Dependencies

Suppose that either \(r\) or \(\ell \) is odd. Two conjugation-invariant MPUs \(\mathcal U_1\) and \(\mathcal U_2\) are strictly equivalent under conjugation if and only if

If either \(r\) or \(\ell \) is odd, all conjugation-invariant MPUs with these source ranks are equivalent under conjugation symmetry.

Two conjugation-invariant MPUs are equivalent under conjugation symmetry if and only if they belong to the same case of Proposition 19.9.1.7.

For \(i=1,2\), suppose the endpoint tensor \(\mathcal U_i\), its conjugate comparison \(\overline{\mathcal U_i}\), the actual two-site block \(\mathcal U_{i,2}\), and its conjugate comparison \(\overline{\mathcal U_{i,2}}\) are simple and each carries reduced full-support source data. Suppose that either \(r\) or \(\ell \) is odd and that the resulting standard-form interpolation carries an admissible path datum of blocking length one. Then \(\mathcal U_1\) and \(\mathcal U_2\) are admissibly equivalent under conjugation symmetry.

Let \(k_0\) be the common blocking length in an admissible equivalence datum for a continuous path \(\mathcal U(p)\) of MPU tensors, and suppose that this supplied length satisfies \(k_0\leq D^4\). Then there are continuous paths of unitaries \(u(p)\) and \(v(p)\) such that the MPU \(U^{(2k_0N)}(p)\) has standard form given by \(u(p)\) and \(v(p)\) for every \(p\in [0,1]\). This is the reduced-source version of [ CPGSV17 , Corollary IV.7 ] .

For \(i=1,2\), suppose the original one-site endpoint tensor \(\mathcal U_i\), its conjugate comparison \(\overline{\mathcal U_i}\), the actual two-site block \(\mathcal U_{i,2}\), and its conjugate comparison \(\overline{\mathcal U_{i,2}}\) are simple and each carries reduced full-support source data. Suppose also that the resulting standard-form interpolation carries an admissible path datum of blocking length one. Then \(\mathcal U_1\) and \(\mathcal U_2\) are admissibly equivalent under conjugation symmetry if and only if they belong to the same case of Proposition 19.9.1.8.

Let two time-reversal-invariant MPUs be compared by an admissible symmetry path \(p\mapsto \mathcal W(p)\) whose path datum has blocking length one. Suppose that the five path families \(\mathcal W(p)\), \(\mathcal W(p)_2\), \(\mathcal W(p)_4\), \(\mathcal W(p)_2^\sharp \), and \(\mathcal W(p)_4^\sharp \) are simple and carry continuously varying reduced full-support source data. Then admissible strict equivalence requires equality of both \(\sigma ^{(1)}\) and \(\sigma ^{(2)}\).

For admissible equivalence after adjoining ancillas and applying the source blocking, let \(p\mapsto \mathcal W_{\mathrm{anc}}(p)\) be the actual ancilla-enlarged blocked witness path, not a reduced or equivalent representative, and regard it as a length-one admissible path. Require continuous reduced full-support source data along \(\mathcal W_{\mathrm{anc}}(p)\), its actual two- and four-site blocks, and the physical-adjoint comparisons of those two blocks. The conclusion is only

No equality between \(\sigma ^{(2)}\) of the original two MPUs is asserted. Such a conclusion requires preservation under adjoining identity ancillas, which is not established here.

In the admissible Case II of Proposition 19.9.1.8, one can write

where \(\widetilde u\) and \(\widetilde v\) are orthogonal and

Strictly equivalent time-reversal-invariant MPUs have the same \(\sigma ^{(1)}\) and \(\sigma ^{(2)}\). Equivalent time-reversal-invariant MPUs have the same \(\sigma ^{(2)}\).

In Case II of Proposition 19.9.1.7, one can write

where \(\widetilde u\) and \(\widetilde v\) are orthogonal and

Let an MPU in standard form be invariant under complex conjugation. Its standard-form unitaries can be chosen in exactly one of the following two cases. In Case I,

uniquely up to local orthogonal transformations. In Case II,

uniquely up to local symplectic transformations, where \(\Sigma _x=\mathbb {1}_{x/2}\otimes \left(\begin{smallmatrix} 0 & 1 \\ -1 & 0 \end{smallmatrix}\right)\). Case II can occur only when \(r\) and \(\ell \) are even.

For every continuous path \(p\mapsto \mathcal U(p)\) of MPU tensors, not necessarily in canonical form, there are an integer \(k_0\leq D^4\) and continuous paths of unitaries \(u(p)\) and \(v(p)\) such that \(U^{(2k_0N)}(p)\) has standard form given by \(u(p)\) and \(v(p)\).

Given \(r\) blocks with bond dimensions \((D_k)_{k=1}^r\), per-block tensors \(\{ A_k^i\} \), and scaling factors \((\mu _k)_{k=1}^r\), the associated block-diagonal tensor is the tensor of total bond dimension \(D = \sum _{k=1}^r D_k\) defined by

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

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

The norm completion of a complex normed star algebra satisfying the C-star identity carries the continuously extended involution and a C-star algebra structure. The canonical map into the completion is a star-algebra homomorphism.

Let \(A\) and \(B\) be complex normed star algebras, and let \(f\colon A\to B\) be a star-algebra equivalence. If \(f\) and \(f^{-1}\) are continuous, then \(f\) extends to a star-algebra equivalence

between their norm completions. Its underlying map is the continuous extension of \(f\). Writing \(j_A\) and \(j_B\) for the canonical maps into the completions, the extension satisfies

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.

In canonical form II of [ CPGSV16 , Appendix A ] , there are positive retained dimensions \(D_k\), normal tensors \(A_k\), weights \(\mu _k\), and, for \(S=\sum _kD_k\le D\), a matrix \(U\in \mathbb {C}^{S\times D}\) such that, for every physical index \(i\),

Thus omitted bond-space coordinates are literally zero. Every retained block additionally satisfies

where \(\Lambda _k\) is diagonal and positive definite. The two displayed equations are the only additional blockwise conditions; no ordering, injectivity, gauges, or basis-of-normal-tensors assumptions are included. The line-246 convention (26) remains a separate condition on the underlying canonical-form data.

Let \(\mathcal U\) and \(\mathcal V\) have physical dimensions \(d_a\) and \(d_b\). They are equivalent under a symmetry \(\mathcal S\) if there are a positive blocking length \(k\) and coprime positive ancilla dimensions \(p_a,p_b\) with \(p_ad_a=p_bd_b\) such that \((\mathcal U^{(p_a)})_k\) and \((\mathcal V^{(p_b)})_k\) are strictly equivalent under \(\mathcal S\).

Fix a virtual bond dimension \(D\). Let \(\mathcal U\) and \(\mathcal V\) be MPU tensors with physical dimensions \(d_a\) and \(d_b\). They are equivalent if there are a positive blocking length \(k\) and coprime positive integers \(p_a,p_b\) with

such that \((\mathcal U^{(p_a)})_k\) and \((\mathcal V^{(p_b)})_k\) are strictly equivalent. Here the positive-dimensional identity ancillas are attached before the common blocking, and

This is the fixed-bond interpretation of [ CPGSV17 , lines 716–724 ] ; it does not introduce a stabilization relation for unequal raw bond dimensions.

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 \((i_1,\ldots ,i_L)\).

Let \(k\) be any blocking length for which \(\mathcal U_k\) is simple, and let \(r\) and \(\ell \) be its right and left source-cut ranks. The source index is

This is the unrestricted definition of [ CPGSV17 , Definition IV.1 ] .

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

Equivalently, its Kraus map is trace-preserving.

The mixed transfer operator (or cross transfer operator) for two MPS tensors \(A\) and \(B\) of the same bond dimension \(D\) is the linear map \(F_{AB}:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) defined by

The product tensor of two MPO tensors contracts the bra index of the first factor with the ket index of the second and takes the tensor product of the bond spaces:

In the vertical reading of [ CPGSV16 , Section 4.5 ] this is the tensor obtained by joining two vertically read tensors along the original horizontal bond, the tensor written \(M_\alpha M_\beta \) in [ CPGSV16 , Theorem 4.14(iii) ] .

For an MPO tensor \(K\), define \(K^\sharp \) by

Thus \(K^\sharp [\beta ,\alpha ]=K[\beta ,\alpha ]^*\), while the virtual indices retain their order. For every periodic chain,

For an integer \(L\geq 0\), group \(L\) adjacent physical sites and denote the resulting local tensor by \(M^{[L]}\). It has a length-\(L\) ket word \(I=(i_1,\ldots ,i_L)\) and a length-\(L\) bra word \(J=(j_1,\ldots ,j_L)\) as its physical indices, and

Thus the ket and bra words are grouped separately. For \(L=0\), both words are empty and the unique local matrix is the identity. For \(L{\gt}0\), this is the MPO version of the physical blocking used for the basis of normal tensors in [ CPGSV16 , lines 317–345 ] . It is a local tensor, not the closed-chain operator \(O_L(M)\) of [ CPGSV16 , lines 962–967 ] .

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 ] .

For words \(u=(i_1,\ldots ,i_N)\) and \(v=(j_1,\ldots ,j_N)\) in \(\{ 0,\ldots ,d{-}1\} ^N\), the word evaluation is

The empty pair of words evaluates to the identity, \(M^{\varnothing ,\varnothing }=\mathbb {1}_D\), and word pairs of different lengths evaluate to \(0\).

The operator generated by \(M\) on \(N\) sites is the matrix \(\rho ^{(N)}(M)\) indexed by configurations \(\sigma ,\tau \in \{ 0,\ldots ,d{-}1\} ^N\) and given by

The cyclic shift of configurations on \(N\) sites is the bijection that relabels each site by one position, \(\sigma \mapsto \sigma \circ r\), where \(r\) is the cyclic rotation \(i\mapsto i+1\) of \(\{ 0,\ldots ,N-1\} \).

Placing one MPO layer on top of another and contracting the shared physical index gives the layer product of two MPO tensors \(M\) and \(M'\) with the same physical dimension, \((MM')^{ij}=\sum _kM^{ik}\otimes M'^{kj}\), an MPO tensor on the tensor product of the virtual spaces. The \(p\)-fold vertical stack \(M^{[p]}\) of \(M\) is the iterated layer product of \(p\) copies of \(M\); the empty stack \(M^{[0]}\) is the trivial-bond tensor with letters \(\delta _{ij}\), which generates the identity operator at every system size.

An MPO tensor is a family of matrices \(\{ M^{ij}\} _{i,j=0}^{d-1}\) with \(M^{ij} \in M_{D}(\mathbb {C})\), indexed by a ket index \(i\) and a bra index \(j\). Diagrammatically,

By combining the pair \((i,j)\) into a single physical index in \(\{ 0,\ldots ,d^2-1\} \), one obtains an MPS tensor with physical dimension \(d^2\) and bond dimension \(D\), with \(M^{\mathrm{MPS}}_{(i,j)}=M^{ij}\), identifying \(\{ 0,\ldots ,d{-}1\} \times \{ 0,\ldots ,d{-}1\} \) with \(\{ 0,\ldots ,d^2{-}1\} \) via the product encoding.

The transfer map associated to \(M\) is the linear map \(\mathcal{E}_M:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) defined by

Let \(A=\{ A^i\} _{i\in \{ 0,\ldots ,d{-}1\} }\) and \(B=\{ B^k\} _{k\in \{ 0,\ldots ,e{-}1\} }\) have auxiliary dimensions \(D\) and \(E\), respectively. Write

for the standard finite-product coordinate bijection. Their independent tensor product \(A\boxtimes B\) has physical dimension \(de\), auxiliary dimension \(DE\), and local matrices determined by

For every \(N\), the same coordinates give the pointwise splitting

This is infrastructure for the tensoring clause in the proof of the MPU Index Theorem IV.6(ii) [ CPGSV17 , lines 824–845 ] , not a separately stated theorem of that paper. The notation \(A=\{ A^i\} _i\) agrees with the canonical-form block notation in [ CPGSV16 , Section 2.3, lines 214–245 ] .

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

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

The tensor \(\mathcal U\) generates matrix product unitaries if \(U^{(N)}\) is unitary for every integer \(N{\gt}1\). This is the matrix product unitary condition in [ CPGSV17 , Section III, equation (10) ] .

An admissible strict-equivalence datum for a path \([0,1]\ni p\mapsto \mathcal W(p)\) consists of a common positive blocking length \(m\) such that the actual blocked path tensor \(\mathcal W(p)_m\) is simple and carries a reduced full-support source datum for every \(p\). The blocked tensors and their source data vary continuously with \(p\), after fixed identifications of the finite-dimensional spaces involved. No equivalent or reduced representative may replace \(\mathcal W(p)_m\). In particular, the actual blocked endpoint tensors carry their own source data.

When the path is used under a specified symmetry \(\mathcal S\), the datum also includes continuously varying reduced full-support source data for the fixed family of comparison paths

Every tensor in this family is simple. This requirement is part of the datum independently of which comparisons a later proof uses; no preservation under conjugation, physical adjoint, transposition, or \(\mathcal S\) is assumed.

Two tensors are admissibly strictly equivalent if their strict equivalence is witnessed by such a path. They are admissibly equivalent if the blocked, ancilla-enlarged endpoints in Definition 19.9.28 are joined by an admissible strict-equivalence path. The source arguments motivating these additional path hypotheses are Proposition IV.5, Theorem IV.6, Corollary IV.7, and the lemma following the symmetry-equivalence definitions in [ CPGSV17 ] ; the datum itself is introduced here and has no source-labelled counterpart.

Let \(k\) be a blocking length for which \(\mathcal U_k\) is simple and is equipped with a reduced full-support source datum, and let \(r\) and \(\ell \) be its right and left source-cut ranks. Define

Since \(r\ell =d^{2k}\), this is equivalently

Let \(\mathcal U\) be a simple tensor equipped with a reduced full-support source datum. The admissible standard form of the two-site blocking \(\mathcal U_2\) is the depth-two form determined by the source unitaries

On four consecutive blocked legs its unitary is

Two tensors \(\mathcal U\) and \(\mathcal V\) are strictly equivalent under a symmetry \(\mathcal S\) if they have the same physical dimension and there is a continuous path \(\mathcal W(p)\) of MPU tensors from \(\mathcal U\) to \(\mathcal V\) such that

for every \(p\in [0,1]\) and every chain length \(N\) under consideration. The path need not be in canonical form. The equivalence is admissible if its witnessing path carries an admissible equivalence datum for the actual blocked path tensor and the fixed conjugate, physical-adjoint, transposed, and \(\mathcal S\)-image comparison paths, and if the common blocking length \(m\) of that datum satisfies \(m\leq D^4\).

Let \(\mathcal U\) and \(\mathcal V\) have physical dimensions \(d_a\) and \(d_b\). They are equivalent under a symmetry \(\mathcal S\) if there are a positive blocking length \(k\) and coprime positive ancilla dimensions \(p_a,p_b\) with \(p_ad_a=p_bd_b\) such that \((\mathcal U^{(p_a)})_k\) and \((\mathcal V^{(p_b)})_k\) are strictly equivalent under \(\mathcal S\). Thus the ancillas are attached before the common blocking. The equivalence is admissible if this strict-equivalence witness carries an admissible equivalence datum for the actual blocked path and the fixed conjugate, physical-adjoint, transposed, and \(\mathcal S\)-image comparison paths, with common blocking length at most \(D^4\).

If \(U\) is the reconstruction coisometry and \(J_k\) is the inclusion of the \(k\)th retained block, define

Let \(A\succeq 0\) have support projection \(P\), and let \(A^+\) denote its inverse on the support. Define

Define also

This construction gives explicit witnesses for the invertible matrices required in Proposition IV.5 of [ CPGSV17 ] ; the paper requires those matrices but does not state these formulas.

A tensor \(A\) of physical dimension \(d\) and bond dimension \(D\) is in MPU canonical form if there are positive block dimensions \(D_k\), weights \(\mu _k\in \mathbb C\), tensors \(A_k\) of bond dimension \(D_k\), and a coisometry

such that

Every \(A_k\) is irreducible and the spectral radius of its transfer map is one. The blocks may be periodic, so no uniqueness condition is imposed on their peripheral eigenvalues. This is the canonical form of [ CPGSV17 , lines 259–267 ] ; normal tensors form a strictly stronger class.

Let \(A\) have canonical-form-II data with retained weights \(\mu _k\), blocks \(A_k\), and positive definite diagonal fixed matrices \(\Lambda _k\). For every positive integer \(L\), the blocked tensor \(A^{[L]}\) has canonical-form-II data with weights \(\mu _k^L\), blocks \(A_k^{[L]}\), and the same matrices \(\Lambda _k\). The retained blocks remain left-canonical:

A bijective relabeling of the physical alphabet likewise preserves canonical-form-II data, including both the matrices \(\Lambda _k\) and the left-canonical identities.

Consequently, if the normalized flattening \(\widetilde{\mathcal U}\) of an MPO tensor has canonical-form-II data, then for every \(L{\gt}0\) the normalized flattening of \(\mathcal U^{[L]}\) has the data obtained by first blocking and then relabeling by \(\vartheta _L\) from Theorem 19.6.5. Theorem 19.6.6 preserves the left-canonical identity under both operations. For each retained block, Lemma D.4.8 gives

The second identity follows by iterating \(\mathcal{E}_{A_k}(\Lambda _k)=\Lambda _k\). Thus the same positive definite diagonal matrix is fixed after blocking. A physical bijection leaves the transfer map unchanged. The final construction applies these two operations to (276).

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

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

For an index \(i_0\) in a finite set \(I\) of cardinality \(n\), let \(M_{i_0}\) be the \(n\times n\) diagonal matrix with a \(1\) at position \(i_0\) and zeros elsewhere.

For an MPO tensor \(\mathcal U\), let \(\mathcal W\) be the double-layer tensor obtained by placing the physical adjoint above \(\mathcal U\) and contracting the adjacent physical legs:

The bond space of \(\mathcal W\) is the product of the two original bond spaces. The physical adjoint exchanges and conjugates the physical indices without reversing the order of the virtual chain, exactly as in the barred upper layer of Figures II_Simple1.png and II_Simple2.png of [ CPGSV17 ] .

Let \(q:\{ 0,\ldots ,D{-}1\} \times \{ 0,\ldots ,D{-}1\} \to \{ 0,\ldots ,D^2{-}1\} \) be the standard enumeration of a pair. For an MPO tensor \(\mathcal W\) and a doubled-virtual matrix \(E\in \mathbb C^{D^2\times D^2}\), define

If \(E(\mathcal W)\) denotes the normalized physical diagonal of the physical-adjoint double layer of \(\mathcal W\), define

This is the contraction shown in Figure IV_index4.png of [ CPGSV17 , Proposition IV.5 ] .

If \(\mathcal U\) is an MPU and \(N{\gt}1\), then \(U_a^{(N)}\) is unitary and defines the star-algebra automorphism

The factor order is the one in [ CPGSV17 , lines 2300–2306 ] .

Let \(\Lambda \subseteq I_{a,N}\) and \(A\in \mathcal A_\Lambda \). Set \(A_L=\iota _{\Lambda ,I_{a,N}}(A)\). For \(N{\gt}1\), the finite-chain term in [ CPGSV17 , lines 2300–2306 ] is the star-algebra map

No stabilization in \(N\) is asserted.

A finite-chain operator symmetry \(\mathcal S\) specifies a set \(A\) of applicable chain lengths and, for every physical dimension \(d\) and length \(N\), a map

We suppress the subscripts on \(\mathcal S_{d,N}\) when \(d\) and \(N\) are understood. An MPO tensor \(\mathcal W\) is invariant under \(\mathcal S\) if

for every \(N\in A\).

For \(a\in \mathbb Z\) and \(N\in \mathbb N\), let \(I_{a,N}=[a,a+N)\cap \mathbb Z\). The increasing bijection \(\{ 0,\ldots ,N-1\} \simeq I_{a,N}\) identifies the periodic configuration basis with \(\operatorname {Config}_d(I_{a,N})\). In this basis, write \(U_{a}^{(N)}\) for the matrix obtained by simultaneously reindexing the rows and columns of \(U^{(N)}\).

Let

be CPSV canonical-form data in ambient bond dimension \(D\), where every \(A_k\) has positive bond dimension \(D_k\). The data have full support when

Let \(\mathcal U\) and \(\mathcal V\) have physical dimensions \(d,e\) and auxiliary dimensions \(D,E\), respectively. Their independent tensor product \(\mathcal U\boxtimes \mathcal V\) has dimensions \(de\) and \(DE\) and entries

A product-valued physical configuration splits sitewise into one configuration for \(\mathcal U\) and one for \(\mathcal V\).

The left rank is

In standard Euclidean coordinates, let \(\widetilde P\) be the matrix of the reduced projection associated with two square matrices \(P\) and \(Q\).

For an MPO tensor \(U\) of physical dimension \(d{\gt}0\), its normalized flattening is the doubled-physical-index MPS tensor

This is the normalization used in the normal-tensor argument of [ CPGSV17 , Section III, equation (13) ] .

Given canonical-form-II data for \(\widetilde{\mathcal U}\), define the transformed tuple by

These data reconstruct \(\widetilde{\mathcal U^\sharp }\).

On the flattened physical alphabet, define the involution

This acts on physical coordinates and is distinct from the doubled-bond exchange used in reflected transfer calculations.

For a bijection \(e\) between two physical index sets, the reindexed tensor and its action on length-\(N\) configurations are

Let \(\widetilde{\mathcal U}=d^{-1/2}\mathcal U\) be the normalized flattening, and let \(E\) be its transfer matrix. A reduced full-support source datum for \(\mathcal U\), with physical dimension \(d{\gt}0\) and positive bond dimension, consists of an integer \(J{\gt}0\), a positive definite matrix \(\rho \) with \(\operatorname{tr}(\rho )=1\), and \((\Phi \rvert =\operatorname {vec}(\mathbb {1}_D)^{\mathsf T}\) such that

These data are supplied in one of two ways.

1. If \(D=1\), take \(J=1\) and \(\rho =\mathbb {1}_1\) using Theorem 19.8.7.

2. If \(D{\gt}1\), choose canonical-form-II data for the normalized flattening of this same tensor, with its blocks filling the ambient bond space. Theorem 19.8.5 supplies the fixed pair with \(J=D^2-1\).

For \(D{\gt}1\), this states the fixed-pair convention in [ CPGSV17 , equations labelled Erightleft and II_UTransfer, and proof of Proposition III.2(ii) ] . The \(D=1\) branch is the direct one-dimensional specialization. This definition does not assert that an arbitrary representative with \(D{\gt}1\) admits the required canonical-form-II data and does not replace \(\mathcal U\) by another tensor.

The doubled-bond coordinate reflection exchanges the two virtual factors,

Independently, spatial reflection reverses a blocked word of length \(K\),

These are support coordinates for the proof of [ CPGSV17 , Lemma III.7 in Section III.B ] , rather than separate definitions printed there. The blocked-word reversal is not used in the transfer transport below; conjugate transposition there reverses only the order of the two blocked letters.

Let \(\mathcal U\) have bond dimension \(D\). Contract the double layer against arbitrary physical matrices, subtract the normalized diagonal coefficient, and let \(\mathcal S\) be the resulting set of residual matrices in \(M_{D^2}(\mathbb C)\). The residual algebra \(\mathfrak S\) is the not necessarily unital algebra generated by \(\mathcal S\). This is the algebra generated by the matrices \(S'_\alpha \) in [ CPGSV17 , Proposition III.3(ii) ] .

The right rank is

The right shift on \((\mathbb C^d)^{\otimes N}\) is

It has bond dimension \(d\) and local matrices \(A^{ij}=\lvert i\rangle \langle j\rvert \). The physical adjoint of \(A\) defines the left-shift tensor.

The tensor \(\mathcal U\) is simple if there are vectors \(a,b\) in the double-layer bond space such that, for all physical indices \(i,j,k,\ell \),

Thus (11) closes one double-layer letter to the physical identity, while (12) inserts the rank-one operator \(|b)(a|\) between adjacent letters. These are precisely the two diagrams in [ CPGSV17 , Definition III.2 ] .

The matrix \(M_1\) is obtained by combining the left auxiliary index with the lower physical index, and the upper physical index with the right auxiliary index. Thus

Transporting the row and column indices along the standard product enumerations gives a matrix \(\widetilde M_1\) indexed by \(\{ 0,\ldots ,Dd-1\} \times \{ 0,\ldots ,dD-1\} \). Its entries are obtained by applying \(M_1\) to the inverse images of the enumerated indices, and

The matrix \(M_2\) is obtained by combining the left auxiliary index with the upper physical index, and the lower physical index with the right auxiliary index. Thus

Transporting the row and column indices along the standard product enumerations gives a matrix \(\widetilde M_2\) indexed by \(\{ 0,\ldots ,Dd-1\} \times \{ 0,\ldots ,dD-1\} \). Its entries are obtained by applying \(M_2\) to the inverse images of the enumerated indices, and

Choose compact singular-value decompositions

with intermediate dimensions \(r\) and \(\ell \), respectively. For \(\rho \in M_{D}(\mathbb {C})\), define the compressed positive metric

The inverse square root \(A^{-1/2}\), rather than \(A\) itself, is the normalization factor used in the first weighted source factor. This is the source-factor construction of [ CPGSV17 , Section III ] .

For a tail of length \(K\), define

The expression contains exactly one normalization factor \(d^{-K}\).

For

define the coefficient indexed by \(l,r,l',r'\) to be

This is the coefficient obtained from the two open-tail expansions in the normalized Gram contraction in the proof of the source-tensor isometry lemma in [ CPGSV17 , Section III ] . The external factor \(u_{(l,r),q}\) is unstarred, while \(u_{(l',r'),p}\) is starred. Thus the displayed coefficient contains the unique normalization factor \(d^{-K}\).

Given source-cut decompositions \(M_1=X_1Y_1\) and \(M_2=X_2Y_2\), define

by

The source legs of \(u\) have order \(\ell \times r\), whereas those of \(v\) have order \(r\times \ell \). These are the two source tensors introduced after the source-cut decompositions in [ CPGSV17 , Section III ] . The entry formulas also fix the conjugate-transpose orientation. A supplied source-factor witness provides these decompositions. The fixed compact-SVD factors specialize the formulas to the previously defined tensors.

Put

with source-bond order \(r\times \ell \). Thus \(Y\) maps \((\mathbb {C}^d\otimes \mathbb {C}^D)^{\otimes 2}\) to \(\mathbb {C}^r\otimes \mathbb {C}^\ell \), and \(Z\) maps in the opposite direction. The regrouping of the open indices is

where \(\operatorname {flat}:\{ 0,\ldots ,D{-}1\} \times \{ 0,\ldots ,D{-}1\} \simeq \{ 0,\ldots ,({-}1\} D^2)\) is the canonical product enumeration. The dressed source-\(v\) Gram equation is

These are the tensorized factors surrounding \(v^\dagger v\) in [ CPGSV17 , Theorem III.8, Section III.B, equations (31)–(32) ] .

For \(\rho \in M_{D}(\mathbb {C})\), let

In the virtual–physical row order \((\alpha ,j)\) of \(M_1\), define

Thus the source expression \(\mathbb {1}_d\otimes \rho \) is first written in the physical–virtual order \((j,\alpha )\) and then reindexed by \(\chi \) to the virtual–physical order \((\alpha ,j)\). The two Kronecker products are not identified without this reindexing. This is the source weight in [ CPGSV17 , equation labelled Y1Y1X1X1 ] .

For a specified tensor \(\mathcal U\) whose right and left source-cut ranks \(r\) and \(\ell \) are positive, define

This definition makes no choice of a simple blocking.

A blocked word of \(L\) product letters separates canonically into two blocked words:

The map decodes the length-\(L\) word, splits every letter into its \(d\) and \(e\) components, and re-encodes the two resulting words.

The source-cut shuffle sends

The analogous map applies to column indices.

Let \(E\) be the normalized diagonal after the first blocking and let \(\mathfrak S\) be its residual nonunital algebra. Define

This is the basis-independent sum of the three forms in equation sprimeforms of [ CPGSV17 , Proposition III.3(ii) ] .

For a block \(k\), define its weighted transfer eigenvalue by

For compression spectral data \((x_i,\mu _i)\), define

For the compression defect blocks \((B_i)_i\), define

The index \(i\) ranges over every vector in the chosen compression eigenbasis, including repeated eigenvalues and the endpoint eigenvalues \(0\) and \(1\).

For orthogonal projections \(P\) and \(Q\) on a finite-dimensional complex inner-product space, let \((x_i)_i\) be an orthonormal eigenbasis of the compression of \(Q\) to \(\operatorname {ran}P\), with ordered real eigenvalues \((\mu _i)_i\).

Let \(P\) and \(Q\) be orthogonal projections on a finite-dimensional complex inner-product space. Define \(\widetilde P\) as the orthogonal projection onto \(\operatorname {ran}(PQ)\).

For an MPO tensor \(\mathcal U=(U^{ij})_{0\leq i,j{\lt}d}\) of bond dimension \(D\) and matrices \(A,B\in \mathbb C^{D\times D}\), define

Let \(\rho \in M_{D}(\mathbb {C})\) be positive definite. Define

These are the factors in [ CPGSV17 , Section III ] .

The MPV overlap between two tensors \(A\) (of bond dimension \(D_1\)) and \(B\) (of bond dimension \(D_2\)) at system size \(N\) is

Diagrammatically, the overlap contracts the MPV ring of \(A\) against the conjugate ring of \(B\) along their shared physical indices, both virtual rings closed by the trace:

This pairing is linear in the first MPV coefficient and conjugate-linear in the second. By Lemma 2.7.4, it is the complex conjugate of the Hilbert-space inner product from Definition 2.7.2.

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 \(\sigma _\partial (\mathcal{E}_A)=\{ 1\} \).

Let \(d{\gt}0\) and let \(\omega \colon \mathcal A\to \mathcal A\) be a star-algebra automorphism. The automorphism \(\omega \) is a one-dimensional quantum cellular automaton if it is translation covariant and has finite forward propagation. Thus there exists a finite set \(\mathcal N\subset \mathbb Z\) such that, for every finite \(\Lambda \subset \mathbb Z\),

This is the definition at line 2298 of [ CPGSV17 ] . It imposes only the displayed forward propagation condition and makes no locality claim for \(\omega ^{-1}\).

For \(d{\gt}0\) and \(L{\gt}0\), forward blocking gives a star-algebra equivalence

This formalizes only the algebraic-local change of site grouping used at lines 2308 and 2313–2320 of [ CPGSV17 ] . It does not assert the QCA-to-MPU converse at line 2308, extend blocking to the quasi-local completion, intertwine translations, provide a propagation estimate or a circuit representation, or transport a quantum cellular automaton.

Let \(L{\gt}0\). The compatible finite blocking maps induce a star-algebra homomorphism

In the reverse direction there is a star-algebra homomorphism

A representative on an arbitrary original region \(\Delta \) is first included into \(\widehat{\operatorname {Hull}_L(\Delta )}_L\) and then finite blocking is inverted. Thus the reverse construction does not require \(\Delta \) itself to be an expanded region.

The finite-region local algebras form a directed system under the maps \(\iota _{\Lambda ,\Gamma }\). Its algebraic direct limit is the algebraic local algebra

For every finite region \(\Lambda \), write \(\iota _\Lambda \colon \mathcal A_\Lambda \to \mathcal A_{\mathrm{loc}}\) for the canonical inclusion. Whenever \(\Lambda \subseteq \Gamma \), these inclusions satisfy

This is the union \(\bigcup _{\Lambda \Subset \mathbb Z}\mathcal A_\Lambda \) in the Appendix of [ CPGSV17 ] . No norm completion is taken.

The normed star algebra of local observables satisfies the C-star axiom. This structure does not include completeness.

For \(d{\gt}0\), the local algebra with the operator norm is a normed complex star algebra. No completeness assertion is made.

Assume \(d{\gt}0\). If a local observable \(A\in \mathcal A_{\mathrm{loc}}\) is represented by \(A_\Lambda \in \mathcal A_\Lambda \) on a finite region \(\Lambda \), define its operator norm by

This does not depend on the finite region or representative. It is the norm whose topology is completed in the Appendix of [ CPGSV17 ] to obtain the quasi-local algebra.

For every \(a\in \mathbb Z\), the homomorphism \(\tau _a\) is a star-algebra automorphism of \(\mathcal A_{\mathrm{loc}}\), and on finite-region representatives

For \(a\in \mathbb Z\), finite-region translations induce a star-algebra homomorphism

determined on every finite-region representative by

Define the finite local image by

It is a star-subalgebra star-isomorphic to \(\mathcal A_\Lambda \). If \(\Lambda +\mathcal N=\Gamma \cup \Delta \) with \(\Gamma \cap \Delta =\varnothing \), then

is the bipartite matrix star-subalgebra to which the left and right support-algebra constructions apply. No support algebra is defined in this statement.

Let \(\Gamma ,\Delta \subset \mathbb Z\) be disjoint finite regions. Writing \(\mathcal C_\Gamma \) and \(\mathcal C_\Delta \) for their configuration spaces, restriction to the two regions gives

with the \(\Gamma \)-configuration first. Reindexing matrix rows and columns gives a star-algebra equivalence

For \(L{\gt}0\) and a finite original region \(\Delta \subset \mathbb Z\), define its block hull by

This is the finite set of length-\(L\) blocks that meet \(\Delta \).

Let \(d,L{\gt}0\), let \(\overline{\mathcal B}_L\colon \mathcal A^{(d^L)}\simeq \mathcal A^{(d)}\) be the quasi-local blocking equivalence, and let \(\omega \) be a star-algebra automorphism of \(\mathcal A^{(d)}\). The induced automorphism of the blocked chain is

Let \(L{\gt}0\) and let \(\mathcal N\subset \mathbb Z\) be a finite original-lattice neighborhood. Write \(\widehat{\{ 0\} }_L=\{ 0,\ldots ,L-1\} \) for the sites in the reference block. Define the induced blocked-lattice neighborhood by

The full reference block is included because the block reached after a displacement can depend on the starting residue. This is the finite-region geometry implicit in the grouping of sites in [ CPGSV17 , lines 2308 and 2313–2320 ] .

Compatible finite-local automorphism data determine a star-algebra equivalence

whose forward and inverse maps are \(f\) and \(g\), respectively.

For each finite region \(\Lambda \), let

be star-algebra homomorphisms such that, whenever \(\Lambda \subseteq \Gamma \),

Let \(f,g\colon \mathcal A_{\mathrm{loc}}\to \mathcal A_{\mathrm{loc}}\) be the maps induced by these compatible families. Compatible finite-local automorphism data require

for every \(A\in \mathcal A_\Lambda \).

The compatible families induce star-algebra homomorphisms

Assume \(d{\gt}0\). The forward local norm hypothesis is

for every finite region \(\Lambda \) and every \(A\in \mathcal A_\Lambda \).

For a finite neighborhood \(\mathcal N\subset \mathbb Z\), the forward family has support bound \(\mathcal N\) if

for every finite region \(\Lambda \) and every \(A\in \mathcal A_\Lambda \).

The forward family commutes with unit translation if, for every finite region \(\Lambda \) and every \(A\in \mathcal A_\Lambda \),

Under the forward local norm hypothesis, the algebraic equivalence extends to a star-algebra equivalence

of the quasi-local completion.

Let \(S\) and \(C\) be finite sets, let \(n=|C|{\gt}0\), and choose an equivalence \(e\colon C\simeq \mathbb Z/n\mathbb Z\) and a primitive \(n\)-th root of unity \(\zeta \). If \(W_{a,b}\) is the corresponding reindexed Weyl operator on \(\mathbb C^C\), define

For on-site dimension \(d\), configurations of \(d^L\)-level blocked spins on \(\Lambda \) are equivalent to configurations of \(d\)-level spins on \(\widehat\Lambda _L\). The blocked label at \(x\) is decoded into its ordered length-\(L\) word, and its \(r\)th letter is placed at \(Lx+r\).

Let \(L{\gt}0\) be a block length and let \(\Lambda \subset \mathbb Z\) be finite. The expansion of \(\Lambda \) is

Every \(z\in \widehat\Lambda _L\) has unique block coordinates \((x,r)\) with \(x\in \Lambda \) and \(0\leq r{\lt}L\). Increasing \(r\) gives the consecutive-site order within the block. This is the finite-region site grouping used in [ CPGSV17 , lines 2308 and 2313–2320 ] ; it does not assert the QCA-to-MPU converse stated at line 2308.

Translation gives bijections

Their inverses translate sites and configurations by \(-a\).

Relabelling configurations defines a star-algebra equivalence

This is the finite-region translation associated with the lattice translation \(i\mapsto i+a\).

Let \(d{\gt}0\) and let \(\omega \colon \mathcal A\to \mathcal A\) be a star-algebra automorphism. The automorphism \(\omega \) has finite forward propagation if there exists a finite set \(\mathcal N\subset \mathbb Z\) such that \(\omega \) propagates within \(\mathcal N\).

Let \(d{\gt}0\), let \(\omega \) be a star-algebra automorphism of \(\mathcal A^{(d)}\), and suppose that \(\omega \) propagates within the finite neighborhood \(\mathcal N\). For every finite region \(\Lambda \), define the finite-region restriction

to be the unique star-algebra homomorphism satisfying

For submodules \(S\subseteq M_I(\mathbb C)\) and \(T\subseteq M_J(\mathbb C)\), define

This is the tensor-product subspace used below, rather than a conjunction of two one-sided containment conditions.

Let \(\mathcal A\subseteq M_{I\times J}(\mathbb C)\) be a star-subalgebra. Its set of left coefficients is

The left support algebra of \(\mathcal A\) is the star-subalgebra

generated by all left coefficients.

Fix a nonnegative integer \(d\). For a finite region \(\Lambda \subset \mathbb Z\), the configuration space and local observable algebra are

The algebra \(\mathcal A_\Lambda \) carries its operator norm. This is the finite-region construction used in the QCA appendix of [ CPGSV17 ] ; no completion is taken here.

Reindexing matrix rows and columns by the configuration blocking equivalence gives a star-algebra equivalence

This is the finite-region equivalence underlying the algebraic-local blocking construction below. No quasi-local completion, compatibility with translations, propagation statement, circuit representation, or transport of quantum cellular automata is asserted here.

If \(\Lambda \subseteq \Gamma \) are finite regions, define

Here product configurations are identified with configurations on \(\Gamma \). This is a unital star-algebra homomorphism.

For \(x\in \mathbb Z\), set

The order \(L_x,P_x\) is the left–right factor order in equation RR2x of [ GNVW12 , lines 1251–1266 ] . The same four-site localization occurs in [ CPGSV17 , lines 2313–2318 ] .

Let \(d{\gt}0\), let \(\omega \colon \mathcal A\to \mathcal A\) be a star-algebra automorphism, and let \(\mathcal N\subset \mathbb Z\) be finite. The automorphism \(\omega \) propagates within \(\mathcal N\) if, for every finite region \(\Lambda \) and every \(x\in \mathcal A\),

For \(d{\gt}0\), the quasi-local observable algebra is the operator-norm completion

The involution on \(\mathcal A\) is the unique continuous extension of the involution on \(\mathcal A_{\mathrm{loc}}\). It makes \(\mathcal A\) a complex C-star algebra, and the canonical star-algebra homomorphism

is the completion map.

For \(d{\gt}0\) and \(L{\gt}0\), site blocking extends uniquely by continuity to a star-algebra equivalence

This is only the completion of the change of site grouping used at [ CPGSV17 , lines 2308 and 2313–2320 ] . This slice does not define scaled translations or carry-aware neighborhoods, prove propagation estimates, construct blocked automorphisms, transport the property of being a quantum cellular automaton, realize a circuit, or reconstruct an MPU from a quantum cellular automaton.

For every finite region \(\Lambda \), define

For \(d{\gt}0\), a quasi-local observable \(x\in \mathcal A\) is supported in a finite region \(\Lambda \subset \mathbb Z\) if it belongs to the image of the canonical embedding \(j_\Lambda \colon \mathcal A_\Lambda \to \mathcal A\). Equivalently,

In particular, \(j_\Lambda (A)\) is supported in \(\Lambda \) for every \(A\in \mathcal A_\Lambda \).

For \(d{\gt}0\) and every \(a\in \mathbb Z\), algebraic-local translation extends to a star-algebra automorphism

of the quasi-local algebra. On the dense algebraic-local subalgebra and on finite-region observables it satisfies

For finite regions \(\Lambda ,\mathcal N\subset \mathbb Z\), define their sumset by

Hence \(j\in \Lambda +\mathcal N\) exactly when \(j=i+a\) for some \(i\in \Lambda \) and \(a\in \mathcal N\). The sumset is monotone in both arguments, either empty argument gives the empty region, and the singleton \(\{ 0\} \) is an identity. A singleton displacement recovers translation,

Translation commutes with either argument, and sumsets are associative:

These identities support composition of propagation bounds. This is the finite region \(\Lambda +\mathcal N\) in the propagation condition of the QCA appendix of [ CPGSV17 ] .

Let \(\mathcal A\subseteq M_{I\times J}(\mathbb C)\) be a star-subalgebra. Its set of right coefficients is

The right support algebra of \(\mathcal A\) is the star-subalgebra

generated by all right coefficients.

Let \(\omega \) be a star-algebra automorphism of the homogeneous quasi-local algebra which propagates within \(\{ -1,0,1\} \). Write the finite image of \(\mathcal A_{E_x}\) in the ordered \(L_x\times P_x\) coordinates as

Define

After the canonical finite-region embeddings, these form one family of star-subalgebras of the quasi-local algebra. Here the matrix embedding is the composition

Under this identification, the canonical local inclusion in [ CPGSV17 , lines 2292–2298 ] is exactly the displayed composition. Formula (874) is equation RR2x in [ GNVW12 , lines 1261–1266 ] . Cirac et al. use the homogeneous even factor in [ CPGSV17 , lines 2322–2328 ] ; the odd factor and unified family are supplied by the GNVW construction.

Scope restriction (homogeneous chain): GNVW allow the cell size \(d(y)\) to depend on \(y\), whereas the present definition fixes one positive size \(d\). The site-dependent statement is not claimed here; see [ con26l ] .

A local observable \(A\in \mathcal A_{\mathrm{loc}}\) is supported in the finite region \(\Lambda \) if it lies in the image of \(\iota _\Lambda \colon \mathcal A_\Lambda \to \mathcal A_{\mathrm{loc}}\).

For a finite region \(\Lambda \subset \mathbb Z\) and \(a\in \mathbb Z\), define

Thus

The map \(\Lambda \mapsto \Lambda +a\) is injective. If \(\Lambda \subseteq \Gamma \), then \(\Lambda +a\subseteq \Gamma +a\).

For \(d{\gt}0\), a star-algebra automorphism \(\omega \) of the quasi-local algebra \(\mathcal A\) is translation covariant if

for every \(a\in \mathbb Z\). This is the covariance condition in the definition of a one-dimensional QCA [ CPGSV17 , line 2298 ] .

Let \(d{\gt}0\), and let \(\omega \) be a star-algebra automorphism of \(\mathcal A\). A finite set \(\mathcal N\subset \mathbb Z\) is a two-sided propagation neighborhood when

and

for every finite \(\Lambda \subset \mathbb Z\). The automorphism has two-sided finite propagation when \(\omega \) and \(\omega ^{-1}\) each have a finite forward propagation neighborhood; the two neighborhoods need not initially agree.

These are derived convenience predicates. The definition in [ CPGSV17 , line 2298 ] and Schumacher–Werner, quant-ph/0405174, Definition 1, both define locality by the forward inclusion for \(\omega \) alone. In particular, these predicates do not assert that forward locality implies locality of the inverse.

For tensors \(A\) and \(B\) of possibly different bond dimensions, we write \(\mathcal{V}^+(A) = \mathcal{V}^+(B)\) if \(|V^{(N)}(A)\rangle = |V^{(N)}(B)\rangle \) for every \(N {\gt} 0\).

Fix a virtual bond dimension \(D\). Two canonical-form tensors \(\mathcal U\) and \(\mathcal V\) are strictly equivalent under a symmetry \(\mathcal S\) if they have the same physical dimension and there is a continuous path \(\mathcal W(p)\) of fixed-bond MPU tensors from \(\mathcal U\) to \(\mathcal V\), not necessarily in canonical form, such that

for every \(p\in [0,1]\) and every applicable chain length \(N\).

Fix a virtual bond dimension \(D\). Let \(\mathcal U\) and \(\mathcal V\) be canonical-form MPU tensors with physical dimensions \(d_a\) and \(d_b\). Given an equality \(h\colon d_a=d_b\), reindex \(\mathcal U\) from \(d_a\) to \(d_b\) along \(h^{-1}\). The tensors are strictly equivalent with respect to \(h\) if there is a continuous path \(\mathcal W(p)\), \(p\in [0,1]\), in the fixed-bond MPU locus such that

Only the canonical-form requirement is relaxed along the path. This is the fixed-bond interpretation of [ CPGSV17 , lines 708–714 ] ; it does not compare unequal raw bond dimensions.

Assume \(d{\gt}0\). Equip \(U_1\) with the tensor product of the two bond-one identity source factorizations. Equip \(U_2\) with the tensor product of the left-shift and right-shift source factorizations, in that order, and equip \(U_3\) with the tensor product in the reverse order. In all three cases the virtual weight is the identity. These are explicit supplied source-factor witnesses; no identification with an arbitrary compact singular-value decomposition is asserted.

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.

The word span at length \(n\) is the subspace

This is  [ SPGWC10 , Section II ] .

For a time-reversal-invariant MPU in standard form, the sign \(\sigma \) is

where \(\mathbb S_{12,34}\) exchanges the pairs of sites \(12\) and \(34\).

Let \(u_+=v_+=\mathbb {1}_{r\ell }\) and \(u_-=v_-=\operatorname {diag}(-1,1,\ldots ,1)\). If at least one of \(r\) and \(\ell \) is odd, \((u_+,v_+)\) and \((u_+,v_-)\) realize the two classes of Theorem 19.9.2.9. If both are even, the four pairs \((u_\pm ,v_\pm )\) realize the four Case I classes. Multiplying by the matrix \(R\) of Corollary 19.9.1.9 realizes the four Case II classes.

The standard forms

realize, respectively,

Assume the standard-form tensors displayed below carry reduced full-support source data and that the comparison paths used to classify them carry admissible path data of blocking length one. Let

If at least one of \(r\) and \(\ell \) is odd, the pairs \((u_+,v_+)\) and \((u_+,v_-)\) represent the two classes of Theorem 19.9.2.10. If \(r\) and \(\ell \) are both even, the four pairs \((u_\pm ,v_\pm )\) represent the four Case I classes of Theorem 19.9.2.14. With

the four pairs \((u'_\pm ,v'_\pm )\) represent the four Case II classes.

Suppose the CZX tensors \(\mathcal U\), \(\mathcal U_2\), and \(\mathcal U_4\), together with the physical-adjoint comparison tensors \(\mathcal U_2^\sharp \) and \(\mathcal U_4^\sharp \), are simple and each carries its own reduced full-support source datum. The CZX MPU is defined by

Its tensor is

It has \(\sigma ^{(1)}=\sigma ^{(2)}=-1\).

For each displayed example, suppose the tensors \(\mathcal U\), \(\mathcal U_2\), and \(\mathcal U_4\), together with the physical-adjoint comparison tensors \(\mathcal U_2^\sharp \) and \(\mathcal U_4^\sharp \), are simple and each carries its own reduced full-support source datum. The standard forms

realize, respectively,

The CZX MPU is defined by

where \(CZ=\operatorname {diag}(1,1,1,-1)\). Let

be the Hadamard matrix. The corresponding tensor is

It has \(\sigma ^{(1)}=\sigma ^{(2)}=-1\).

Given two simple tensors \(\mathcal U\) and \(\mathcal V\) with standard forms, they generate the same MPU for every \(N\) if and only if there are unitaries \(x\), \(y\), and \(z\) relating the two standard forms by the local gauge equations of [ CPGSV17 , Eqns. (SFuu) and (SFvv) ] . This is the unrestricted statement of the source’s Fundamental Theorem of MPU.

The source index is independent of the chosen simple blocking. More precisely, if \(k\geq k_0\) and both \(\mathcal U_{k_0}\) and \(\mathcal U_k\) are simple, then

This is the unrestricted statement of [ CPGSV17 , Proposition IV.2 ] .

For MPU tensors under the source’s standing canonical-form convention, take the index defined from any simple blocking. This index has the following properties.

1. It does not change under blocking.

2. It is additive under tensor products and composition.

3. It is constant under continuous deformations through MPU tensors.

4. Two MPU tensors have the same index if and only if they are equivalent.

This is the unrestricted statement of [ CPGSV17 , Theorem IV.6 ] .

Let \((A_i)_{i\in I}\) span \(M_{D}(\mathbb {C})\), and let \(C:\mathbb {C}^{m_2}\otimes \mathbb {C}^D\longrightarrow \mathbb {C}^{m_1}\otimes \mathbb {C}^D\) satisfy, for every \(i\in I\),

Then there is a matrix \(F:\mathbb {C}^{m_2}\to \mathbb {C}^{m_1}\) such that \(C=F\otimes \mathbb {1}_D\).

Let \(A\) be left-canonical, tensor-irreducible, and have primitive transfer map. Then \(A\) has primitive fixed-point data \((\rho ,P_\rho ,N)\) with \(\rho {\gt}0\) and \(N^n\to 0\). For every blocking length \(P\), the tensor \(A^{[P]}\) is left-canonical, its transfer map is \(\mathcal{E}_A^P\), and \(\rho \) is fixed by this blocked map. For every \(n\ge 1\),

Finally, a channel with a positive-definite fixed point whose positive semidefinite fixed points are all proportional is irreducible as a map; irreducibility of a transfer map implies tensor irreducibility.

Two MPUs described by simple tensors are strictly equivalent under \(\mathcal S\) if and only if their standard-form unitaries are joined by continuous families \(\widetilde u(p)\) and \(\widetilde v(p)\) whose generated MPU is invariant under \(\mathcal S\) for every \(p\in [0,1]\).

For a configuration \(\sigma =(\sigma _0,\ldots ,\sigma _{N-1})\), let \(w_\sigma =[\sigma _0,\ldots ,\sigma _{N-1}]\). Then \(A^{w_\sigma } =A^{\sigma _0}A^{\sigma _1}\cdots A^{\sigma _{N-1}}\).

Let \(\zeta A\) denote the tensor with matrices \(\{ \zeta A^i\} _{i=0}^{d-1}\). Then, for any word \(w\), \((\zeta A)^w = \zeta ^{|w|} A^w\), where \(|w|\) is the length of \(w\).

Let \(B\) be an irreducible MPS tensor of positive bond dimension with \(\mathcal{E}_B(\rho )=r\rho \) for some \(\rho {\gt}0\) and \(r{\gt}0\), and suppose every eigenvalue of \(\mathcal{E}_B\) of modulus \(r\) equals \(r\). Then \(r^{-1/2}B\) is a normal tensor.

Let \(B\) be an irreducible MPS tensor of positive bond dimension, not all of whose matrices vanish. Then there are \(\rho {\gt}0\) and \(r{\gt}0\) with \(\mathcal{E}_B(\rho )=r\rho \).

Let \(E_k:H_k\to \bigoplus _lH_l\) be the canonical inclusion and let \(B=\bigoplus _lB_l\). For distinct \(k\) and \(\ell \), the corresponding inclusions have orthogonal ranges. The identities are

In particular, \(B\) commutes with the range projection \(E_kE_k^\dagger \).

For matrices \(A,B\in \mathbb C^{n\times n}\) and the exchange matrix \(\mathbb S\) on \(\mathbb C^n\otimes \mathbb C^n\),

In particular, if \(U\) is unitary, then

For ket and bra configurations \(\sigma ,\tau \), \(V^{(N)}(\widehat M)_{(\sigma ,\tau )} =H^{(N)}(M)_{\sigma ,\tau }\). If \(p_0=(a,b)\) is the first doubled index and \(\sigma ',\tau '\) are the ket and bra configurations on the remaining \(N\) sites, contraction with \(Y_L\), \(Y_R\), and \(Y_C\) gives, respectively,

If \(\mathcal U\) generates matrix product unitaries and \(N{\gt}1\), then

This is the unitarity equation displayed in [ CPGSV17 , Section III, equation (12) ] .

Suppose there is a positive integer \(k\) such that \(\mathcal U_k\), \(\mathcal U'_k\), and the composition tensor \(\mathcal W_k\) are simple and each carries its own reduced full-support source datum. Suppose in addition that \(\mathcal U_{2k}\), \(\mathcal U'_{2k}\), and \(\mathcal W_{4k}\) are simple and each carries its own reduced full-support source datum. Define the three indices initially from \(\mathcal U_k\), \(\mathcal U'_k\), and \(\mathcal W_k\); the supplied data at \(2k\) and \(4k\) make the later comparison admissible. If \(\mathcal W\) generates the composition of the MPUs generated by \(\mathcal U\) and \(\mathcal U'\), then

This is the six-block reduced-source version of [ CPGSV17 , Theorem IV.6(ii) ] .

Suppose there is one positive blocking length \(k\) such that \(\mathcal U_k\), \(\mathcal V_k\), and \((\mathcal U\otimes \mathcal V)_k\) are simple and each carries its own reduced full-support source datum. Define all three indices from these blocks. Then

This is the common-block reduced-source version of [ CPGSV17 , Theorem IV.6(ii) ] .

Under the hypotheses of Lemma 19.9.2.19, let \(x\) and \(y\) be unitaries and change the standard-form gauge by

Then the trace formula has the same value for \((u',v')\) as for \((u,v)\). Consequently the sign \(\sigma \) is independent of the standard-form gauge.

For a time-reversal-invariant MPU in admissible standard form, suppose its one-site tensor \(\mathcal U\), its actual two-site block \(\mathcal U_2\), and the physical-adjoint comparison tensor \(\mathcal U_2^\sharp \) are simple and each carries its own reduced full-support source datum. Then the sign \(\sigma \) in Proposition 19.9.1.12 is

where \(\mathbb S_{12,34}\) exchanges the pairs of sites \(12\) and \(34\).

Let \(\mathcal U\) be an MPU tensor with physical dimension \(d{\gt}0\) and bond dimension \(D{\gt}0\). Let \(\rho {\gt}0\) satisfy \(\operatorname{tr}(\rho )=1\), put \((\Phi \rvert =\operatorname {vec}(\mathbb {1}_D)^{\mathsf T}\), and suppose that for some \(J{\gt}0\) the normalized transfer matrix satisfies

Let \(r\) and \(\ell \) be the two source-cut ranks, and construct the source factors and \(u\) from \(\mathcal U\) using \(\rho \). Then

Consequently \(u\) is an isometry and \(r\ell \geq d^2\). This is the reduced-source version of [ CPGSV17 , Lemma III.7 ] .

Fix one of the three symmetry comparisons and let \(p\mapsto \mathcal W(p)\) be the actual source-constructed interpolation used in that comparison. Suppose it carries an admissible symmetry-path datum of blocking length one. Thus the path \(\mathcal W(p)\) and the four fixed comparison paths \(\overline{\mathcal W(p)}\), \(\mathcal W(p)^\sharp \), \(\mathcal W(p)^{\mathsf T}\), and \(\mathcal S[\mathcal W(p)]\) are simple and carry continuously varying reduced full-support source data. Suppose also that every one- and two-site endpoint tensor used in the local comparison carries its own datum. Then \(U_i^{(N)}\) and \(U_j^{(N)}\) are in the same admissible phase under time reversal combined with the local swap, transposition combined with the local swap, or conjugation if and only if the corresponding \(\widetilde U_i^{(N)}\) and \(\widetilde U_j^{(N)}\) are in the same admissible phase under time reversal, transposition, or conjugation, respectively.

Fix a positive integer \(m\leq D^4\). The actual \(m\)-site blocked endpoint MPUs \(U_1^{(mN)}\) and \(U_2^{(mN)}\) are admissibly strictly equivalent under \(\mathcal S\) if and only if there are continuous families of unitaries \(\widetilde u(p)\) and \(\widetilde v(p)\) whose endpoints generate \(U_1^{(mN)}\) and \(U_2^{(mN)}\), whose generated MPU is invariant under \(\mathcal S\) for every \(p\in [0,1]\), and whose tensor path and fixed conjugate, physical-adjoint, transposed, and \(\mathcal S\)-image comparison paths carry continuously varying reduced full-support source data. These families arise from the actual blocked path \(\mathcal W(p)_m\) in an admissible equivalence datum of common length \(m\); the blocked path itself is the length-one admissible witness for the displayed endpoint families. When \(m{\gt}1\), this criterion does not assert strict equivalence of the unblocked families \(U_1^{(N)}\) and \(U_2^{(N)}\). This is the reduced-source form of [ CPGSV17 , Lemma following the symmetry-equivalence definitions ] .

Suppose that \(\mathcal U\) generates matrix product unitaries. For every positive integer \(L\), the tensor \(\mathcal U^{[L]}\) obtained by blocking \(L\) adjacent sites also generates matrix product unitaries. This is the blocking operation of [ CPGSV17 , Definition II.5 and equation (9) ] ; preservation of the MPU property under blocking is used in the proof of [ CPGSV17 , Proposition III.3 ] .

The matrix \(M_{i_0}\) from definition 19.4.1 satisfies

• \(M_{i_0}^k = M_{i_0}\) for every \(k\ge 1\),

• \(\operatorname{tr}(M_{i_0}^k) = 1\) for every \(k\ge 1\),

• \(\chi _{M_{i_0}}(X) = X^{\, n-1}(X-1)\).

Let \(\mathcal U\) and \(\mathcal V\) be arbitrary matrix product operator tensors of common physical dimension \(d\). For their product tensor, four-site blocking satisfies

No unitarity or simplicity assumption is required.

Let \(\mathcal W\) generate the composition of the MPUs generated by \(\mathcal U\) and \(\mathcal U'\). Under the source’s standing canonical-form convention,

This is the unrestricted composition statement in [ CPGSV17 , Theorem IV.6(ii) ] .

For MPU tensors under the source’s standing canonical-form convention,

This is the unrestricted tensor-product statement in [ CPGSV17 , Theorem IV.6(ii) ] .

Let \(e\) be the canonical bijection which groups a word of length \(mn\) into \(n\) consecutive words of length \(m\). Iterating blocking at lengths \(m\) and \(n\), then applying this physical reindexing, agrees with direct blocking at length \(mn\):

If \(\mathcal U\) generates matrix product unitaries and \(N{\gt}1\), then

Let \(\mathcal U\) and \(\mathcal V\) be MPO tensors with the same positive physical dimension \(d\), and let

be their composition tensor. Under the canonical identification of its auxiliary space with \(\mathbb {C}^{D_U}\otimes \mathbb {C}^{D_V}\), the normalized flattenings satisfy

This identity combines the raw composition tensor used in the proof of [ CPGSV17 , Theorem IV.6(ii) ] with the normalization in [ CPGSV17 , Section III, equation (13) ] ; it is not a separately stated result of that proof.

Under \(u'=(x\otimes y)u\) and \(v'=v(y^\dagger \otimes x^\dagger )\), the trace formula in Lemma 19.9.2.18 has the same value for \((u',v')\) as for \((u,v)\). Thus \(\sigma \) is independent of the standard-form gauge.

Let \(I\) be a finite set of odd cardinality and let \(A\in \operatorname {Mat}_I(\mathbb C)\) satisfy \(A^T=-A\). Then

Let \(I\) be finite and let \(A\in \operatorname {Mat}_I(\mathbb C)\) satisfy \(A^T=-A\) and \(\det A\ne 0\). Then \(|I|\) is even.

Strict equivalence under a symmetry implies strict equivalence.

Let \(\rho \succeq 0\). If

then

Let \(U=(U^{ij})\) be an MPO tensor, and let \(\widehat U^{ij}=AU^{ij}B\). Then

Let \(\zeta A\) denote the tensor with matrices \(\{ \zeta A^i\} _{i=0}^{d-1}\). Then, for every \(N\) and every configuration \(\sigma = (i_1, \ldots , i_N)\), \(V^{(N)}(\zeta A)_\sigma = \zeta ^{N} V^{(N)}(A)_\sigma \).

If \(V^{(N)}(A)_\sigma = V^{(N)}(A')_\sigma \) and \(V^{(N)}(B)_\sigma = V^{(N)}(B')_\sigma \) for all \(N {\gt} 0\) and all \(\sigma \), then \(O_{AB}(N) = O_{A'B'}(N)\) for all positive \(N\).

Let \(d{\gt}0\) and \(L{\gt}0\). The equivalence agrees with the forward lift, and its inverse agrees with the block-hull unblocking map. For \(A\in \mathcal A^{(d^L)}_\Lambda \) and \(C\in \mathcal A^{(d)}_\Delta \),

In particular, if \(C\in \mathcal A^{(d)}_{\widehat\Lambda _L}\), then

The forward and inverse maps are operator-norm isometries. In particular, each preserves the norm.

Let \(L{\gt}0\). The forward map applies finite blocking to a representative on \(\Lambda \) and regards the result as supported on \(\widehat\Lambda _L\). The reverse map enlarges a representative on \(\Delta \) to the expansion of its block hull, applies the inverse finite blocking equivalence, and regards the result as supported on \(\operatorname {Hull}_L(\Delta )\). Their composites have the orientations

The first identity holds for every on-site dimension \(d\); the second also assumes \(d{\gt}0\).

Let \(d,L{\gt}0\). If \(\mathcal B_L\) denotes the algebraic-local blocking equivalence, then for every \(a\in \mathbb Z\),

This is the translation rule induced by the blocking of sites used in [ CPGSV17 , lines 2308 and 2313–2320 ] .

For \(d{\gt}0\) and every local observable \(A\),

The automorphisms form an additive action of \(\mathbb Z\):

The homomorphisms satisfy

If \(d{\gt}0\), then every translation preserves the operator norm and distance:

A local observable \(A\) is supported in a finite region \(\Lambda \) if and only if \(\tau _a(A)\) is supported in \(\Lambda +a\).

Let \(L{\gt}0\). Every finite original region is contained in the expansion of its block hull,

If \(\Delta \subseteq \Theta \), then \(\operatorname {Hull}_L(\Delta )\subseteq \operatorname {Hull}_L(\Theta )\). Expansion and block hull are exact in the blocked direction:

Let \(L{\gt}0\). For every finite blocked region \(\Lambda \) and every finite original-lattice neighborhood \(\mathcal N\),

This is the exact finite-region carry identity associated with the site grouping in [ CPGSV17 , lines 2313–2320 ] .

Let \(d,L{\gt}0\). If \(\omega \) is a one-dimensional QCA on \(\mathcal A^{(d)}\), then

is a one-dimensional QCA on \(\mathcal A^{(d^L)}\).

For every \(X\in \mathcal A^{(d^L)}\),

The inverse of the blocked automorphism is the blocked inverse:

For every finite \(\mathcal N\subset \mathbb Z\), define

Then \(L_{\mathcal N}{\gt}0\) and

With the finite-supremum convention, \(L_{\varnothing }=1\).

Let \(L{\gt}0\), let \(\mathcal N\subset \mathbb Z\) be finite, and let \(b\in \mathbb Z\). Then

The quotient is the Euclidean quotient by the positive integer \(L\). Consequently these formulas include negative displacements and both positive and negative block carries. In general, \(\operatorname {Hull}_L(\mathcal N)\) alone omits some carries caused by the starting residue. This makes explicit the boundary geometry implicit in [ CPGSV17 , lines 2313–2320 ] .

Let \(L{\gt}0\) and let \(\mathcal N\subset \mathbb Z\) be finite. If \(|n|{\lt}L\) for every \(n\in \mathcal N\), then

Let \(L{\gt}0\) and let \(\mathcal N\subset \mathbb Z\) be finite. If

then

For every \(A\in \mathcal A_{\mathrm{loc}}\),

Under the forward local norm hypothesis, every \(A\in \mathcal A_{\mathrm{loc}}\) satisfies

Under the forward local norm hypothesis, for all \(A,B\in \mathcal A_{\mathrm{loc}}\),

Under the forward local norm hypothesis, every \(A\in \mathcal A_{\mathrm{loc}}\) satisfies

If the finite-local unit-translation formula holds, then every \(A\in \mathcal A_{\mathrm{loc}}\) satisfies

For every finite region \(\Lambda \) and \(A\in \mathcal A_\Lambda \),

Let \(j\colon \mathcal A_{\mathrm{loc}}\to \mathcal A\) be the canonical completion map. For every \(A\in \mathcal A_{\mathrm{loc}}\),

For every finite region \(\Lambda \) and \(A\in \mathcal A_\Lambda \),

For every \(M\in \operatorname {Mat}_{S\times C}(\mathbb C)\),

Let \(S\) be a nonempty finite set. The average is contractive in the operator norm. Moreover, if every complementary Weyl conjugation fixes \(X\), then

Let \(\Gamma ,\Lambda ,\mathcal N\subset \mathbb Z\) be finite. Then

Let \(L{\gt}0\), let \(a\in \mathbb Z\), and let \(\Lambda \subset \mathbb Z\) be a finite blocked region. Then

This is the translation rule for the site grouping used in [ CPGSV17 , lines 2308 and 2313–2320 ] .

If \(\Lambda \subseteq \Gamma \), then for every \(A\in \mathcal A_\Lambda \),

Let \(d\) be a nonnegative integer, let \(\Lambda \subset \mathbb Z\) be finite, and let \(a\in \mathbb Z\). For every \(A\in \mathcal A_\Lambda \),

Consequently, \(\tau _{a,\Lambda }\) is an isometry from \(\mathcal A_\Lambda \) onto \(\mathcal A_{\Lambda +a}\).

If \(\omega \) has finite forward propagation, then \(\omega ^{[L]}\) has finite forward propagation.

Every finite set \(\mathcal N\subset \mathbb Z\) is contained in \([-R,R]\cap \mathbb Z\) for some \(R\in \mathbb N\). This includes \(\mathcal N=\varnothing \), for which one may take \(R=0\). If \(\omega \) propagates within a fixed set \(\mathcal N\), then \(R\) may be chosen so that

Consequently, finite forward propagation admits a symmetric-interval witness:

Translation by zero acts trivially on sites, configurations, and finite-region observables. Successive translations by \(a\) and \(b\) agree with translation by \(a+b\). Pointwise, for \(i\in \Lambda \), \(x,y\in \operatorname {Config}_d(\Lambda )\), and \(A\in \mathcal A_\Lambda \),

These identities express the additive action without identifying the distinct configuration spaces before applying their canonical translation bijections.

Let \(d{\gt}0\), let \(\omega ,\eta \colon \mathcal A\to \mathcal A\) be star-algebra automorphisms, and let \(\mathcal N,\mathcal M\subset \mathbb Z\) be finite. Suppose that \(\omega \) propagates within \(\mathcal N\) and \(\eta \) propagates within \(\mathcal M\). The composite \(\eta \circ \omega \) first applies \(\omega \) and then \(\eta \), so

Hence \(\eta \circ \omega \) propagates within \(\mathcal N+\mathcal M\). In particular, if \(\omega \) and \(\eta \) have finite forward propagation, then \(\eta \circ \omega \) has finite forward propagation. This is the composition closure of the finite-neighborhood condition in Appendix line 2298 of [ CPGSV17 ] .

Let \(L{\gt}0\). For finite blocked regions \(\Lambda \subseteq \Gamma \), the square formed by finite blocking and the two canonical local inclusions commutes:

For \(d{\gt}0\), the map \(\iota _{\Lambda ,\Gamma }\) is injective.

For \(d{\gt}0\), the map \(\iota _{\Lambda ,\Gamma }\) is an operator-norm isometry.

If \(d{\gt}0\), then for every \(A\in \mathcal A_\Lambda \),

For every finite region \(\Lambda \), \(\iota _{\Lambda ,\Lambda }\) is the identity on \(\mathcal A_\Lambda \).

If \(\Lambda \subseteq \Gamma \subseteq \Delta \), then

For \(d{\gt}0\) and every finite region \(\Lambda \), the canonical map

is an operator-norm isometry.

For the neighbourhood \(\mathcal N=\{ -1,0,1\} \),

Let \(\mathcal N\subset \mathbb Z\) be finite. If \(\omega \) propagates within \(\mathcal N\), then \(\omega ^{[L]}\) propagates within the carry-aware blocked neighborhood \(\mathcal C_L(\mathcal N)\). Explicitly, for every finite blocked region \(\Lambda \) and every \(X\in \mathcal A^{(d^L)}\) supported in \(\Lambda \), the observable \(\omega ^{[L]}(X)\) is supported in \(\Lambda +\mathcal C_L(\mathcal N)\).

Let \(d,L{\gt}0\). Suppose that \(\omega \) propagates within the finite neighborhood \(\mathcal N\) and that \(|n|{\lt}L\) for every \(n\in \mathcal N\). For every finite blocked region \(\Lambda \) and every observable \(X\in \mathcal A^{(d^L)}\) supported in \(\Lambda \), the observable \(\omega ^{[L]}(X)\) is supported in

Let \(d{\gt}0\), let \(\omega \colon \mathcal A\to \mathcal A\) be a star-algebra automorphism, and let \(\mathcal N\subset \mathbb Z\) be finite. The following conditions are equivalent:

Thus condition (c) is precisely the range-inclusion formula from line 2298 of [ CPGSV17 ] , expressed inside the quasi-local algebra, and condition (d) is its finite-region witness form.

Let \(d{\gt}0\) and let \(\omega \colon \mathcal A\to \mathcal A\) be a star-algebra automorphism. If \(\mathcal N\subseteq \mathcal M\) and \(\omega \) propagates within \(\mathcal N\), then \(\omega \) propagates within \(\mathcal M\). More explicitly, for every finite region \(\Lambda \),

Consequently, if \(\omega \) has finite forward propagation, then for every prescribed finite set \(\mathcal N\subset \mathbb Z\) there is a finite set \(\mathcal M\subset \mathbb Z\) such that

This is the neighborhood-enlargement consequence of the propagation condition in Appendix line 2298 of [ CPGSV17 ] .

Let \(d{\gt}0\) and \(L{\gt}0\). Write \(j^{(d)}\) for the canonical map from the algebraic-local algebra into its completion. The underlying map of \(\overline{\mathcal B}_L\) is the continuous extension of \(\mathcal B_L\), and on the two dense algebraic-local subalgebras,

For \(A\in \mathcal A^{(d^L)}_\Lambda \) and \(C\in \mathcal A^{(d)}_\Delta \), these identities become

If \(C\in \mathcal A^{(d)}_{\widehat\Lambda _L}\), the inverse formula reduces exactly to

Finally, for all quasi-local observables \(X\) and \(Y\) on the blocked chain,

Let \(d{\gt}0\) and \(L{\gt}0\). If a blocked-chain observable \(X\) is supported in \(\Lambda \), then \(\overline{\mathcal B}_L(X)\) is supported in \(\widehat\Lambda _L\). If an original-chain observable \(Y\) is supported in \(\Delta \), then \(\overline{\mathcal B}_L^{-1}(Y)\) is supported in \(\operatorname {Hull}_L(\Delta )\). In particular, support transport is exact on expanded regions:

Let \(d,L{\gt}0\). If \(\overline{\mathcal B}_L\) denotes the quasi-local blocking equivalence, then for every \(a\in \mathbb Z\),

This is the completion of the translation rule induced by the site grouping in [ CPGSV17 , lines 2308 and 2313–2320 ] .

For every finite region \(\Lambda \), the map \(j_\Lambda \) is an injective isometry. If \(\Lambda \subseteq \Gamma \), then

Let \(d{\gt}0\). If \(x\in \mathcal A\) is supported in \(\Lambda \) and \(\Lambda \subseteq \Gamma \), then \(x\) is supported in \(\Gamma \).

For \(d{\gt}0\), the quasi-local automorphisms form an additive action of \(\mathbb Z\):

For \(d{\gt}0\), every quasi-local translation preserves the operator norm and distance:

Let \(L{\gt}0\). For every finite blocked region \(\Lambda \) and every finite original-lattice neighborhood \(\mathcal N\),

If \(\omega \) is translation covariant, then \(\omega ^{[L]}\) is translation covariant.

For a finite set \(\mathcal N\subset \mathbb Z\), write

Reflection fixes every symmetric interval:

If \(\mathcal N\subseteq \mathcal M\), then every common two-sided bound by \(\mathcal N\) is also a common bound by \(\mathcal M\). More generally, if \(\omega \) propagates within \(\mathcal N\) and \(\omega ^{-1}\) propagates within \(-\mathcal N\), then both propagate within

Finally, any two-sided finite-propagation pair admits a common finite neighborhood and therefore a common symmetric-interval neighborhood. Thus there is an \(R\in \mathbb N\) such that

and

for every finite \(\Lambda \subset \mathbb Z\).

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

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

Let \(x\) be unitary and satisfy \(x=x^T\) or \(x=-x^T\). There are a symmetric unitary \(S\) and a matrix \(\widetilde\Lambda \) such that

In the symmetric case, \(\widetilde\Lambda \) is real and symmetric. In the skew-symmetric case, it is real and skew-symmetric. In both cases, \(\det (\widetilde\Lambda )=1\).

Two MPUs \(U_i^{(N)}\) and \(U_j^{(N)}\) are in the same phase under time reversal combined with the local swap, transposition combined with the local swap, or conjugation if and only if the corresponding \(\widetilde U_i^{(N)}\) and \(\widetilde U_j^{(N)}\) are in the same phase under time reversal, transposition, or conjugation, respectively.

For any MPU tensor \(\mathcal U\) under the source’s standing canonical-form-II convention, let \(r\) and \(\ell \) be the two source-cut ranks and let \(u\) be the source operator. Then

Consequently \(u\) is an isometry and \(r\ell \geq d^2\). This is the unrestricted statement of [ CPGSV17 , Lemma III.7 ] .

An MPU in standard form is invariant under complex conjugation if and only if there are unitaries \(x\) and \(y\) such that either \(x=x^T\) and \(y=y^T\), or \(x=-x^T\) and \(y=-y^T\), and

Suppose \(r\) and \(\ell \) are even. Two conjugation-invariant MPUs in the standard forms of Proposition 19.9.1.7 are strictly equivalent under conjugation if and only if they belong to the same case and

Let \([0,1]\ni x\mapsto \mathcal W(x)\) be a continuous path of tensors generating MPUs, without requiring the path to be in canonical form. For each \(x\), take the canonical form and the two source-cut ranks \(r(x)\) and \(\ell (x)\) of its associated simple tensor. Then \(r(x)\) and \(\ell (x)\) are constant on \([0,1]\).

For a time-reversal-invariant MPU in standard form, the MPS sign \(G\overline G=\pm \mathbb {1}\) equals the sign \(\sigma ^{(2)}\) obtained after blocking two sites.

The MPUs \(U_1^{(N)}\) and \(U_2^{(N)}\) lie in different strict conjugation phases exactly when \(d=4k+2\) for a nonnegative integer \(k\). Otherwise they lie in the same strict phase. All these phases are in Case I, so the two MPUs are equivalent after blocking and adjoining ancillas.

Let an MPU be in standard form, with one-site physical space \(\mathbb C^{d_0}\otimes \mathbb C^{d_0}\). It satisfies \(U^{(N)}=U^{(N)T}\) for every even \(N\) if and only if there are a unitary \(x\) and a phase \(e^{i\phi }\) such that

For a conjugation-invariant MPU, the MPS sign \(G\overline G=\pm \mathbb {1}\) agrees with Case I and Case II, respectively, of Proposition 19.9.1.7.

Let an MPU in admissible standard form be invariant under complex conjugation, and suppose every standard-form tensor compared below carries reduced full-support source data. Its standard-form unitaries can be chosen in exactly one of the following two cases. In Case I,

and the choice is unique up to local orthogonal transformations. In Case II,

where

The choice is unique up to local transformations \(S_x\) satisfying \(S_x^T\Sigma _xS_x=\Sigma _x\). Case II can occur only when \(r\) and \(\ell \) are even.

Let a tensor and its conjugate comparison tensor both be equipped with reduced full-support source data and written in admissible standard form. The generated MPU is invariant under complex conjugation if and only if there are unitaries \(x\) and \(y\) such that either \(x=x^T\) and \(y=y^T\), or \(x=-x^T\) and \(y=-y^T\), and

This is the reduced-source version of the conjugation characterization in [ CPGSV17 , Proposition V.1 ] .

Let \([0,1]\ni x\mapsto \mathcal W(x)\) be a continuous path of tensors generating MPUs, without requiring the path to be in canonical form. Suppose that the path carries an admissible equivalence datum with blocking length \(m\). Then the ranks \(r(x)\) and \(\ell (x)\) of the actual blocked path tensors \(\mathcal W(x)_m\) are constant on \([0,1]\).

The admissible index is independent of the chosen admissible simple blocking. More precisely, if \(k\geq k_0\), both \(\mathcal U_{k_0}\) and \(\mathcal U_k\) are simple, and each is equipped with a reduced full-support source datum, then

Let an MPU be in admissible standard form. Suppose its one-site tensor \(\mathcal U\), its actual two-site block \(\mathcal U_2\), and the physical-adjoint comparison tensor \(\mathcal U_2^\sharp \) are simple and each carries its own reduced full-support source datum. Write the one-site physical space as \(\mathbb C^{d_0}\otimes \mathbb C^{d_0}\). Time-reversal symmetry then forces \(\operatorname {ind}(\mathcal U)=0\). Under these hypotheses, \(U^{(N)}=U^{(N)\dagger }\) for every even \(N\) if and only if there is a unitary \(x\) such that

Let an MPU be in admissible standard form. Suppose its one-site tensor \(\mathcal U\), its actual two-site block \(\mathcal U_2\), and the transposed comparison tensor \(\mathcal U_2^{\mathsf T}\) are simple and each carries its own reduced full-support source datum. Write the one-site physical space as \(\mathbb C^{d_0}\otimes \mathbb C^{d_0}\). Transposition symmetry then forces \(\operatorname {ind}(\mathcal U)=0\). Under these hypotheses, \(U^{(N)}=U^{(N)T}\) for every even \(N\) if and only if there are a unitary \(x\) and a phase \(e^{i\phi }\) such that

Let \(\mathcal U\) be conjugation invariant. Suppose \(\mathcal U\) and its conjugate comparison tensor are simple and carry their own reduced full-support source data. Let \(p\mapsto \mathcal W(p)\) be the actual standard-form interpolation from \(\mathcal U\) to its Case I or Case II representative. Suppose this interpolation carries an admissible conjugation-symmetry path datum of blocking length one: the path \(\mathcal W(p)\) and its fixed conjugate, physical-adjoint, transposed, and conjugation-image comparison paths are simple and carry continuously varying source data. Then the sign \(G\overline G=\pm \mathbb {1}\) in the MPS classification agrees with Case I and Case II, respectively, of Proposition 19.9.1.8.

Let an MPU in admissible standard form be invariant under time reversal. Suppose \(\mathcal U\), \(\mathcal U_2\), and \(\mathcal U_4\), together with the physical-adjoint comparison tensors \(\mathcal U_2^\sharp \) and \(\mathcal U_4^\sharp \), are simple and each carries its own reduced full-support source datum. Then the MPS sign \(G\overline G=\pm \mathbb {1}\) equals the sign \(\sigma ^{(2)}\).

For every blocking length \(k\) under consideration, suppose the endpoint tensor \(\mathcal U_k\), its actual two-site block \(\mathcal U_{2k}\), and the physical-adjoint comparison tensor \(\mathcal U_{2k}^\sharp \) are simple and each carries its own reduced full-support source datum. Let \(\sigma ^{(k)}\) be the sign of the resulting standard form. Then

Suppose a time-reversal-preserving path carries an admissible equivalence datum of blocking length one. At every path point, require the one-site tensor, its actual two-site block, and the physical-adjoint comparison of that two-site block to be simple and to carry continuously varying reduced full-support source data. Then the sign \(\sigma =\pm 1\) is independent of the admissible standard-form gauge and is constant along the path.

After adding one ancilla of dimension \(d\) per site and performing the source blocking, denote the actual enlarged blocked endpoints by \(\widehat{\mathcal U}_1\) and \(\widehat{\mathcal U}_2\). For either time reversal or transposition, let \(p\mapsto \mathcal W_{\mathcal S}(p)\) be the source-constructed interpolation from \(\widehat{\mathcal U}_1\) to \(\widehat{\mathcal U}_2\). Suppose this interpolation carries an admissible symmetry-path datum of blocking length one. Thus the path \(\mathcal W_{\mathcal S}(p)\) and its fixed conjugate, physical-adjoint, transposed, and \(\mathcal S\)-image comparison paths are simple and carry continuously varying reduced full-support source data. Suppose also that the one- and two-site enlarged endpoint tensors used in the local comparison carry their own data. Then \(\widehat{\mathcal U}_1\) and \(\widehat{\mathcal U}_2\) are admissibly strictly equivalent under both time reversal and transposition. No equivalence of the original ancilla-free tensors is asserted, because the equal ancilla dimensions used here need not satisfy the coprimality condition in Definition 19.9.2.6.

For each endpoint tensor \(\mathcal A\) among \(U_1\), \(U_2\), \(\widetilde U_1\), and \(\widetilde U_2\), suppose that \(\mathcal A\), its conjugate comparison \(\overline{\mathcal A}\), its actual two-site block \(\mathcal A_2\), and the conjugate comparison \(\overline{\mathcal A_2}\) are simple and carry their own reduced full-support source data. For every strict comparison used below, let \(p\mapsto \mathcal W(p)\) be the actual source-constructed interpolation between the corresponding endpoint tensors. Suppose it carries an admissible conjugation-symmetry path datum of blocking length one: the path and its fixed conjugate, physical-adjoint, transposed, and conjugation-image comparison paths are simple and carry continuously varying source data. Require a length-one admissible path datum also for the actual blocked interpolation used in the final equivalence claim. Under these hypotheses, \(U_1^{(N)}\) and \(U_2^{(N)}\) are in different admissible strict phases under conjugation exactly when \(k\) is a nonnegative integer and

Otherwise they are in the same admissible strict phase. All these standard forms lie in Case I, so \(U_1^{(N)}\) and \(U_2^{(N)}\) are admissibly equivalent after blocking and adding ancillas.

Suppose the two-site blocked tensors carry reduced full-support source data and the standard-form comparison paths carry admissible path data of blocking length one. Then the actual two-site blocked tensors of \(U_2\) and \(U_3\) are both in the trivial admissible strict phase under conjugation. Consequently \(U_2\) and \(U_3\) lie in the same admissible equivalence class.

For each of time reversal and transposition, let \(p\mapsto \mathcal W_{\mathcal S}(p)\) be the actual source-constructed interpolation from \(\widetilde U_2\) to \(\widetilde U_3\). Suppose it carries an admissible symmetry-path datum of blocking length one. Thus the path \(\mathcal W_{\mathcal S}(p)\) and its fixed conjugate, physical-adjoint, transposed, and \(\mathcal S\)-image comparison paths are simple and carry continuously varying reduced full-support source data. Suppose also that the one- and two-site endpoint tensors used in the local comparison carry their own data. Then \(\widetilde U_2^{(N)}\) and \(\widetilde U_3^{(N)}\) are in the same admissible phase under both time reversal and transposition.

For a time-reversal-invariant MPU in standard form, the sign \(\sigma =\pm 1\) is independent of the gauge and is constant under continuous deformations that preserve time-reversal symmetry.

Let \(\mathcal U\) be a time-reversal-invariant MPU in standard form, and let \(\sigma ^{(k)}\) be the sign obtained from a standard form of its \(k\)-site block. Then

Assume \(d{\gt}0\). For the fixed standard-form source factors, the corresponding unitaries satisfy

The preceding theorems establish these matrices for explicit supplied source-factor witnesses. It remains to identify that supplied choice with the fixed standard-form choice.

After adding one ancilla of dimension \(d\) per site, \(\widetilde U_1^{(N)}\) and \(\widetilde U_2^{(N)}\) are strictly equivalent under both time reversal and transposition.

After blocking two sites, \(U_2^{(N)}\) and \(U_3^{(N)}\) are both in the trivial phase under conjugation.

The MPUs \(\widetilde U_2^{(N)}\) and \(\widetilde U_3^{(N)}\) are in the same phase under both time reversal and transposition.

Let an MPU be in standard form, with one-site physical space \(\mathbb C^{d_0}\otimes \mathbb C^{d_0}\). It satisfies \(U^{(N)}=U^{(N)\dagger }\) for every even \(N\) if and only if there is a unitary \(x\) such that

Let \(\mathcal U\) be a simple tensor under the source’s standing canonical-form-II convention. Its standard form is the depth-two form of the two-site block \(\mathcal U_2\) determined by the source unitaries \(u\) and \(v\). This is the unrestricted definition of [ CPGSV17 , Definition III.9 ] .

The canonical star-algebra homomorphism from a normed star algebra into its completion has dense range.

Let \(A\) be a tensor in CPSV canonical form, with ambient bond dimension \(D\). Then there is a tensor \(B\) of the same bond dimension and an \(G\in \mathrm{GL}(D,\mathbb {C})\) such that, for every physical index \(i\),

Moreover, \(B\) is in canonical form II and the total retained block dimension is unchanged. This is the nonsingular gauge assertion of [ CPGSV16 , Appendix A ] .

Let \(A\) have canonical-form data

For every \(L\geq 1\), \(A^{[L]}\) has canonical-form data with the same coisometry \(U\), the blocked normal tensors \(A_k^{[L]}\), and the weights \(\mu _k^L\):

For a tensor or matrix, an overline denotes entrywise conjugation. If \(A\) and \(B\) are gauge equivalent through \(X\), then \(\overline A\) and \(\overline B\) are gauge equivalent through \(\overline X\). If \(A\) is left-canonical, then \(\overline A\) is left-canonical. If \(A\) is algebraically normal, then \(\overline A\) is algebraically normal. Independently, if \(A\) is a CPSV normal tensor, then \(\overline A\) is a CPSV normal tensor.

Let \(\sigma \) denote entrywise conjugation. For every tensor \(A\), as operators on the matrix space,

evaluating at \(\overline X\) gives, for every matrix \(X\),

If \(\Lambda \) is positive definite and diagonal, then

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 \(e:\{ 0,\ldots ,d'-1\} \to \{ 0,\ldots ,d-1\} \) be a bijection and set

The transfer maps of \(A\) and \(e^*A\) agree. In particular, physical relabeling preserves left-canonical normalization and CPSV normality. If \(A\) has canonical-form data

then \(e^*A\) has the same weights and coisometry and the relabeled normal blocks:

If \(A\) and \(B\) are gauge equivalent, then they generate the same MPV family.

If \(d{\gt}0\), the identity matrix extends to a basis of \(M_{d}(\mathbb {C})\) whose other elements are traceless.

Let \(B\) be an irreducible tensor, let \(X\) be invertible, and let \(c\ne 0\). Then the rescaled conjugated tensor with letters \(cX^{-1}B^vX\) is irreducible. In particular, the rescaled unital gauge of an irreducible tensor is irreducible; this is the normalization of [ CPGSV16 , Section 2.3 ] applied to an irreducible corner.

Let \(D{\gt}0\). If \(A\) is algebraically normal and left-canonical, then

If a finite matrix family satisfies

then every blocked family satisfies the same identity.

Let \(A{\gt}0\) be a square complex matrix and let \(V\) be a possibly rectangular matrix satisfying \(VV^\dagger =\mathbb {1}\). Then

Setting \(B=A\) recovers the ordinary transfer map: \(F_{AA}=\mathcal{E}_A\).

Let \(\varphi :\{ 0,\ldots ,d^2-1\} \to \{ 0,\ldots ,d-1\} ^2\) be the canonical bijection. Then \((K^{[2]})^{i,j} =(K^{[L]})^{\varphi (i),\varphi (j)}\bigm |_{L=2}\). Consequently, for every nonempty chain length \(N\), the closed MPO of \(K^{[2]}\) is obtained from the closed MPO of the general length-two block by applying \(\varphi \) independently at every ket and bra site. In particular, if \(K\) generates matrix product density operators, then so does \(K^{[2]}\).

For every integer \(L\geq 0\), pairing the ket and bra letters site by site gives the canonical identification

Under this identification, the doubled-index MPS tensor of \(M^{[L]}\) is the physical reindexing of the \(L\)-blocked doubled-index MPS tensor of \(M\). In particular, one is injective if and only if the other is.

For every integer \(L\geq 0\) and two MPO tensors \(M,N\) with the same physical dimension, let juxtaposition denote the MPO tensor product obtained by contracting the intermediate physical index. Then \((MN)^{[L]}=M^{[L]}N^{[L]}\). For \(L=0\), each blocked tensor is the unique empty-word local matrix, namely the identity, so the equality reduces to the product of two identity local tensors.

Let \(L\in \mathbb {N}\), and let

concatenate the \(N\) blocked words. For blocked configurations \(\sigma \) and \(\tau \),

The algebraic identity holds for every \(L\in \mathbb {N}\). For \(L\geq 1\), positivity of the original nonempty chain of length \(NL\) shows that blocking \(L\) adjacent physical sites of an MPDO again gives an MPDO. This is the closed-chain identification implicit when the one-site tensor and its two-site blocking are placed in vertical canonical form in [ CPGSV16 , Appendix C.4, lines 1952–2017 ] .

Let \(K\) be an MPO tensor. For every chain length \(N\), the self-overlap of the doubled-index MPS family equals the Hilbert–Schmidt trace of the closed operator:

This is a project derivation preparing the Lemma C.5 route of [ CPGSV16 , Appendix C.2, lines 1473–1499 ] ; the pure-state analogue is the normalized-MPV self-overlap lemma of [ CPGSV16 , lines 1080–1100 ] . It is not a statement of that source. Until a trace normalization is established, the right-hand side is the raw squared Hilbert–Schmidt norm of the MPO family, not a normalized-state purity.

For every chain length \(L\), the operator generated by the product tensor is the product of the generated operators: \(O_L(M\mathbin {\cdot }N)=O_L(M)O_L(N)\).

If \(w\) has length \(n\) and

then

Let \(A\) and \(B\) be CPSV normal tensors, and suppose that

Then \(A\boxtimes B\) is a CPSV normal tensor and is left-canonical:

This is the retained-normal-block infrastructure for the tensoring sentence in the proof of the MPU Index Theorem IV.6(ii) [ CPGSV17 , lines 824–845 ] , not a separate theorem stated there. The retained-normal-block terminology is that of the canonical-form decomposition in [ CPGSV16 , Section 2.3, lines 214–245 ] ; the two left-canonical identities are explicit infrastructure hypotheses, not an additional assertion attributed to that passage.

Let \(S_N(A)=\operatorname{span}_{\mathbb {C}}\{ A^w:|w|=N\} \). The pointwise splitting of length-\(N\) words gives

In particular, if \(S_N(A)=M_{D}(\mathbb {C})\) and \(S_N(B)=M_{E}(\mathbb {C})\), then \(S_N(A\boxtimes B)=M_{DE}(\mathbb {C})\). If \(A\) and \(B\) are algebraically normal, then \(A\boxtimes B\) is algebraically normal.

This is the homogeneous-word infrastructure for the tensoring sentence in the proof of the MPU Index Theorem IV.6(ii) [ CPGSV17 , lines 824–845 ] , not a separate theorem stated there. It applies to the retained blocks in the canonical-form decomposition [ CPGSV16 , Section 2.3, lines 214–245 ] .

Suppose that \(r\) and \(\ell \) are both even and that the standard-form interpolation determined below carries an admissible path datum of blocking length one. Choose admissible standard forms of two conjugation-invariant MPUs as in Proposition 19.9.1.8. They are admissibly strictly equivalent under conjugation symmetry if and only if they belong to the same case of Proposition 19.9.1.8 and

For \(i=1,2\), suppose the original one-site endpoint tensor \(\mathcal U_i\) and its conjugate comparison tensor are simple and each carries reduced full-support source data. Suppose that either \(r\) or \(\ell \) is odd and that the standard-form interpolation determined below carries an admissible path datum of blocking length one. Then \(\mathcal U_1\) and \(\mathcal U_2\) are admissibly strictly equivalent under conjugation symmetry if and only if

Let \(\mathcal U\) and \(\mathcal V\) be simple tensors equipped with reduced full-support source data and written in admissible standard form, with source unitaries \((u_{\mathcal U},v_{\mathcal U})\) and \((u_{\mathcal V},v_{\mathcal V})\). They generate the same MPU for every \(N\),

if and only if their admissible standard forms are related by the local unitary gauges of [ CPGSV17 , Eqns. (SFuu) and (SFvv) ] : there are unitaries \(x\), \(y\), and \(z\) on the three corresponding source and canonical-form-II legs. In particular, the source unitaries obey the local relation

while the two canonical-form-II tensors are related on their virtual leg by \(z\) and on their source legs by \(x\) and \(y\).

Assume the following operation-specific admissibility hypotheses. Every pair of simple blockings compared for blocking independence carries reduced full-support source data. For a tensor product \(\mathcal U\otimes \mathcal V\), there is one positive length \(k\) such that \(\mathcal U_k\), \(\mathcal V_k\), and \((\mathcal U\otimes \mathcal V)_k\) are simple and carry their own data. For a composition tensor \(\mathcal W\) of \(\mathcal U\) and \(\mathcal U'\), there is one positive length \(k\) such that

are simple and carry their own data. Continuous-deformation and equivalence claims use the path data of Definition 19.9.29. Under these hypotheses, the MPU index has the following properties.

1. It does not change under blocking.

2. It is additive under tensor products and composition.

3. It is constant under continuous deformations through MPU tensors.

4. Admissibly equivalent MPU tensors have the same index. Conversely, tensors with the same index are admissibly equivalent whenever the standard-form interpolation is supplied with an admissible path datum.

These are the conclusions of [ CPGSV17 , Theorem IV.6 ] , restricted to the explicit admissibility hypotheses above.

Let \(\mathcal U\) be an MPU tensor equipped with a reduced full-support source datum, and let \(\rho \) and \((\Phi \rvert \) be its normalized fixed pair. Compute the source-cut ranks \(r\) and \(\ell \) and the source operators \(u\) and \(v\) from this same tensor \(\mathcal U\) using \(\rho \). Then the following conditions are equivalent:

1. \(\mathcal U\) is simple;

2. \(r\ell =d^2\);

3. \(u\) is unitary;

4. \(v\) is unitary.

This is the reduced-source version of [ CPGSV17 , Theorem III.8 ] .

Let \(\mathcal U\) generate an MPU, and suppose its direct length-\(k\) block is simple for some \(k{\gt}0\). Then its direct length-\(k'\) block is simple for every \(k'\geq k\), without any divisibility condition on \(k'\) by \(k\). This is the corollary following [ CPGSV17 , Proposition III.3 ] .

For every retained block and physical index \(i\),

The matrices \(X_A\) and \(Z_A\) are mutually inverse, and

Consequently, \(X_A\) is a unit in the matrix algebra, with inverse \(Z_A\), and \(\det (X_A)\) is a unit in \(\mathbb C\).

In the notation of Proposition IV.5 of [ CPGSV17 ] , let \(L\) and \(R\) be the positive semidefinite outer factors with support projections \(P\) and \(Q\), respectively. The required witnesses are

Thus

or equivalently \(XL=P\) and \(RZ=Q\). Here the paper’s matrix \(Z\) is \(X_R\), not the inverse-extension matrix \(Z_R\) from Definition 19.9.57.

If a block has bond dimension \(D\), then the ambient tensor \(A\) is irreducible.

For every blocking length \(L\), separating each blocked product letter by the canonical equivalence \(q\) gives the exact identity

The ordering of the two component words agrees with the independent tensor-product convention.

Assume \(d{\gt}0\), and let \(E\) be the normalized transfer matrix of \(\mathcal U\). After blocking \(L\) sites, the normalized physical diagonal of the double layer is exactly \(E^L\). More precisely, if

is the standard product encoding, then its entries are

Thus the row and column reindexings have the same explicit orientation. If \(\mathcal U\) is an MPU and \(D{\gt}1\), then at the stabilization exponent \(J\) this diagonal is the same reindexing of the rank-one projector \(|\rho )(\Phi |\).

Every MPU of positive physical and bond dimensions has a positive blocking length \(k\leq D^4\) for which the blocked tensor is simple. If \(D{\gt}1\), one may choose

If \(D=1\), the length-one block is simple. Simplicity is preserved under every further iterated blocking by a positive factor and under a bijective relabeling of the physical basis.

For every chain length \(N\), closing the double layer gives

Let \(d,D{\gt}0\). Every bond-\(D\) MPU tensor becomes simple after blocking exactly \(D^4\) sites. For each fixed \(L\), the map

is continuous. Consequently, if \(x\mapsto \mathcal W(x)\) is a continuous family of bond-\(D\) MPU tensors, then \(x\mapsto \mathcal W(x)_{D^4}\) is a continuous family of simple tensors. This is only the common-blocking step in the proof of Proposition 19.9.59; it does not choose reduced representatives or assert constancy of their source ranks.

Let \(M\in M_{m,n}(\mathbb {C})\) be a complex matrix of rank \(r\). There are matrices \(V\in M_{r,m}(\mathbb {C})\) and \(U\in M_{r,n}(\mathbb {C})\) and strictly positive real numbers \(s_0,\ldots ,s_{r-1}\) such that, for \(D=\operatorname{diag}(s_0,\ldots ,s_{r-1})\),

Thus no zero singular values occur in the intermediate space. This is the compact decomposition used in the standard-form construction of [ CPGSV17 , Section III ] .

Let \(\mathcal U\) and \(\mathcal V\) be matrix product unitary tensors with the same physical dimension. Contracting the output index of \(\mathcal U\) with the input index of \(\mathcal V\) gives a matrix product unitary tensor whose periodic operator is \(U^{(N)}V^{(N)}\), in that order. This is the concatenation tensor \(\mathcal W\) in the proof of [ CPGSV17 , Theorem IV.6(ii) ] .

Let \(\mathcal U\) be an MPU with physical dimension \(d{\gt}0\), and let \(N{\gt}1\). Every closed product of \(N\) identity and residual coefficients that contains at least one residual coefficient has zero trace. If the normalized diagonal is rank one, then every nonempty residual word \(P\) satisfies

The same conclusion holds after blocking \(J{\gt}0\) sites whenever the \(J\)th normalized transfer power is \(|\rho )(\Phi |\), with both virtual indices reindexed by \(q^{-1}:\{ 0,\ldots ,D^2{-}1\} \longrightarrow \{ 0,\ldots ,D{-}1\} \times \{ 0,\ldots ,D{-}1\} \).

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

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

For every blocking length \(L\), physical adjunction commutes with blocking, and consequently

The virtual order is unchanged on both sides.

For every physical-index bijection \(e\), taking the double layer commutes with physical reindexing:

If \(Y\) has a right inverse \(Z\), so that \(YZ=\mathbb {1}\), then for every square matrix \(G\) on the row space of \(Y\),

For ket and bra words \(i,j\), physical reindexing gives

Let \(L,R\in \mathbb C^{D\times D}\) and let \(\operatorname {vec}\) stack matrix columns, so

Then the rank-one doubled-virtual matrix satisfies

No adjoint or normalization factor occurs in the contraction result. The binary map \((\mathcal W,E)\mapsto C(\mathcal W,E)\), the map \(\mathcal W\mapsto E(\mathcal W)\), and consequently the map \(\mathcal W\mapsto \widehat{\mathcal W}_{\mathrm{ff}}\) are continuous. These assertions neither choose \(L\) and \(R\) continuously nor infer their positivity. They make no claim about reduced representatives or constancy of ranks.

At every length \(N\), the sitewise configuration bijection identifies

For every system size \(N\), the sitewise configuration bijection identifies the finite-size operator with

Let \(r,\ell \colon X\to \mathbb N\) be lower semicontinuous. Suppose that \(c{\gt}0\) and

for every \(x\in X\). Then both \(r\) and \(\ell \) are locally constant.

Under the hypotheses of Theorem 19.9.49, if \(S\subseteq X\) is preconnected, then for all \(x,y\in S\),

In particular, both functions are constant when \(X\) is preconnected and when \(X=[0,1]\).

Suppose that canonical-form data have full support:

Then every positive physical blocking of these data also has full support.

A bijective relabeling of the physical alphabet preserves full support of canonical-form data.

For a complex square matrix \(A\), write \(G_1(A)\) for its maximal generalized eigenspace at the eigenvalue \(1\). Then

Let \(x{\gt}0\). After adjoining an identity ancilla of dimension \(x\), the normalized flattening is the physical reindexing of the original flattening with a normalized diagonal ancilla alphabet. Its transfer map is exactly the original transfer map. Consequently, chosen canonical-form-II data with full support determine canonical-form-II data with full support for \(\mathcal U^{(x)}=\mathcal U\otimes \mathbb {1}_x\).

The product-index equivalence has the orientation

This result preserves the normalized transfer data. It does not identify the old and new source cuts, ranks, or chosen compact-SVD factors.

Let \(A\) and \(B\) be finite matrices over a field. Then

Let a Lie algebra \(L\) act nilpotently on a finite-dimensional vector space \(V\) over a field \(K\). Write \(C_r(L,V)\) for the \(r\)th term of the lower central series of this action. Then

Suppose that an MPS tensor \(A\) is left-canonical:

Then for every \(L\geq 0\) its blocked tensor is left-canonical:

If \(e\) is a bijection of physical alphabets, then \(e^*A\) is left-canonical if and only if \(A\) is:

If the physical and auxiliary dimensions are \(d\) and \(D\), respectively, then

Let \(\mathcal U\) be an MPU of positive physical and bond dimensions. Suppose that \(\rho \) has trace one and that, for some positive integer \(J\), the normalized transfer matrix satisfies

Put \(K=JD^2\). After the standard product reindexing of the physical coordinates, the direct length-\(K\) block satisfies the contractions (11) and (12) with the specific witnesses \(\rho \) and \(\mathbb {1}\) from this transfer-power identity.

If \(P\) and \(Q\) are orthogonal projections, then \(\widetilde P\) is an orthogonal projection and

There is also an invertible matrix \(Y\) satisfying \(\widetilde P QY=\widetilde P\).

Suppose a length-\(N\) closed tensor equals the physical identity. For arbitrary physical matrices \(X_0,\ldots ,X_{N-1}\),

In particular this applies to the double layer of an MPU whenever \(N{\gt}1\).

Let \(A\) be a not necessarily unital associative subalgebra of \(n\times n\) matrices over a field. If every matrix in \(A\) is nilpotent, then every ordered product of \(n\) matrices from \(A\) vanishes.

Let \(V\) be a finite-dimensional vector space over a field \(K\), and let \(A\subseteq \operatorname {End}_K(V)\) be a not necessarily unital associative subalgebra. If every element of \(A\) is nilpotent, then every ordered product of \(\dim _K(V)\) elements of \(A\) is zero. The empty product in dimension zero is also zero because the identity endomorphism of the zero space is zero.

Let \(R\) be the normalized physical diagonal of the double layer of an MPO tensor. Suppose

Then the two boundaries of the stabilized diagonal tail are fixed:

This is the fixed-pair normalization from the canonical-form-II fixed-point equations in [ CPGSV17 , equations (6a)–(6b) ] and underlies the tail contractions in [ CPGSV17 , Theorem III.8, Section III.B, equations (31)–(32) ] .

Let \(\mathcal W\) be an MPO tensor of physical dimension \(d\), and let

be its normalized physical diagonal. Here and in the two diagonal-tail identities below, scalar inverses are the totalized complex-field inverses, so \(0^{-1}=0\). For every \(K\geq 0\),

At \(K=0\), the word product is the identity matrix. This is the algebraic diagonal-tail expansion used in the source-factor networks of [ CPGSV17 , Theorem III.8, Section III.B, equations (31)–(32) ] .

Let \(\mathcal U\) be an MPO tensor of physical dimension \(d\), and let \(L\geq 0\). For the normalized-tensor interpretation assume \(d{\gt}0\), as in Definition 19.6.3; with the convention \(0^{-1}=0\), the algebraic identity below also holds for \(d=0\). Pairing the ket and bra letters at each site gives a canonical bijection

If \(\widetilde{\mathcal U}=d^{-1/2}\mathcal U\) denotes normalized flattening, then

Thus normalized flattening commutes with blocking, up to the canonical doubled-index relabeling. This is the normalized form of the blocking operation in [ CPGSV17 , Definition II.5 and equation (9) ] .

Let \(U\) be an MPU tensor with positive physical and bond dimensions. Suppose that the normalized flattening is equipped with CPSV canonical-form data whose blocks have full support in the ambient bond space. Then the normalized flattening is a normal tensor.

This is the full-support representative form of [ CPGSV17 , Proposition III.1 ] . The full-support hypothesis is essential for this ambient representative: adjoining a zero virtual direct summand preserves every periodic operator but introduces a nontrivial invariant projection. Thus the theorem does not assert that an arbitrary, possibly nonminimal MPU representative is normal.

Let \(d{\gt}0\), let \(\mathcal U\) be an MPU, and let \(K\geq 0\). For \(p,q\in \{ 0,\ldots ,d{-}1\} \times \{ 0,\ldots ,d{-}1\} \), separate the first two input sites from a common input tail \(\tau \in \{ 0,\ldots ,K{-}1\} \to \{ 0,\ldots ,d{-}1\} \). Then

The global contraction is the input-first equation \((U^{(K+2)})^\dagger U^{(K+2)}=\mathbb {1}\). This is the input-oriented tail contraction supporting the network calculation in [ CPGSV17 , Theorem III.8, Section III.B, equations (31)–(32) ] .

Assume \(d{\gt}0\). For any MPO tensor \(\mathcal U\), tail length \(K\), and \(p,q\in \{ 0,\ldots ,d{-}1\} \times \{ 0,\ldots ,d{-}1\} \), let \(W_{\mathrm{out}}\) be its output-first double layer and let \(E_{\mathrm{out}}\) be the normalized physical diagonal of \(W_{\mathrm{out}}\). Then

The retained letters occur in \((q,p)\) order, so \(q\) is unstarred and \(p\) is starred; their doubled bonds are in \((\text{ket},\text{bra})\) order. This is the normalized closed output tail in [ CPGSV17 , Eq. uUnitary and lines 550–556 ] .

Let \(d{\gt}0\), let \(\mathcal U\) be an MPU, and let \(K\geq 0\). For \(p,q\in \{ 0,\ldots ,d{-}1\} \times \{ 0,\ldots ,d{-}1\} \), separate the first two output sites from a tail \(\tau \in \{ 0,\ldots ,K{-}1\} \to \{ 0,\ldots ,d{-}1\} \). Then

The retained matrix entry has the reversed order \((q,p)\), and the global contraction is the output-first equation \(U^{(K+2)}(U^{(K+2)})^\dagger =\mathbb {1}\).

If the physical dimension satisfies \(d{\gt}0\) and the bond dimension is \(D=1\), then the normalized transfer matrix satisfies

Consequently its powers stabilize at the positive exponent one.

Let \(U\) be an MPU tensor with positive physical and bond dimensions, and let \(E\) be the transfer matrix of its normalized flattening. Then

This is the transfer-matrix conclusion of [ CPGSV17 , Proposition III.1 ] . The first set equality alone does not assert algebraic multiplicity.

Let \(U\) be an MPU tensor with physical dimension \(d{\gt}0\) and bond dimension \(D{\gt}1\), and let \(E\) be the transfer matrix of its normalized flattening. There is an integer \(J\) satisfying

and vectors \(\rho ,\Phi \) satisfying

such that

Moreover, for every integer \(k\geq J\),

Thus blocking \(J\) sites removes the entire generalized zero-eigenspace. The one-dimensional case is stated separately below.

This is the Jordan-elimination step in [ CPGSV17 , proof of Proposition III.2(ii) ] .

Let \(U\) be an MPU tensor with positive physical and bond dimensions, and let \(E\) be the transfer matrix of its normalized flattening. For every integer \(N{\gt}1\),

This is the trace identity in the proof of the normal-tensor result [ CPGSV17 , Proposition III.1 ] .

Suppose \(E^2=E\), \(EAE=0\) for every \(A\in \mathfrak S\), and every ordered product of \(N{\gt}0\) elements of \(\mathfrak S\) vanishes. For scalars \(c_k\) and elements \(S_k\in \mathfrak S\),

Products are taken in increasing site order and need not commute.

Order the left doubled bond as \((\alpha ,\gamma )\) and the right doubled bond as \((\beta ,\delta )\). Then the double layer whose lower tensor is \(\mathcal U^\sharp \) has entries

If \(\widetilde{\mathcal U}\) is in canonical form II, then \(\widetilde{\mathcal U^\sharp }\) is in canonical form II.

If canonical-form-II data for \(\widetilde{\mathcal U}\) have full support, then the data from Definition 19.4.21 also have full support.

The physical adjoint is involutive. If \(\mathcal U\) generates matrix product unitaries, then so does \(\mathcal U^\sharp \), and for every chain length \(N\),

The normalized flattening of the physical adjoint satisfies

Physical adjunction exchanges the two source cuts and conjugates every entry:

Physical adjunction exchanges the two source ranks:

Let \(\mathcal U\) be an MPU with physical dimension \(d{\gt}0\). Every closed product of positive length that contains at least one residual coefficient has zero trace. In particular, this includes a single residual coefficient. This is the mixed-trace assertion preceding equation ESE=0 in [ CPGSV17 , Proposition III.3(ii) ] .

For finite index sets \(M,N\), matrix rank defines a lower-semicontinuous map

Consequently, if \(A\colon X\to \operatorname {Mat}_{M\times N}(\mathbb C)\) is continuous, then \(x\mapsto \operatorname {rank}A(x)\) is lower semicontinuous.

Let \(W\) be the double layer of an MPO tensor, and let \(R\) be its normalized physical diagonal. If

then, for all physical indices \(i,j,k,\ell \),

This is the exact diagonal-word tail between the two retained letters in [ CPGSV17 , Theorem III.8, Section III.B, equations (31)–(32) ] .

Let \(d{\gt}0\), \(D{\gt}1\), and let \(\mathcal U\) be an MPU whose normalized flattening has chosen canonical-form-II data with full support. Put \(J=D^2-1\) and \(W=\mathcal U_{JD^2}\). There is a matrix \(\rho {\gt}0\) with \(\operatorname{tr}(\rho )=1\) such that

With the corresponding vectorized pair \(\rho ',\Phi '\), the same blocked tensor \(W\) satisfies the two aligned simple contractions

This theorem supplies only the fixed pair and the two simple contractions. It does not assert a source-cut factorization or source-\(v\) isometry.

Let \(d,D{\gt}0\) and let \(U\) be an MPU tensor. There are \(D_{\mathrm{red}}{\gt}0\) and an MPU tensor \(U_{\mathrm{red}}\) of bond dimension \(D_{\mathrm{red}}\) such that, for every integer \(N{\gt}0\),

The normalized doubled-index tensor of \(U_{\mathrm{red}}\) has canonical-form-II data whose retained block dimensions sum to \(D_{\mathrm{red}}\). No equality is asserted at length zero, and no virtual gauge is asserted between the two potentially unequal-dimensional bond spaces.

Suppose \(d{\gt}0\), \(D{\gt}1\), and \(\mathcal U\) is an MPU whose normalized flattening is presented by chosen canonical-form-II data with blocks filling the bond space. There is a unique block \(k\). Normalize its diagonal positive definite fixed matrix \(\Lambda _k\) to have trace one, and let \(V_k\) be its inclusion into the ambient bond space. Then

where \((\Phi \rvert \) is the vectorized identity. Consequently,

The diagonal matrix is the block matrix \(\Lambda _k\) in the canonical-form-II fixed-point equations in [ CPGSV17 , equations (6a)–(6b) ] ; its ambient transport \(\rho \) need not be diagonal. This is a statement about the chosen reduced representative and the blocking step in [ CPGSV17 , proof of Proposition III.2(ii) ] ; it does not assert that every ambient MPU tensor is itself in canonical form II.

Let \(d,D{\gt}0\), let \(U\) be an MPU tensor of physical dimension \(d\) and bond dimension \(D\), and let \(\widetilde U=d^{-1/2}U\) be its normalized doubled-index tensor. There are an integer \(D_{\mathrm{red}}{\gt}0\), a normal tensor \(A_{\mathrm{red}}\) of bond dimension \(D_{\mathrm{red}}\), and canonical-form-II data for \(A_{\mathrm{red}}\) such that, for every integer \(N{\gt}0\) and every word \(\sigma \) of length \(N\),

This organizes the dimension-bounded canonical replacement used in [ CPGSV17 , lines 257–294 and 319–326 ] . Its irreducible reduction and canonical-form-II gauge are those of [ CPGSV16 , lines 195–255 and 1058–1077 ] . The shifted transfer argument is the one used in [ CPGSV17 , lines 344–356 ] .

Let \(W=(W^{ij})\) be an MPO tensor. Let \(L,R\succeq 0\) satisfy

and let \(P\) and \(Q\) be their support projections. Assume, letter by letter,

Set \(T=\widetilde P\) and \(\widetilde W^{ij}=TW^{ij}T\). Then, for every \(N\geq 0\),

Moreover, if \(V^\dagger V=I\) and \(VV^\dagger =T\), then

Let \(W=(W^{ij})\) be an MPO tensor, let \(L\), \(R\), \(X\), \(Z\), \(Y\) be square matrices of the same size, let \(P\) be an orthogonal projection, let \(Q\) be a square matrix of the same size, and set \(\widetilde P\) equal to their reduced projection. Assume, letter by letter, \(W^{ij}P=W^{ij}=QW^{ij}\). Set \(\widehat W^{ij}=LW^{ij}R\) and \(\widetilde W^{ij}=\widetilde P W^{ij}\widetilde P\). If \(X,Z,Y\) satisfy \(XL=P\), \(RZ=Q\), and \(PQY=\widetilde P\), then

If in addition \(X\), \(Z\), and \(Y\) are invertible, then

If \(L,R\succeq 0\) and \(P=\operatorname {supp}(L)\), \(Q=\operatorname {supp}(R)\), such invertible matrices may be chosen as \(X=L^++(\mathbb {1}-P)\), \(Z=R^++(\mathbb {1}-Q)\), together with an invertible right factor \(Y\) of the reduced projection.

The doubled-bond equivalence and the blocked-word reversal are involutions. The latter sends the letter in position \(k\) to the letter in position \(K-1-k\) and reverses the encoded word.

Let \(R(\mathcal U)\) denote the normalized physical diagonal of the double layer of an MPO tensor \(\mathcal U\), and let \(s(a,b)=(b,a)\) be the doubled-bond swap. Then

This identity requires neither the MPU property nor a positivity or positive-dimension hypothesis. It is a coordinate identity supporting the calculation in the proof of [ CPGSV17 , Lemma III.7 in Section III.B ] , not a separate theorem printed there.

Let \(\mathcal U\) be an MPU with \(d,D{\gt}0\). Let \(E\) be the transfer matrix of its normalized flattening, and suppose

where \(\rho {\gt}0\), \(\operatorname{tr}(\rho )=1\), and \(J{\gt}0\). Set \(K=JD^2\). For \(n\in \{ K,K+2\} \), let \(\widehat W_n\) be the double layer of the \(n\)-site block of \(\mathcal U^\sharp \). Then, for all blocked physical indices,

The same supplied vectors occur at both lengths; the \(K+2\) identities do not replace them by existentially chosen witnesses.

Let \(\mathcal U\) be an MPO tensor. Write \(E\) for the transfer matrix of its normalized flattening, \(E^\sharp \) for that of \(\mathcal U^\sharp \), and \(E_{\mathrm{out}}\) for the normalized diagonal of the double layer of \(\mathcal U^\sharp \). If \(d{\gt}0\) and

then

No MPU, positivity, trace, positive bond-dimension, or positivity of \(J\) is needed for this implication.

Suppose in addition that \(D{\gt}0\) and \(\operatorname{tr}(\rho )=1\), and set \(K=JD^2\). Then the normalized diagonal \(R_K\) of the double layer of the \(K\)-site block of \(\mathcal U\) satisfies

More generally, if \(d{\gt}0\), \(\rho {\gt}0\), and the normalized diagonal \(R_L\) of an \(L\)-site block satisfies \(R_L=|\rho )(1|\), write \(R_L^\sharp \) for the normalized diagonal of the corresponding block of \(\mathcal U^\sharp \). Then

Consequently, under the combined assumptions \(d,D{\gt}0\), \(\rho {\gt}0\), \(\operatorname{tr}(\rho )=1\), and \(E^J=|\rho )(1|\),

Here all rank-one operators use the column-stacking identification \(M_D(\mathbb C)\cong \mathbb C^{D^2}\).

Let \(I\) and \(J\) be finite sets, let \(e\colon I\to J\) be a bijection, and let \(A\) be a unitary matrix indexed by \(I\). The matrix indexed by \(J\) whose \((j,k)\) entry is

is unitary.

If \(\mathcal U\) is an MPU with positive physical dimension, then every nonempty product of residual generators has zero trace. Consequently, for every \(A\in \mathfrak S\) and every integer \(N{\gt}1\),

and every element of \(\mathfrak S\) is nilpotent.

Let \(\mathcal U\) be an MPU of bond dimension \(D\) and positive physical dimension. Every ordered product of exactly \(D^2\) elements of \(\mathfrak S\) vanishes. In particular, for arbitrary residual generators,

This is equation Sprime=0 in [ CPGSV17 , Proposition III.3(ii) ] , with the corrected exact length \(D^2\) in place of the unsupported strict intermediate bound.

If the physical and auxiliary dimensions are \(d\) and \(D\), respectively, then

For every complex square matrix \(A\),

Let the first-block tensor be an MPU of bond dimension \(D{\gt}0\). Assume its normalized double-layer diagonal \(E\) is a rank-one idempotent. After a second blocking of exactly \(D^2\) sites, every residual coefficient belongs to \(\mathcal T\).

The conclusion is membership in the sum of the three subspaces. It does not assign one exclusive type to a coefficient in an arbitrary physical basis.

Let \(\mathcal V\) be an MPU of positive physical and bond dimensions whose normalized double-layer diagonal is

Then blocking \(\mathcal V\) by exactly \(D^2\) sites produces a simple tensor with witnesses \(a=\Phi \) and \(b=\rho \).

For a matrix product tensor \(A\), a complex number \(c\), and a nonnegative integer \(N\), scaling every local matrix by \(c\) gives

Consequently, if \(d{\gt}0\) and \(U\) is an MPO tensor of physical dimension \(d\), then

Let \(A\) be the right-shift tensor and let \(z_{(\alpha ,i)}=\delta _{\alpha i}\) be the vectorized identity. Its raw source cuts are

For the left-shift tensor \(A^\sharp \) they are

In every physical dimension, including \(d=0\),

If \(d{\gt}0\), the remaining two ranks are

These are the raw source-cut quantities from [ CPGSV17 , Section III ] ; no choice of source factors is involved.

For every \(N{\gt}0\), the tensors in Definition 19.9.3.7 generate \(T^{(N)}\) and \(T^{(N)\dagger }\), respectively. Both tensors are matrix product unitaries.

Let \(A\) be an \(n\times n\) complex matrix. If \(\operatorname{tr}(A^k)=1\) for every integer \(k{\gt}1\), then

Thus \(1\) is the sole nonzero eigenvalue and has algebraic multiplicity one. This is the exact shifted-moment conclusion of [ CPGSV17 , Proposition III.1 ] .

Let \(A\) be an \(n\times n\) complex matrix such that \(\operatorname{tr}(A^k)=1\) for every \(k{\gt}1\), and let \(m{\gt}1\). Then \(\operatorname{tr}((A^m)^r)=1\) for every \(r\ge 1\).

Let \(A\) be an \(n\times n\) complex matrix. If \(\operatorname{tr}(A^k)=1\) for every integer \(k{\gt}1\), then

Thus \(1\) occurs and every nonzero spectral value equals \(1\). This is the set-spectrum consequence of [ CPGSV17 , Proposition III.1 ] ; set equality alone does not assert algebraic multiplicity.

Let \(T:M_D(\mathbb C)\to M_D(\mathbb C)\) be a linear map with \(D{\gt}0\). If \(\operatorname{tr}(\widehat T^{\, N})=1\) for every integer \(N{\gt}1\), then

This is the radius-and-primitivity part of [ CPGSV17 , Proposition III.1 ] .

Let \(A\) have literal CPSV canonical-form data with \(r\) blocks. If its transfer matrix \(E\) satisfies \(\operatorname{tr}(E^N)=1\) for every integer \(N{\gt}1\), then

This is the block multiplicity consequence used in [ CPGSV17 , Proposition III.1 ] .

Let \(A\) be an \(n\times n\) complex matrix such that \(\operatorname{tr}(A^k)=0\) for every \(k{\gt}1\), and let \(m{\gt}1\). Then \(\operatorname{tr}((A^m)^r)=0\) for every \(r\geq 1\). This auxiliary result supports the generalization below to a weaker hypothesis; the source itself supplies vanishing for every positive exponent.

Let \(A\) be an \(n\times n\) complex matrix. If \(\operatorname{tr}(A^N)=0\) for every integer \(N{\gt}1\), then \(A\) is nilpotent and

The source obtains the all-positive trace hypothesis in [ CPGSV17 , proof of Proposition III.3 ] ; its specialization is the elementwise nilpotence assertion used before the nil-matrix theorem. The result above is a stronger generalization because it assumes vanishing only for \(N{\gt}1\). It is useful independently and requires no positivity, normality, or diagonalizability.

For every MPU of positive physical and bond dimensions there are normalized left and right fixed vectors \(\Phi \) and \(\rho \) of the transfer matrix such that the unblocked double layer satisfies

The unblocked tensor need not satisfy the rank-one insertion identity.

If \(\mathcal U\) is simple and \(N{\gt}1\), then

Every simple tensor generates matrix product unitaries. This is [ CPGSV17 , Proposition III.3(i) ] .