sector-supported rank-one map \(\mathcal R_{j;R,l}(X)=I_j(R)\operatorname{tr}(lC_j(X))\) has operator trace

Finally, if \(\mathcal{E}_j^*(l)=\lambda l\), then

Let \((A_k)_{k}\) be a finite family of block tensors of physical dimension \(d\), and fix a length \(L\); write \(A_k^w\) for the length-\(L\) word evaluation of \(A_k\) along a word \(w \in \{ 0,\ldots ,d{-}1\} ^L\). The finite-length spanning hypothesis is that the tuple-valued evaluations span the full product algebra of block matrices,

The corresponding finite-length separation property is that some length \(L\) makes the trace pairing against block words nondegenerate: there is a length \(L\) such that, for every tuple \((\Delta _k)_k\) of block matrices,

Spanning at length \(L\) gives separation at the same length: a trace functional that vanishes on every tuple \((A_k^w)_k\) vanishes on their span, hence on the whole product algebra, so nondegeneracy of the product trace pairing forces every \(\Delta _k=0\). Combined with injectivity, left-canonicality, and nonzero block weights, this finite-length separation supplies the older per-copy trace-separation hypotheses for the block-diagonal tensor.

The spanning hypothesis has an equivalent entrywise form. For a block index \(k\) and matrix entry \((a,b)\), let \(w\longmapsto (A_k^w)_{a,b}\) be the corresponding scalar word family, indexed jointly by the triple \((k,a,b)\). Linear independence of this scalar family already implies the spanning condition, hence the separation property, and hence the older per-copy trace-separation hypotheses directly.

The block-injectivity proposition of [ CPGSV16 , lines 340–345 ] gives a concrete sufficient criterion for the spanning condition: a common length \(L\) with every block \(L\)-block-injective, together with a length \(S\) at which, for every \(k\), there is a coefficient family \((c_w)_{w\in \{ 0,\ldots ,d{-}1\} ^S}\) with

Concatenating the injective prefix with this finite selector family produces the spanning condition at length \(L+S\), hence the separation property and the older per-copy trace-separation hypotheses.

Let \(A\) be the tensor with physical index pairs \((\alpha ,\beta )\in \{ 0,1\} ^{2}\) and letters

the canonical normal-tensor renormalization fixed point with maximally mixed spectrum \(\Lambda =(1/2,1/2)\): each node carries two spins, and the pair \((|00\rangle +|11\rangle )/\sqrt2\) is shared between the second spin of one node and the first spin of the next. The transfer map is

which is idempotent since \(\operatorname{tr}\operatorname {diag}(\Lambda )=1\), and \(A\) is a single normal tensor, so no copy weights are involved. With the Pauli matrix \(Z\) on the second spin of one node and on the first spin of the next node, the two regions are adjacent, and the two-region expectation is

while after a shift by one site, which leaves one free site on each complementary arc, it is

Thus \(A\) is a renormalization fixed point whose correlations are not independent of distance in the unrestricted sense of Definition 18.5.15. The block \(\mathcal{E}_A^{0}=1\) between adjacent observables is not governed by idempotence.

Let \((B_k)_k\) be a family of tensors and let \(c\in \mathbb {C}\). If

then the corresponding block transfer sum \(\mathcal S_B\) satisfies, for every bond matrix \(Y\),

Every one-letter MPS blocking witness induces the exact doubled-MPO virtual-gauge condition.

Let \(P\) be an auxiliary BNT sector decomposition with distinct basis tensors \((A_j)_j\). The multiplicity-one unit-weight tensor \(A=\bigoplus _j A_j\) is in literal CPSV canonical form, with retained weights all equal to \(1\), retained blocks \((A_j)_j\), and the identity ambient coisometry. This statement does not reconstruct the raw weighted repeated-copy tensor represented by \(P\).

Let \(P\) be an auxiliary BNT sector decomposition with distinct basis tensors \((A_j)_j\), and let \(A=\bigoplus _j A_j\) be their direct sum with one unit-weight copy of each sector. Then \((A_j)_j\) is a basis of normal tensors for \(A\): every \(A_j\) is a normal tensor, at every positive system length \(N\)

and the states \(\{ |V^{(N)}(A_j)\rangle \} _j\) are linearly independent for all sufficiently large \(N\).

Let \((A_j)_j\) be a family of distinct irreducible left-canonical blocks, no two of which are gauge-phase equivalent, and suppose the direct sum \(\bigoplus _j A_j\) is a renormalization fixed point. Then for every pair of distinct components the mixed transfer operator vanishes, \(\mathcal{E}_{j,j'}=\sum _iA_j^i\otimes \overline{A_{j'}^i}=0\). This is the cross-block (\(\delta _{j,j'}\)) content of the isometry condition of [ CPGSV16 , Theorem 3.11 and Corollary 3.12 ] at the level of the normal-tensor blocks. The passage to the residual tensors \(U_j\) is given by Theorem 18.4.16.

Let \(P\) be an auxiliary BNT sector decomposition of total bond dimension \(D\), and form the direct sum \(A=\bigoplus _j A_j\) with one unit-weight copy of each BNT basis tensor. There is a positive length \(L\leq 3D^5\) such that, for every basis sector \(j\) and every \(R,l\in M_{D_j}(\mathbb {C})\), a physical observable on \(L\) sites realizes the sector-supported insertion (5). This is the multiplicity-one specialization of the block-injective physical-observable assertion used in [ CPGSV16 , proof of Theorem 3.8 ] , equations (1252) and (1256) in the local source. It does not realize the weighted copy-pair insertions for the full weighted tensor represented by \(P\).

Let \(A=\bigoplus _k A_k\), and suppose the simultaneous length-\(L\) word evaluations span \(\bigoplus _kM_{D_k}(\mathbb {C})\). For every sector \(j\) and matrices \(R,l\in M_{D_j}(\mathbb {C})\), there is an observable \(O\) on \(L\) sites whose inserted transfer map vanishes on every sector pair except \((j,j)\) and satisfies

In particular, arbitrary virtual matrix-unit maps supported on one sector are realized by physical observables.

Suppose the simultaneous length-\(L\) word evaluations of \((A_j)_j\) span \(\bigoplus _jM_{D_j}(\mathbb {C})\). There are coefficients \(C_{(j,a,b),w}\) such that, for every sector \(k\),

Let \((\mu _k,A_k)_{k=1}^r\) be nonzero weights and injective left-canonical blocks with positive bond dimensions, non-increasing moduli \(|\mu _1|\ge \cdots \ge |\mu _r|{\gt}0\), and \(\sigma _\partial (\mathcal{E}_{A_k})=\{ 1\} \). Then \((\mu _k,A_k)_{k=1}^r\) satisfies the auxiliary hypotheses of Definition 18.3.1.1, and \(O_{A_kA_k}(N)\to 1\) for every \(k\).

Let

be literal CPSV canonical-form data. There are phase classes \(j\), copy multiplicities \(r_j\), and an equivalence

Each class has a representative \(C_j\). If the copy \((j,q)\) corresponds to the block index \(k\), then its bond dimension equals that of \(C_j\) and there are \(\zeta _k\in \mathbb {C}\), with \(|\zeta _k|=1\), and \(X_k\in \mathrm{GL}_{D_k}(\mathbb {C})\) such that

The copy retains the original coefficient \(\mu _k\), and the representatives \((C_j)_j\) form a basis of normal tensors for \(A\).

Enumerate \(\mathcal G\). The induced equivalence \(\Phi \) between the dependent grouped block coordinates and the original flattened listed coordinates gives

For \(x=(j,q)\), let \(\widehat\mu _x\) be the original weight of that copy and let \(\widehat B_x^i=\zeta _xC_j^i\), with the required dimension identification. Define the grouped tensor by

If \(G=\bigoplus _kX_k\) is the block-diagonal gauge, then, letter by letter,

and

Let \(\omega =e^{2\pi i/3}\) and consider the tensor with one physical component

This tensor is in canonical form, with two one-dimensional normal components of weights \(1\) and \(\omega \). For the matrix unit \(e_{21}\),

The right-hand side alternates between \(\omega e_{21}\) and \(\omega ^2 e_{21}\). Hence the dyadic renormalization flow does not converge. This refutes the unrestricted canonical-form convergence assertion in [ CPGSV16 , lines 417 and 1211–1244 ] ; it does not affect the primitive single-component convergence above.

CPSV asserts that a tensor \(A\) in canonical form is a renormalization fixed point if and only if it can be written as

where \(|\mu _{j,q}|=1\), each \(\rho _j\) is positive diagonal with \(\operatorname{tr}(\rho _j)=1\), and

The source specifies that \(\bigoplus _{j,q}\) is a simultaneous direct sum in the physical and virtual spaces, not an ordinary virtual block diagonal at each fixed physical letter \(i\) [ CPGSV16 , Theorem 3.11, lines 561–562 ] . It does not provide a coordinate map for the physical routing of \(q\), a corresponding dimension hypothesis, or a \(q\)-indexed isometry equation. Consequently the display does not yet determine a source-faithful coordinate-level predicate. The routed characterization therefore remains not ready pending a source clarification or erratum.

Local fix (square-root diagonal). CPSV prints a bare \(\Lambda _j\) in Theorem 3.11, whereas the reference tensor at line 1300 has coefficients \(\sqrt{(\rho _j)_{\alpha ,\alpha }}\), which forces the factor \(\sqrt{\rho _j}\) used above. This repair is documented in [ con26p ] .

Every normal tensor has positive bond dimension.

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

Let \(A\) be a normal tensor in the sense of [ CPGSV16 , Section 2.2 ] . Its bond dimension is necessarily positive, and there is a positive definite matrix \(\sigma \) such that the gauge obtained from \(\sigma \) is trace-preserving, primitive, and irreducible. No scalar rescaling is needed.

Let \(A\) be a CPSV canonical-form renormalization fixed point. Choose one representative \(C_j\) from each matrix-product-vector phase class. Then there are invertible matrices \(X_j\), positive numbers \(\lambda _{j,\alpha }\) with \(\sum _\alpha \lambda _{j,\alpha }=1\), and tensors \(U_j\) such that

The conclusion also covers the case in which there are no classes.

Local fix (square-root diagonal). CPSV prints a bare \(\Lambda _j\) in Corollary 3.12, whereas the reference tensor at line 1300 has coefficients \(\sqrt{(\rho _j)_{\alpha ,\alpha }}\), which forces the factor \(\sqrt{\lambda _{j,\alpha }}\) used above. This repair is documented in [ con26p ] .

Let \(\omega =e^{2\pi i/3}\), and set

Then \(\omega ^3=1\), \(\omega \ne 1\), and

The doubled MPO tensor generates positive semidefinite operators on every nonempty chain, is in CPSV canonical form, and has four scalar-block weights satisfying the source normalization convention. It also satisfies the literal one-letter virtual-gauge condition in the source diagram, but its transfer map is not idempotent. Hence the canonical-restricted pure-state equivalence claimed after [ CPGSV16 , Appendix D, equation RFP-gauge ] is false.

CPSV asserts that a tensor \(A\) in canonical form has physical BNT zero correlation length if and only if

and, more generally, that for such \(A\) the following are equivalent: \(A\) is a renormalization fixed point; \(A\) has physical zero correlation length; and, for every \(N{\gt}2\), the vector \(|V^{(N)}(A)\rangle \) belongs to the ground space of a nearest-neighbor commuting parent Hamiltonian [ CPGSV16 , Theorems 3.8 and 3.10 ] . Both directions of the biconditional fail as printed, and with them the three-way equivalence.

For the reverse implication, let \(C^0=(1)\) and form the direct sum of two scalar copies with weights \(1\) and \(1/2\). The resulting tensor is

The scalar tensor \(C\) is normal, and the weights obey the source convention \(|\mu _k|\leq 1\), with at least one weight of modulus one. The singleton BNT family \((C)\) is locally orthogonal and gives \(A\) physical BNT zero correlation length, whereas

Thus zero correlation length does not imply transfer idempotence under the source’s raw weight normalization.

Independently, the Bell-pair chain of Theorem 18.2.9 is a single normal block. Taking its weight to be \(1\) and the ambient coisometry to be the identity gives a literal CPSV canonical-form tensor. Its transfer map is idempotent, so it is a renormalization fixed point, but its adjacent two-region expectation is \(1\), while the expectation after an allowed one-site shift is \(0\). Hence transfer idempotence does not imply physical zero correlation length for the unrestricted quantification over disjoint regions, regardless of the commuting-parent condition.

Source gap (adjacent regions). The printed forward argument uses \(\mathcal{E}_A^n=\mathcal{E}_A\), which follows from idempotence only for \(n\geq 1\). Adjacent regions insert \(\mathcal{E}_A^0=\mathbb {1}\) and are included in the source definition. This gap and its scope are recorded in [ con26e ] . Theorem 18.5.18 proves the repaired forward implication when both complementary gaps are positive. It does not prove the unrestricted biconditional of [ CPGSV16 , Theorem 3.8 ] or the unrestricted three-way equivalence of [ CPGSV16 , Theorem 3.10 ] . The converse and equivalence results below require their stated unit-weight, multiplicity-one, or spectral hypotheses. The printed statements are recorded here as refuted, not as theorems awaiting proof.

Suppose \(A\) has CPSV canonical-form data

where every \(A_k\) is normal. If \(A\) is a renormalization fixed point, then every block \(k\) satisfies

Suppose every bond dimension is positive. The transfer map of \(\bigoplus _k B_k\) is idempotent if and only if, for every pair \(j,j'\), the mixed transfer operator is idempotent:

Let \((B_k)_k\) be a family of tensors, let \(\Phi \) be the canonical identification of the block-labelled bond space with the total bond space, and let \(\mathcal S_B\) be the block transfer sum. Then, for every block matrix \(Y\),

Let \(R\) be a Hausdorff topological monoid with continuous multiplication. If \(E^{2^n}\to P\), then \(P^2=P\).

Fix an injective, trace-preserving MPS tensor \(A\) with positive definite fixed point \(\rho \). Let

be the corresponding fixed-point projection. Then there exist \(C {\gt} 0\) and \(0 {\lt} \delta \le 1\) such that, for all \(n\ge 0\) and \(X\in M_{D}(\mathbb {C})\),

The transfer-map gap \(\delta \) exists by primitivity (the corresponding theorem in the companion quantum-channel volume  [ LTC26 , “Complementary transfer-map gap for primitive maps” ] , the complementary transfer-map gap).

Let \(A\) be an irreducible MPS tensor with \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\) and \(D{\gt}0\). Then there exist a unitary \(U\) and a diagonal positive definite \(\Lambda \) such that, for \(B^i:=U^\dagger A^iU\),

This proves the diagonal Perron normalization required in CFII. It does not prove that \(A\) is in the CPSV canonical form or that its block is normal; those require the corresponding primitivity and canonical-form hypotheses.

If \(A\) and \(B\) are gauge equivalent and \(A\) is normal, then \(B\) is normal.

Let \(P\) be the auxiliary BNT sector decomposition with one scalar basis tensor \(C^0=(1)\), two one-dimensional copies, and weights \(1\) and \(1/2\). Let \(A=\mathcal A(P)\), so

Then \(P\) satisfies the auxiliary BNT sector hypotheses, \(A=\mathcal A(P)\), and \(A\) and \(\mathcal A(P)\) generate the same matrix-product vectors. Moreover, the singleton family \((C)\) is a physical BNT zero-correlation-length family for \(A\), while

Its length-\(N\) matrix-product-vector coefficient is \(V^{(N)}(A)=1+2^{-N}\).

Let \(a\) be a nonzero element of a complex normed algebra. If \(a^2=a\), then

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

Let \((A_j)_j\) be nonzero-dimensional, irreducible, left-canonical, pairwise gauge-phase-distinct BNT components. If the direct-sum tensor \(A=\bigoplus _j A_j\) satisfies \(\mathcal{E}_A^2=\mathcal{E}_A\), then \(A\) has positive-gap physical CID and \(\mathcal{E}_{j,j'}=0\) for all \(j\ne j'\). Thus this explicit direct-sum representative has positive-gap BNT zero correlation length. This does not include the adjacent-region cases in [ CPGSV16 , Theorem 3.8 ] .

If \(\mathcal{E}_A^2=\mathcal{E}_A\), then physical correlations are independent of distance whenever both complementary gaps are positive.

A tensor in the square-root fixed-point form is a renormalization fixed point. This is the backward direction of the structural characterization of pure-state renormalization fixed points in [ CPGSV16 , Section 3 ] , which the source states as immediate.

Let \((B_k)_k\) be a family of tensors, each in the square-root fixed-point form, whose cross-block mixed transfer operators vanish, \(\mathcal{E}_{j,j'}=0\) for \(j\ne j'\). Then the direct sum \(\bigoplus _k B_k\) is a renormalization fixed point. This is the backward direction of the structural characterization of pure-state renormalization fixed points [ CPGSV16 , Section 3 ] , in the distinct-blocks case where each normal tensor appears once; the cross-block vanishing is the \(j\ne j'\) part of the isometry condition of that characterization, so it is part of the source joint isometry condition.

Suppose \(\rho ^R{\gt}0\) is a right fixed point of the transfer map. If the virtual-insertion condition holds for every positive-definite right fixed point, then the transfer map is idempotent.

A normal left-canonical renormalization fixed-point tensor has the square-root fixed-point form. With \(\lambda _\alpha =\rho _{\alpha ,\alpha }/\operatorname{tr}\rho \) and \(\rho _0=\operatorname{diag}(\lambda _\alpha )\), it admits \(A^i=X\sqrt{\rho _0}\, U^iX^{-1}\), where \(\rho _0\) is positive and trace-normalized and \(U\) is a unit pair-index isometry. Trace normalization is the identity \(\sum _\alpha \rho _{\alpha ,\alpha }=\operatorname{tr}\rho \). This is the normal-tensor statement corresponding to lines 1278 and 1281–1283, with the square-root correction dictated by the reference tensor at line 1300 [ CPGSV16 , Appendix B ] .

Each block of a multi-block tensor whose blocks are normal, left-canonical renormalization fixed points has the square-root fixed-point form, with a trace-normalized diagonal weight and a unit pair-index isometry. This is the single-block trace-normalized form applied to each block; it omits the cross-block orthogonality between distinct normal-tensor blocks.

The transfer map of \(A\) is idempotent if and only if there exists an isometry \(V\colon \mathbb {C}^d\to \mathbb {C}^{d^2}\), written as coefficients \(V_{(i_1,i_2),j}\) with \(V^\dagger V=\mathbb {1}\), such that

for all \(i_1,i_2\in \{ 0,\ldots ,d-1\} \).

Let \(A\) have positive bond dimension and a nonzero idempotent transfer map. Let \(\mu _q\) have unit modulus. If the literal bond direct sum \(\bigoplus _q \mu _qA\) has an idempotent transfer map, then \(\mu _q=\mu _{q'}\) for every pair \(q,q'\).

Let \((B_k)_k\) be a family of tensors and let \(c\in \mathbb {C}\). If

then, for every ordered pair of blocks \((j,j')\) whose two bond dimensions are positive,

The MPV of a block-diagonal tensor decomposes as

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

A normal tensor is a renormalization fixed point if and only if it has the square-root fixed-point form of Definition 18.3.1.12. Its bond dimension is necessarily positive, rather than subject to an additional hypothesis, and no left-canonical hypothesis is imposed on the original tensor. This is the corrected form of Lemma B.1 in [ CPGSV16 , Appendix B, lines 1274–1301 ] .

Let \(B\) be a normal tensor and let \(c\in \mathbb {C}\setminus \{ 0\} \). If \(cB\) is a renormalization fixed point, then

Let \(A\) be a left-canonical MPS tensor with \(D{\gt}0\), tensor-irreducible, and with peripheral spectrum \(\{ 1\} \). Tensor irreducibility implies irreducibility of the completely positive transfer map, so \(A\) is normal.

Suppose there exists an isometry \(V\colon \mathbb {C}^d\to \mathbb {C}^{d^2}\), written as coefficients \(V_{(i_1,i_2),j}\), such that

for all \(i_1,i_2\in \{ 0,\ldots ,d-1\} \). Then \(\mathcal{E}_A^2=\mathcal{E}_A\).

The transfer map of an MPS tensor is idempotent if and only if the tensor has a physical blocking isometry.

Under the spectral-pair hypotheses of Theorem 18.2.20, physical BNT zero correlation length for \(A=\bigoplus _j A_j\) implies \(\mathcal{E}_A^2=\mathcal{E}_A\).

Let \(P\) be an auxiliary BNT sector decomposition in BNT canonical form and \(A=\bigoplus _j A_j\). If \(A\) has physical BNT zero correlation length, then \(\mathcal{E}_A^2=\mathcal{E}_A\).

Let \(P\) be an auxiliary BNT sector decomposition and \(A=\bigoplus _j A_j\). Assume that every non-idempotent block \(\mathcal{E}_j^2\ne \mathcal{E}_j\) has matrices \(r_j,l_j\) and a scalar \(\lambda _j\) satisfying the five spectral conditions in Theorem 18.2.19. Then

The spectral-pair hypothesis is assumed explicitly.

Let \(P\) and \(A=\bigoplus _j A_j\) satisfy the multiplicity-one unit-weight hypotheses of Theorem 18.2.20, including its normalized nonzero subleading spectral pair for every non-idempotent block. If \(A\) has positive-gap BNT zero correlation length, then for every \(N{\gt}2\) its translated two-site parent interactions satisfy all three ground-space conditions:

This is only the conditional corrected implication from (ii) to (iii) of [ CPGSV16 , Theorem 3.10 ] , not a three-way equivalence.

Let \(A_1,\ldots ,A_g\) be the distinct basis tensors of a BNT canonical form, and set \(A=\bigoplus _{j=1}^g A_j\). Then

This statement concerns one unit-weight copy of each distinct sector and positive complementary gaps. It does not include repeated copies, arbitrary raw weights, or adjacent complementary regions.

Let \(P\) be an auxiliary BNT sector decomposition in BNT canonical form and \(A=\bigoplus _j A_j\). Then

Let \(P\) be an auxiliary BNT sector decomposition in BNT canonical form and \(A=\bigoplus _j A_j\). If \(A\) has positive-gap BNT zero correlation length, then \(\mathcal{E}_A^2=\mathcal{E}_A\).

Let \(P\) be an auxiliary BNT sector decomposition in BNT canonical form and \(A=\bigoplus _j A_j\). If \(A\) has positive-gap BNT zero correlation length, then for every \(N{\gt}2\) its translated two-site parent interactions satisfy

Let \(P\) be a BNT canonical form with basis representatives \((A_j)_{j=0}^{g-1}\), and suppose that the representatives have already been blocked so that every \(A_j\) is injective. Then representative word-tuple separation holds at every length \(L\geq 6(g-1)+1\). Consequently, if the first-site actions of \(Y,Z\in M_{d}(\mathbb {C})\) agree on the positive-length MPV family of the tensor represented by \(P\), then \(YA_j=ZA_j\) for \(0\leq j{\lt}g\). The same conclusion holds when the first-site equality is given for any tensor with the same positive-length MPV family as the tensor represented by \(P\).

More generally, let \(P\) be a BNT canonical form before physical blocking, let \(p{\gt}0\), and suppose that every \(p\)-blocked representative \(A_j^{[p]}\) is injective. Then the blocked sector decomposition \(P^{[p]}\) has representative word-tuple separation at every blocked length \(L\geq 6(g-1)+1\).

Without changing the physical alphabet, a BNT canonical form also has eventual representative word-tuple separation. If \(p{\gt}0\) is a common block-injectivity length, then one may take pairwise separating words of length \(6p\), form selectors of length \(6p(g-1)\), and append homogeneous words of any length at least \(p\).

Finally, the BNT canonical-form hypotheses imply that there is a common \(L{\gt}0\) for which every unblocked representative \(A_j\) is \(L\)-block injective. Indeed, irreducibility and left-canonicality give a finite peripheral period \(m_j\). The self-overlap tends to \(m_j\) along multiples of \(m_j\), but by normalization it also tends to \(1\); hence \(m_j=1\). Thus every representative is normal, and finiteness gives a common positive block-injectivity length. If the first-site actions of \(Y,Z\) agree on the tensor represented by \(P\), then \(YA_j=ZA_j\) for \(0\leq j{\lt}g\). The same conclusion holds when the first-site equality is given for any tensor with the same positive-length MPV family. This composes the blocking conclusion of [ CPGSV16 , lines 318–345 ] with Appendix C.3, Lemma L [ CPGSV16 , lines 1835–1858 ] .

No additional common-length hypothesis is required.

Let \(A\) be a normalized MPS tensor with a nonzero PSD fixed point \(\rho \) of the transfer map. If \(\rho _{\operatorname{spec}}(\mathcal{E}_A-P){\lt}1\), where \(P\) is the fixed-point projection, then \(O_{AA}(N)\to 1\) as \(N\to \infty \).

A tensor class appears as a limit of the two-site renormalization flow if and only if it has a physical blocking isometry. Equivalently, there is an isometry \(V\colon \mathbb {C}^d\to \mathbb {C}^{d^2}\) such that

for every \(i_1,i_2\). This is [ CPGSV16 , Theorem 3.1 and lines 1205–1209 ] .

Let \(\mathcal V=\bigsqcup _j\{ j\} \times [D_j]\times [D_j]\) be the disjoint union of the within-sector virtual pairs. For \(x=(j,\alpha ,\beta )\) in \(\mathcal V\), set \(u_x(i)=(U_j^i)_{\alpha ,\beta }\). Then \(\sum _i u_x(i)\overline{u_y(i)}=\delta _{x,y}\).

Let \((A_j)_j\) be a family of normal, irreducible, left-canonical blocks, no two of which are gauge-phase equivalent, with \(\dim _j\ge 1\) for all \(j\), and suppose the direct sum \(\bigoplus _j A_j\) is a renormalization fixed point. Then there are invertible matrices \(X_j\), trace-one positive diagonal matrices \(\rho _j=\operatorname{diag}(\lambda _{j,\alpha })\), and tensors \(U_j\) with

such that \((U_j)_j\) satisfies the joint isometry condition. These are the isometry equations of [ CPGSV16 , Corollary 3.12 ] for the direct sum of distinct normal-tensor blocks.

Every block satisfying the auxiliary hypotheses of Definition 18.3.1.1 is injective. This is an immediate projection of that auxiliary predicate, not a theorem about literal CPSV canonical form.

Let \(A\) and \(B\) be tensors with bond dimensions \(D\) and \(R\). Suppose that \(U\in \mathbb {C}^{R\times D}\) satisfies \(UU^\dagger =I_R\) and, for every physical index \(i\),

Then \(A\) is a renormalization fixed point if and only if \(B\) is a renormalization fixed point.

Let \((B_k)_k\) be a family of normal, irreducible, left-canonical blocks with \(\dim _k\ge 1\) for all \(k\), no two of which are gauge-phase equivalent. Then the direct sum \(\bigoplus _k B_k\) is a renormalization fixed point if and only if each block is in the square-root fixed-point form and the mixed transfer operators between distinct blocks vanish:

This is the distinct-blocks (multiplicity-one, phase-one) case of the structural characterization of pure-state renormalization fixed points [ CPGSV16 , Theorem 3.11 ] , combining its forward and backward directions.

Let \((A_k)_k\) satisfy the auxiliary block-family hypotheses of Definition 18.3.1.1. For each \(k\), the transfer map of \(A_k\) is idempotent if and only if \(A_k\) satisfies the one-block auxiliary conjunction of Definition 18.5.19. The block-family hypothesis is unused; this is Theorem 18.5.21 applied to one block. It should not be read as the BNT-family ZCL assertion in [ CPGSV16 , Theorem 3.10 ] .

Let \(P\) be an auxiliary BNT sector decomposition with distinct basis tensors \((A_j)_j\), and let \(A=\bigoplus _j A_j\) be their direct sum with one unit-weight copy of each sector. If \(A\) is a renormalization fixed point,

then the following restricted conclusion holds: its physical correlations are independent of the separation whenever both complementary gaps are positive, and for all distinct BNT components \(j\ne j'\)

This is a multiplicity-one, unit-weight, positive-gap statement. It does not assert the physical zero-correlation-length conclusion of [ CPGSV16 , Theorem 3.8 ] .

Let \(A_1,\ldots ,A_g\) be the distinct basis tensors of a BNT canonical form, and let \(B=\bigoplus _{j=1}^g A_j\) contain one copy of every basis tensor. If \(B\) is a renormalization fixed point, then for every \(N{\gt}2\) its translated two-site parent interactions commute and

Hence \(B\) satisfies the all-chain nearest-neighbor commuting parent-Hamiltonian ground-space condition. This is the multiplicity-one distinct-sector forward implication in [ CPGSV16 , Definition 3.9 and Theorem 3.10 ] . It does not include repeated copies or arbitrary raw sector weights.

Let \(A_1,\ldots ,A_g\) be the distinct basis tensors of a BNT canonical form, and set \(A=\bigoplus _{j=1}^g A_j\). Then

This statement concerns the multiplicity-one, unit-weight direct sum; it does not include repeated copies or arbitrary raw weights.

Assume \(D {\gt} 0\). Let \(A\) be a normal renormalization fixed point such that \(\sum _i (A^i)^\dagger A^i = \mathbb {1}\). Then there exist a unitary \(U\) and a diagonal positive definite matrix \(\Lambda \) such that, for \(B^i := U^\dagger A^i U\), \(\sum _i (B^i)^\dagger B^i = \mathbb {1}\) and \(\mathcal{E}_B(\Lambda ) = \Lambda \). This is the diagonal fixed-point reduction in [ CPGSV16 , Appendix B ] .

If \(A\) is a normal RFP tensor, then \(A\) is injective.

A normal left-canonical renormalization fixed-point tensor \(A\) admits a decomposition \(A^i=X\Lambda U^iX^{-1}\), where \(\Lambda \) is diagonal positive, \(\sum _i (U^i)^\dagger U^i=I\), and

When each block \(A_k\) of a multi-block tensor is a normal, left-canonical renormalization fixed point, that block admits the isometry decomposition \(A_k^i = X_k \Lambda _k U_k^i X_k^{-1}\), with \(X_k\) invertible, \(\Lambda _k\) diagonal positive, and \(U_k = (U_k^i)\) satisfies \(\sum _i (U_k^i)^\dagger U_k^i = I\) and the pair-index orthonormality condition

This is the blockwise form of the diagonal Kraus decomposition (Theorem 18.3.1.7); see [ CPGSV16 , Section 3 ] . The source additionally imposes the normalization \(\operatorname{tr}(\Lambda _k) = 1\); the statement here gives positive \(\Lambda _k\) without it. That normalization is genuine rather than a conjugation gauge, since rescaling \(\Lambda _k \mapsto \Lambda _k/\operatorname{tr}(\Lambda _k)\) factors out as an overall scalar on \(A_k^i\) (conjugation by \(X_k\) preserves the scale); the statement is thus the unnormalized diagonal Kraus form. Scope restriction (source isometry). Corollary 3.12 also invokes the joint isometry condition

The theorem above records the contracted per-block condition \(\sum _i (U_k^i)^\dagger U_k^i=I\) and the diagonal pair-index equation with right-hand side \(D_k^{-1}\delta _{\alpha ,\alpha '}\delta _{\beta ,\beta '}\). It does not impose the trace-normalization \(\operatorname{tr}(\Lambda _k)=1\), and it does not include the cross-block equations for \(k\ne \ell \).

Under the hypotheses of Theorem 18.3.1.10, each block admits a decomposition \(A_k^i = X_k \Lambda _k U_k^i X_k^{-1}\) with \(\Lambda _k\) diagonal positive and

This is a blockwise unit pair-index decomposition. It still does not impose the source trace-normalization of \(\Lambda _k\) or the cross-block orthogonality equations for distinct blocks.

The decomposition in Theorem 18.3.1.7 additionally records that the diagonal weights satisfy \(\sum _\alpha \Lambda _\alpha ^2 = D\). This square-sum identity is the trace-normalization seed: the explicit weights are \(\Lambda _\alpha = \sqrt{D\, \rho _{\alpha ,\alpha }/\operatorname{tr}\rho }\), and \(\sum _\alpha \rho _{\alpha ,\alpha } = \operatorname{tr}\rho \) for the diagonal fixed point \(\rho \).

A normal left-canonical renormalization fixed-point tensor \(A\) admits a decomposition \(A^i=X\Lambda U^iX^{-1}\), where \(\Lambda \) is diagonal positive and

This is the unit pair-index convention of [ CPGSV16 , Section 3 ] . The statement still does not impose the trace-normalization \(\operatorname{tr}(\Lambda )=1\).

Assume \(D {\gt} 0\). Let \(A\) be a normal renormalization fixed point such that \(\sum _i (A^i)^\dagger A^i = \mathbb {1}\). Then \(A\) is injective. This is the left-canonical injectivity step in [ CPGSV16 , Appendix B ] .

Fix one block \(A_k\) in a family satisfying Definition 18.3.1.1. Then the primitive transfer map of that block satisfies: \(\mathcal{E}_{A_k}^{2^n}\) converges pointwise to an idempotent linear map \(\mathcal{E}_{k,\infty }\) as \(n\to \infty \). Thus there exists \(\mathcal{E}_{k,\infty }\) with \(\mathcal{E}_{k,\infty }^2=\mathcal{E}_{k,\infty }\) such that \(\mathcal{E}_{A_k}^{2^n}(\rho )\to \mathcal{E}_{k,\infty }(\rho )\) for every bond matrix \(\rho \). This is the per-block primitive convergence used in the Appendix B discussion, not convergence of the full weighted canonical-family transfer matrix [ CPGSV16 , Appendix B, lines 1211–1244 ] .

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

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

Let \(a\) belong to a complex Banach algebra and let \(c\in \mathbb {C}\setminus \{ 0\} \). Then

Let \(A\) and \(B\) be irreducible trace-preserving normalized MPS tensors of the same bond dimension that are not gauge-phase equivalent. Then \(\rho _{\operatorname{spec}}(F_{AB}){\lt}1\).

Let \(A\) and \(B\) be irreducible trace-preserving normalized tensors of bond dimensions \(D_1\) and \(D_2\), respectively, with \(D_1\neq D_2\). Then \(\rho _{\operatorname{spec}}(F_{AB}^{\mathrm{rect}}){\lt}1\).

Let \(A\) be an injective left-canonical RFP tensor with positive-definite fixed point \(\rho \) of the transfer map \(\mathcal{E}_A\). Then \(\mathcal{E}_A = P_\rho \), where \(P_\rho (X) = \frac{\operatorname{tr}(X)}{\operatorname{tr}(\rho )} \rho \) is the rank-one fixed-point projection.

The auxiliary conjunction \((\mathcal{E}_A^2=\mathcal{E}_A\ \text{and virtual-insertion distance independence})\) is equivalent to \(\mathcal{E}_A^2=\mathcal{E}_A\). This is a logical simplification of the auxiliary convention, not the physical ZCL theorem of [ CPGSV16 , Theorem 3.8 ] .

Let \(P\) be an auxiliary BNT sector decomposition, let \(A=\bigoplus _k A_k\), and fix a sector \(j\). Suppose there are matrices \(r,l\in M_{D_j}(\mathbb {C})\) and a scalar \(\lambda \ne 0\) with \(|\lambda |{\lt}1\) such that

Then \(A\) does not have positive-gap physical CID.