Dependencies

Let \(P\) satisfy the auxiliary BNT sector hypotheses, with distinct basis tensors \((B_j)_j\). Then the multiplicity-one direct sum \(\bigoplus _j B_j\) 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 all \(j\ne j'\).

In this case, there exist invertible matrices \(X_j\), trace-one positive diagonal matrices \(\rho _j=\operatorname{diag}(\lambda _{j,\alpha })\), and tensors \(U_j\) such that \(B_j^i=X_j\sqrt{\rho _j}\, U_j^iX_j^{-1}\), where \((U_j)_j\) satisfies the joint isometry condition (Definition 17.4.10).

If \(P\) is in BNT canonical form, then every basis block \(P_j\) is normal in the algebraic sense of Definition 2.5.5.

Let \(A=(A^i)_{i=0}^{d-1}\) be an MPS tensor with bond dimension \(D\). The physical-isometry class of \(A\) appears as a limit of the renormalization flow if there exist a physical dimension \(e\) and an MPS tensor \(B=(B^a)_{a=0}^{e-1}\) of 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 may change under blocking, while the bond dimension \(D\) is preserved. Transfer matrices represent the quotient by physical isometries [ 13 , lines 382–394 and 1205–1209 ] .

A tensor \(A\) with one physical component has a blocking witness if there exist 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 \(A^{[L]}\).

In the coefficient form of a basis of normal tensors, each block \(A_j\) is a normal tensor in the spectral sense (Definition 7.6.1), the MPV of \(A\) is given at every positive length by 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 [ 13 , Section 2.3 ] .

The connected two-point function is

Let \(A\) have bond dimension \(D\). It is in the canonical form of [ 13 , 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 [ 13 ] is a separate normalization of these data, not part of canonical-form membership:

Every weight \(\mu _k\) is nonzero. This is the reading of [ 13 , 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 8.1.1; 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 [ 13 ] .

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, (8) is represented by

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

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 [ 13 , Theorem 3.10(iii) ] . For every \(N{\gt}2\), the fixed-chain condition of Definition 12.1.16 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 if for every pair of distinct components \(j\ne j'\),

Diagrammatically,

\( = 0.\)

This is the mixed-sector condition of [ 13 , Definition 3.5 ] .

Let \((A_j)_j\) be a basis of normal tensors for \(A\). The family \((A_j)_j\) satisfies virtual-insertion BNT zero correlation length for \(A\) if \(A\) satisfies virtual-insertion distance independence (Definition 17.5.11) and the components \((A_j)_j\) are locally orthogonal: \(\mathcal{E}_{j,j'}=0\) for all \(j\ne j'\) (Definition 17.2.2). In contrast, [ 13 , Definition 3.6 ] quantifies over physical observables on all disjoint regions (Definition 17.2.6).

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 \(k\).

Condition (6) corresponds to channel primitivity: for a primitive channel \(T\), \(T^n(\rho )\to \rho _\infty \) for every state \(\rho \) [ 244 , Theorem 6.7(3) ] , which implies convergence of the self-overlap to \(1\) (see also [ 244 , Theorem 6.8 ] and Chapter 5).

An MPS tensor satisfies virtual-insertion distance independence if 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 connected correlator satisfies \(C(X,Y;n)=C(X,Y;m)\).

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 if there exist an invertible matrix \(X\) and a tensor \(U\) satisfying the unit pair-index isometry

such that

Local fix (square-root factor in canonical form): The definition uses the square-root diagonal \(\sqrt{\rho }\) rather than the bare diagonal printed in [ 13 , line 1278 ] , as required by the unit pair-index isometry and reference-tensor construction; documented in [ 85 ] .

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 [ 13 , Appendix B, lines 1278, 1281–1283, 1300 ] .

An MPS tensor \(A\) satisfies one-block transfer idempotence if \(\mathcal{E}_A^2=\mathcal{E}_A\). Diagrammatically, two doubled columns equal one:

\( = \)

In [ 13 , Definition 3.5 ] , local orthogonality denotes instead the vanishing of mixed transfer maps between distinct BNT components (Definition 17.2.2).

A family \((A_j)_j\) of MPS tensors has positive-gap BNT zero correlation length for \(A\) if \((A_j)_j\) is a basis of normal tensors for \(A\) (Definition 8.1.2), its components are locally orthogonal (Definition 17.2.2), and the physical correlations of \(A\) are independent of separation whenever both complementary gaps are positive (Definition 17.5.15). Adjacent regions are excluded.

Let \(O_1,O_2\) be observables on disjoint blocks of a periodic chain, with observable transfer maps \(\mathcal{E}_{O_1},\mathcal{E}_{O_2}\) (Definition 17.5.12). The tensor has positive-gap physical correlations independent of distance if for all positive complementary gaps \(n_1,n_2,m_1,m_2{\gt}0\) satisfying \(n_1+n_2=m_1+m_2\),

This restricts Definition 17.5.14 to strictly positive gaps, excluding adjacent regions [ 13 , Definition 3.3 ] .

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

Diagrammatically, contracting \(U_j\) with the conjugate of \(U_{j'}\) over the physical index identifies the virtual legs pairwise:

\( = \delta _{j,j'}\, \) \( \otimes \)

Each block satisfies unit pair-index orthonormality (\(j=j'\)), and the cross-block sums between distinct blocks vanish (\(j\ne j'\)), as in [ 13 , Theorem 3.11 ] .

A tensor \(A\) satisfies the virtual-insertion zero-correlation-length condition if its transfer map is idempotent (\(\mathcal{E}_A^2=\mathcal{E}_A\)) and it satisfies virtual-insertion distance independence (Definition 17.5.11).

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 [ 13 , 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 [ 13 , Theorem 4.9, lines 851–856 ] .

The transfer map \(\mathcal{E}_A\) of an MPS tensor \(A\) satisfies transfer map idempotence when \(\mathcal{E}_A\circ \mathcal{E}_A=\mathcal{E}_A\).

A tensor \(A\) is normal [ 17 ] 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  [ 13 ] , (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 similarity on the virtual indices. For a pure tensor \(A\), this condition is imposed on the doubled tensor

Since the physical algebra is one-dimensional, every physical unitary channel in the diagram is trivial, recovering the relation in [ 13 , Appendix D, lines 2100–2107 ] .

An MPS tensor \(A\) is a renormalization fixed point if there exists an isometry \(V\colon \mathbb {C}^d\to \mathbb {C}^{d^2}\) with coefficients \(V_{(i_1,i_2),j}\) satisfying \(V^\dagger V=\mathbb {1}\) such that for all physical indices \(i_1,i_2\in \{ 0,\ldots ,d-1\} \),

This expresses the physical two-site blocking relation \(AA=A\) of [ 13 , 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 transfer map also depends linearly on the observable \(O\). Diagrammatically, \(O\) is inserted between the two layers of the doubled chain on its \(L\) sites:

This is the observable transfer map of [ 13 , 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

Diagrammatically, the four maps are composed around the ring:

The trace is the operator trace on the virtual matrix space, as in [ 13 , 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.

Let \(P\) be a sector decomposition. In addition to spanning and eventual linear independence, the canonical form specifies irreducible left-canonical representatives, self-overlap normalization, pairwise gauge-phase separation, and weight normalization. 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 [ 13 , Section 2.3 ] canonical-form normalization holds:

The MPV expansion takes the two-layer form

matching [ 13 , Section 2.3 ] and [ 17 , Definition IV.2 ] . The canonical-form normalization accommodates equal-modulus and repeated-copy sectors, such as \(C \oplus (-C)\) and \(C \oplus (1/2) C\).

A basis of normal tensors \((A_j)_j\) for \(A\) has physical BNT zero correlation length if the generated matrix product states have physical correlations independent of distance for all disjoint regions (Definition 17.5.14) and the components \((A_j)_j\) are locally orthogonal (Definition 17.2.2). This is [ 13 , 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 if, for all \(m_1,m_2\geq 0\) satisfying \(n_1+n_2=m_1+m_2\),

This allows either region to be translated along the chain without crossing the other, including adjacent regions (\(n_i=0\)) [ 13 , Definition 3.3 ] .

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 \(G_\rho \) and the matrix of its inverse satisfy

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 \(a=0\).

A family \((A_j)_j\) has positive-gap BNT zero correlation length for \(A\) if and only if \((A_j)_j\) is a basis of normal tensors for \(A\) (Definition 8.1.2), the physical correlations of \(A\) are independent of separation whenever both complementary gaps are positive (Definition 17.5.15), and the components \((A_j)_j\) are locally orthogonal (Definition 17.2.2).

If a BNT family \((A_j)_j\) for \(A\) has zero correlation length for all disjoint physical regions (\(\mathrm{ZCL}_{\mathrm{phys}}\)), then it satisfies the positive-gap zero correlation length condition (\(\mathrm{ZCL}_+\)):

If an MPS tensor satisfies physical distance independence (Definition 17.5.14), then it satisfies the positive-gap condition (Definition 17.5.15).

If \(A \sim _{\mathrm{gauge}} B\) (so \(B^i = Y A^i Y^{-1}\) for invertible \(Y\)), then \(A\) has the square-root fixed-point form if and only if \(B\) does.

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,

The spanning and separation conditions are equivalent after existentially quantifying the length. Spanning at length \(L\) gives separation at that length by nondegeneracy of the product trace pairing. Conversely, if the span were proper, a nonzero linear functional \(f\) would annihilate it. Nondegeneracy of the matrix trace pairing represents \(f\) as

for some tuple \((\Delta _k)_k\), contradicting separation. This finite-word property generalizes the one-letter simultaneous span in the source definition of block-injective canonical form to length-\(L\) words.

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 implies the product-algebra spanning condition, and hence trace separation.

The block-injectivity proposition of [ 13 , 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 product-algebra spanning and trace-separation properties.

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

Diagrammatically, the transfer map is the rank-one map whose output legs carry \(\operatorname {diag}(\Lambda )\) and whose input legs are traced:

\( = \) \(\; \otimes \; \)

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 17.5.14. 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 \(Y\),

If \(A = \zeta X A^{[2]} X^{-1}\) is a one-letter MPS blocking witness with \(|\zeta |=1\), then the doubled tensor \(M = A \otimes \overline{A}\) satisfies

satisfying the doubled-MPO virtual-gauge condition (Definition 17.5.3).

Let \(P\) be an auxiliary BNT sector decomposition with distinct basis tensors \((A_j)_j\). The multiplicity-one unit-weight direct sum \(A=\bigoplus _j A_j\) is in CPSV canonical form (Definition 7.6.7), with unit retained weights, retained blocks \((A_j)_j\), and the identity ambient coisometry.

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 \(j\ne j'\), the mixed transfer operator vanishes:

This establishes the cross-block (\(\delta _{j,j'}\)) orthogonality of [ 13 , Theorem 3.11 and Corollary 3.12 ] . The passage to the residual tensors \(U_j\) is given by Theorem 17.4.16.

Let \(P\) be an auxiliary BNT sector decomposition of total bond dimension \(D\), and let \(A=\bigoplus _j A_j\) be the direct sum of its basis tensors. There exists a 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 establishes the multiplicity-one case of the block-injective observable construction in [ 13 , proof of Theorem 3.8, lines 1252–1256 ] .

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 17.3.1.1, 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 \(\omega ^2 e_{21}\), so the dyadic renormalization flow does not converge. This refutes the unrestricted canonical-form convergence assertion of [ 13 , lines 417 and 1211–1244 ] ; it does not affect the single-component convergence of Theorem 17.5.1.

Cirac et al.  [ 13 , Theorem 3.11 ] assert 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 both the physical and virtual spaces [ 13 , lines 561–562 ] . However, it does not provide an explicit coordinate embedding for the physical routing of \(q\), a dimension constraint, or a \(q\)-indexed isometry equation. Consequently, this statement remains unformalized pending clarification of the coordinate-level routing in the source.

Local fix (square-root factor in canonical form): 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 [ 85 ] .

Every normal tensor has positive bond dimension.

Every normal tensor in the sense of [ 13 , 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 [ 13 , 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

If no classes are present, the statement holds vacuously.

Local fix (square-root factor in canonical form): The formula uses the square-root weights \(\sqrt{\lambda _{j,\alpha }}\) rather than the bare diagonal printed in [ 13 , line 547 ] , as required by the reference-tensor construction; documented in [ 85 ] .

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 in [ 13 , Appendix D, lines 2100–2107 ] is false.

Cirac et al.  [ 13 , Theorems 3.8 and 3.10 ] assert 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. 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 map idempotence under the source weight normalization.

For the forward implication, the Bell-pair chain of Theorem 17.2.9 is a normal tensor with idempotent transfer map, but its expectation on adjacent regions differs from its expectation when the regions are shifted by one site. Thus transfer map idempotence does not imply correlation independence when adjacent regions are included.

Local fix (false source claim for adjacent regions): The printed forward implication in [ 13 , Theorem 3.8 ] asserts correlation independence for all disjoint regions, including adjacent ones. However, the Bell-pair counterexample (Theorem 17.2.9) shows that transfer idempotence does not imply correlation independence for adjacent regions; the source derivation uses \(\mathcal{E}_A^n=\mathcal{E}_A\), which holds only for \(n\ge 1\), whereas adjacent regions insert \(\mathcal{E}_A^0=\mathbb {1}\). Documented in [ 31 ] . Theorem 17.5.17 establishes the forward implication when both complementary gaps are positive.

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  [ 216 , “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 establishes the diagonal Perron normalization required in canonical form II.

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 correlations independent of distance and \(\mathcal{E}_{j,j'}=0\) for all \(j\ne j'\). Thus \(A\) has positive-gap BNT zero correlation length (Definition 17.2.4). As shown in Theorem 17.2.9, this does not extend to adjacent regions.

If \(\mathcal{E}_A^2=\mathcal{E}_A\), then \(A\) satisfies positive-gap physical distance independence (Definition 17.5.15).

If an MPS tensor \(A\) has the square-root fixed-point form, then its transfer map \(\mathcal{E}_A\) is idempotent:

In particular, \(A\) is a renormalization fixed point [ 13 , Section 3 ] .

Let \((B_k)_k\) be a family of tensors in square-root fixed-point form whose cross-block mixed transfer operators vanish (\(\mathcal{E}_{j,j'}=0\) for all \(j\ne j'\)). Then the direct sum \(\bigoplus _k B_k\) is a renormalization fixed point. This establishes the converse direction of the pure-state fixed-point characterization [ 13 , Section 3 ] for distinct blocks.

If the virtual-insertion condition (Definition 17.5.11) holds for every positive-definite right fixed point \(\rho ^R{\gt}0\) of the transfer map, then \(\mathcal{E}_A^2=\mathcal{E}_A\).

Every normal left-canonical renormalization fixed-point tensor has the square-root fixed-point form (Definition 17.3.1.10): setting \(\lambda _\alpha =\rho _{\alpha ,\alpha }/\operatorname{tr}\rho \) and \(\rho _0=\operatorname{diag}(\lambda _\alpha )\), there exist an invertible matrix \(X\) and a unit pair-index isometry \(U\) such that

This is the normal-tensor characterization of [ 13 , Appendix B, lines 1278–1300 ] , incorporating the square-root weight correction.

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

Let \(A\) be a normal tensor. Then its transfer map satisfies \(\mathcal{E}_A^2 = \mathcal{E}_A\) if and only if \(A\) has the square-root fixed-point form

In particular, the bond dimension of \(A\) is positive. This establishes the corrected form of [ 13 , Lemma B.1, 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 \(\mathcal{E}_A\) of an MPS tensor \(A\) is idempotent, \(\mathcal{E}_A^2 = \mathcal{E}_A\), if and only if there exists an isometry \(V \colon \mathbb {C}^d \to \mathbb {C}^d \otimes \mathbb {C}^d\) such that

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 \(M,N\) belong to the span of the length-\(L\) words of a tensor \(A\). There is an operator \(O\) on \(L\) physical sites whose inserted transfer map satisfies \(\mathcal{E}_O(X)=MXN^\dagger \) for every virtual matrix \(X\). This is the two-sided multiplication used in the observable choices of [ 14 , Appendix C ] and [ 219 , Lemma 2 ] .

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 equivalence holds for the multiplicity-one direct sum of distinct BNT basis tensors with positive complementary gaps.

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 [ 13 , lines 318–345 ] with Appendix C.3, Lemma L [ 13 , 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 [ 13 , 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 with bond dimensions \(D_j\ge 1\), no two of which are gauge-phase equivalent. If the direct sum \(\bigoplus _j A_j\) is a renormalization fixed point, then there exist invertible matrices \(X_j\), trace-one positive diagonal matrices \(\rho _j=\operatorname{diag}(\lambda _{j,\alpha })\), and tensors \(U_j\) with

that is,

\( = \)

such that \((U_j)_j\) satisfies the joint isometry condition (Definition 17.4.10). This formalizes the isometry equations of [ 13 , Corollary 3.12 ] for distinct normal blocks.

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 \(D_k\ge 1\) 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 has the square-root fixed-point form and all cross-block mixed transfer operators vanish:

This is the distinct-block characterization of pure-state renormalization fixed points [ 13 , Theorem 3.11 ] .

Let \(P\) be an auxiliary BNT sector decomposition with distinct basis tensors \((A_j)_j\), and let \(A=\bigoplus _j A_j\) be their multiplicity-one direct sum. If \(A\) is a renormalization fixed point (\(\mathcal{E}_A^2 = \mathcal{E}_A\)), then \(A\) has positive-gap BNT zero correlation length: its physical correlations are independent of separation whenever both complementary gaps are positive, and for all distinct BNT components \(j\ne j'\),

This establishes the positive-gap, multiplicity-one formulation of [ 13 , 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, establishing the multiplicity-one distinct-sector forward implication of [ 13 , Theorem 3.10 ] .

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 [ 13 , 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

In the decomposition of Theorem 17.3.1.7, the diagonal weights satisfy \(\sum _\alpha \Lambda _\alpha ^2 = D\).

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 [ 13 , Section 3 ] .

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 [ 13 , Appendix B ] .

Fix one block \(A_k\) in a family satisfying Definition 17.3.1.1. Then the dyadic powers of its transfer map converge pointwise to an idempotent linear map: there exists \(\mathcal{E}_{k,\infty }\) with \(\mathcal{E}_{k,\infty }^2=\mathcal{E}_{k,\infty }\) such that \(\mathcal{E}_{A_k}^{2^n}(\rho )\to \mathcal{E}_{k,\infty }(\rho )\) as \(n\to \infty \) for every virtual matrix \(\rho \). This formalizes the per-block primitive convergence of [ 13 , 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 [ 13 , lines 340–345 ] .

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

If \(\mathcal{E}_A\circ \mathcal{E}_A=\mathcal{E}_A\), then every nonzero eigenvalue of \(\mathcal{E}_A\) equals \(1\), the form in which [ 17 , Section III.A ] states the renormalization fixed-point property.

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.

A tensor satisfies the virtual-insertion zero-correlation-length condition (Definition 17.5.18) if and only if its transfer map is idempotent:

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 correlations independent of distance (Definition 17.5.15).