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 18.9.1.14.

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 [ 16 , 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 18.9.1.15.

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 18.9.1.15, one can write

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

Under the hypotheses of Theorem 19.5.10.2, suppose in addition that

Then there exist unitaries \(W_1\in \operatorname {Mat}_r(\mathbb {C})\) and \(W_2\in \operatorname {Mat}_\ell (\mathbb {C})\) such that, for \(i=1,2\),

This is the isometrically normalized specialization of [ 197 , lines 1052–1066 ] ; the additional isometry hypotheses remove the scalar freedom.

If \(z\in \mathcal A\) satisfies \([z,R]=0\) for every \(y\in \mathbb Z\) and every \(R\in \widehat{\mathcal R}_y\), then \([z,A]=0\) for every \(A\in \mathcal A\). This is the step from commutation with all \(\mathcal R_y\) to commutation in the quasi-local algebra at [ 203 , lines 1279–1282 ] ; see [ 64 ] .

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 18.9.1.14, 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,

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

where

The choice is unique up to local symplectic transformations \(S_x\) satisfying \(S_x^T\Sigma _xS_x=\Sigma _x\). 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 [ 13 , Section 2.3 ] if there are positive integers \(D_1,\ldots ,D_r\), normal tensors \(A_k\) of bond dimension \(D_k\), nonzero weights \(\mu _k\in \mathbb {C}\), and, writing \(S=\sum _kD_k\), a matrix \(U\in \mathbb {C}^{S\times D}\) such that \(S\le D\) and, for every physical index \(i\),

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

The convention in line 246 of [ 13 ] is a separate normalization of these data, not part of canonical-form membership:

Every weight \(\mu _k\) is nonzero. This is the reading of [ 13 , Section 2.3 ] adopted here: the remark in line 219 there places the degenerate case in the block dimensions (\(D_k=0\)), and lines 224–225 choose each \(\mu _k\) so that the transfer map of \(\mu _kA_k\) has spectral radius one, which a vanishing weight cannot satisfy. The reading loses nothing, since if \(\mu _{k_0}=0\) then \(\mu _{k_0}A_{k_0}^i=0\) is the zero matrix of size \(D_{k_0}\) and

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

The basis-of-normal-tensors refinement is Definition 8.1.1; it is not part of canonical form itself.

In canonical form II of [ 13 , 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. In particular, a normal left-canonical tensor of positive bond dimension with a positive definite diagonal fixed matrix has a one-block presentation with weight one and no omitted bond coordinates. The line-246 convention (29) 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 \(\mathcal S\).

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

As in [ 16 , lines 716–724 ] , canonical form of the blocked tensors with ancilla is a condition of the definition.

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 \(\mathcal U\) be a tensor in canonical form generating matrix product unitaries, let \(k\geq 1\) 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

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

This is the definition of [ 16 , Definition IV.1 ] , stated there for a tensor in canonical form with the ranks of its simple blocking, after the source has assumed without loss of generality that the tensor is in canonical form II (lines 294 and 356). The index of every matrix product unitary is taken through a canonical-form-II representative; for a tensor in canonical form this value is the displayed expression in the ranks of its own blocking, since these ranks are invariant under the similarity to canonical form II.

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

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

Equivalently, its Kraus map is trace-preserving.

For \(a\in \mathbb Z\) and \(N\geq 0\), put \(I(a,N)=\{ a,a+1,\ldots ,a+N-1\} \). The site numbering \(i\mapsto a+i\) identifies configurations on \(I(a,N)\) with length-\(N\) configurations. Reindexing both matrix coordinates gives a star-algebra equivalence \(C_{a,N}:\mathcal A_{I(a,N)}\to M_{d^N}(\mathbb C)\). Explicitly, \(C_{a,N}(X)_{\sigma \tau }=X_{\widetilde\sigma \widetilde\tau }\), where \(\widetilde\sigma (a+i)=\sigma _i\). This includes \(N=0\) and the one-dimensional algebra on the empty interval.

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 [ 13 , Section 4.5 ] , this joins two vertically read tensors along an original horizontal bond. It is the tensor written \(M_\alpha M_\beta \) in [ 13 , 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. The doubled-bond coordinate exchange \((a,b)\mapsto (b,a)\), an involution of the doubled virtual space, is recorded here as the coordinate map accompanying this definition. 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 [ 13 , lines 317–345 ] . It is a local tensor, not the closed-chain operator \(O_L(M)\) of [ 13 , lines 962–967 ] .

For a one-site matrix \(V\), write \(V^{\otimes N}\) in the configuration basis as

If the copies of the \(j\)-th normal tensor have common coefficient \(\mu _j\), set \(\mathcal K_j=\mu _j A_j\).

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

This is the blocking used in [ 13 , Theorem 4.9, lines 851–856 ] .

View the MPO tensor \(M\) vertically as the family of physical-space operators \(M_{ab}\) of Definition 16.4.4, indexed by the virtual indices, with matrix elements \((M_{ab})_{ij}=M^{ij}_{ab}\). A matrix \(P\in M_{d}(\mathbb {C})\) then acts on the tensor by

and on an \((N+1)\)-site chain by the first-spin operator \(P\otimes \mathbb {1}^{\otimes N}\). On the letters of the vertically viewed tensor these actions are the left and right matrix products, \((PM)_{ab}=PM_{ab}\) and \((MP)_{ab}=M_{ab}P\). The first-spin operators compose site by site, \(P_1Q_1=(PQ)_1\).

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) [ 16 , 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 [ 13 , Section 2.3, lines 214–245 ] .

Let \(A\) and \(B\) have the same physical alphabet and respective bond dimensions \(D_A\) and \(D_B\). A reduction from \(B\) to \(A\) is a pair \(V\in \mathbb {C}^{D_A\times D_B}\) and \(W\in \mathbb {C}^{D_B\times D_A}\) such that

for every word \(\mathbf a\), including the empty word. This is the reduction of [ 221 , Proposition 20 and the following definition ] .

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 [ 16 , Section III, equation (10) ] .

The remaining sites outside two adjacent child packets and the joining flags are identified with the sites outside the parent packet. Join the two physical input configurations and transport the arbitrary remaining configuration along this site bijection.

Fix one minimal cut basis and balanced metric family. An interval circuit implementation of budget \(K\) consists of a full logical unitary supported on the interval and a neighboring-pair circuit of at most \(K\) gates. The circuit implements that logical unitary with the common workspace returned to zero on every logical input. Its initialized interval columns are the actual weighted interval columns, including arbitrary outside configurations.

Fix the minimal cut bases and one balanced metric family of the given unitary. The input of an interval consists of its physical configuration with all interval auxiliaries initialized to zero, together with an arbitrary configuration outside the interval. Its output consists of the weighted interval amplitudes in the prescribed cut-label encodings, with all strictly interior auxiliaries zero and the outside configuration unchanged. An interval implementation satisfies this complete column identity.

At an internal cut, order the joining packet as the left child’s right bond register, the right child’s left bond register, the dilation flag, and the attenuation flag. Its consecutive site inclusion is determined by the allocated logical registers.

Combine the prescribed child output inclusions on their disjoint interval supports with an arbitrary outside basis inclusion. Setting a selected set of outside coordinates to zero gives its initialized complement inclusion.

For a one-site interval, identify its consecutive packet with the physical site followed by its right and left boundary bond registers. The induced configuration bijection preserves the prescribed physical and boundary coordinates.

Split the physical configuration of an interval \([j,k)\) into its two consecutive restrictions to \(\lbrack j,m)\) and \(\lbrack m,k)\).

Include the physical chain into the logical register with its auxiliaries zero. Include the logical register into the complete register with the common scratch pool zero. Their composition includes the physical chain with every auxiliary zero.

Regroup the physical outputs, outer cut labels, and outside configurations before the two joining labels. Pull a basis inclusion back along a site bijection to place the joining packet at the end.

Give every leaf cost \(K\). At a node whose children have costs \(T_1,T_2\), let \(T_0=T_1+T_2+M\) and assign cost \(\ell (1+2T_0+B+L)+T_0\), where \(M\) bounds the joining operation, \(B,L\) bound the two reflections, and \(\ell \) is the number of amplification rounds at that node. Define \(C=(2D+1)M+D(1+B+L)\).

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 18.9.69 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 [ 16 ] ; the datum itself is introduced here and has no source-labelled counterpart.

Let \(\mathcal U\) be an MPU in canonical form II. Choose any positive blocking length \(k\) for which \(\mathcal U_k\) is simple, and let \(r_k\) and \(\ell _k\) be its right and left source-cut ranks. Such a blocking exists, and both ranks are positive. Define

Since \(r_k\ell _k=d^{2k}\), one also has

This is the canonical-form-II representative case of [ 16 , Definition IV.1 ] .

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 the invertible matrices for Proposition IV.5 of [ 16 ] .

Fix a positive block length \(L\) and an integer anchor \(a\). The number of sites from \(a\) to the next globally aligned block boundary is \(q(a,L)=(-a)\bmod L\). If an interval of length \(N\) has the decomposition \(N=p+mL+s\), its configuration may be specified by a prefix word of length \(p\), a word of \(m\) blocked letters, and a suffix word of length \(s\). Their ordered concatenation defines a configuration on \(I_{a,N}\), including when \(m=0\). This concatenation is a bijection from triples of such words to configurations on the interval.

Fix \(k,p,s\geq 0\) and an integer anchor \(a\). On the interval of length \(N=p+3(2k)+s\), distinguish three consecutive complete blocks of length \(2k\) between an arbitrary prefix and suffix. Let \(H=d^k\). On the output side, the four neighboring half-blocks form a factor \(X\) of dimension \(H^4\); the prefix, suffix, and two exposed outer half-blocks form the exterior factor \(E_{\mathrm{out}}\). On the input side, the central complete block forms a factor \(J\) of dimension \(H^2\); the prefix, suffix, and the two outer complete blocks form \(E_{\mathrm{in}}\). Ordered concatenation gives bijections from \(E_{\mathrm{out}}\times X\) and \(E_{\mathrm{in}}\times J\) to the respective configuration bases of the same finite interval. Write \(M_{a;p,k,s}\) for the finite-chain MPO matrix in these two bases.

For positive \(P\) and its dual \(Q=P^{-1}/r\), the joining contraction is \(C=(\sqrt Q)^{-1}(\sqrt P)^{-1}\).

Write the two child isometries as a tensor product and regroup their outputs into physical coordinates, outer bond coordinates, and the two joining bond coordinates. Apply the normalized bond-reset dilation only to the joining coordinates and its binary flag. The remaining coordinates carry the identity.

An interval of length one is a leaf. Split every longer interval at its midpoint into consecutive children of lengths \(\lfloor n/2\rfloor \) and \(\lceil n/2\rceil \). Label each internal node by its actual joining cut. A consecutive interval partition records these nonempty child intervals and their single-site leaves.

Two injective basis encodings with the same image determine a unique bijection between their label sets, carrying each encoded configuration to itself.

For a blocking length \(k\), the blocked symmetry has applicable lengths

Its action is obtained by conjugating \(\mathcal S_{d,Nk}\) along the canonical identification \((\mathbb C^{d^k})^{\otimes N}\cong (\mathbb C^d)^{\otimes Nk}\).

For a joining bond of dimension \(r\), use a qudit packet of \(\max \{ 2,\lceil \log _d(2r^2)\rceil +1\} \) sites to encode the two copies of the joining-pair space in the dilation.

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

such that \(\sum _k D_k=D\) and

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 [ 16 , lines 259–267 ] ; normal tensors form a strictly stronger class. The nonzero-weight and full-support conventions are recorded in [ 77 , 58 ] .

Let \(\mathcal U\) be an MPO tensor, and write \(\widetilde{\mathcal U}=d^{-1/2}\mathcal U\) and \(E=\mathcal E_{\widetilde{\mathcal U}}\). A canonical-form-II presentation of \(\mathcal U\) consists of the MPU condition, a literal CPSV canonical-form-II representation

The ambient fixed pair has \(\Phi =\mathbb {1}_D\) and positive coefficients \(\rho _0,\ldots ,\rho _{D-1}\). It satisfies

Thus the retained blocks have literal full support, and \(\rho \) is a positive diagonal trace-one right fixed point in the ambient bond coordinates. This is the standing convention of [ 16 , lines 269–281, 344–356, and 488–502 ] .

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 18.6.9. Theorem 18.6.10 preserves the left-canonical identity under both operations. For each retained block, Lemma D.4.9 gives

where (285) 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 (279).

Choose injective placements of the local logical sites in the full logical register and of the local scratch sites in the common scratch pool. Their disjoint union places the entire local circuit, preserving the logical and scratch blocks.

A one-site interval output consists of a physical letter and two bond labels. Encode the two labels separately in their assigned registers, allowing width zero at an endpoint cut. Its initialized input contains only the physical letter, with both bond registers in zero.

The ambient operator \(U\) implements the logical operator \(Z\) with reusable initialized workspace when \(UJ=JZ\). This identity applies to the entire logical space.

For \(a\geq 1\), compute the prefix conjunction, change the phase by minus one when the final temporary coordinate is one, and undo the conjunction. Its gate budget is \(K(d,a)=8c_da^2+4a\).

Let \(\mathcal U\) and \(\mathcal V\) have physical dimensions \(d_a\) and \(d_b\). They are coherently equivalent under a dimension-indexed symmetry family \(\mathcal S\) if they are MPUs and 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 the \(k\)-site blocking of the supplied enlarged-dimension symmetry. The family obeys the identity-ancilla semiconjugacy law, and its applicable blocked lengths are precisely the lengths \(N\) for which \(Nk\in A\). The ancillas are attached before the common blocking.

A coherent symmetry family supplies an action \(\mathcal S_{d,N}\) for every physical dimension \(d\), with one common set \(A\) of applicable lengths. For an ancilla dimension \(x\), let

where \(R_{d,x,N}\) is the canonical sitewise identification of the \(d\)-dimensional and \(x\)-dimensional configurations. Coherence means

Thus the action on the full enlarged operator algebra is supplied data; it is not obtained by extending its values on operators \(X\otimes I\).

Put \(q=\lceil \log _d D\rceil \). At each internal physical cut allocate two \(q\)-site bond registers and two amplification flags. The logical register consists of these sites and the physical chain. Append a scratch pool with as many sites as the logical auxiliary register. Consecutive indexing preserves the order physical sites, logical auxiliary sites, then scratch sites.

Encode the two joining bond labels using the given child bond encodings, followed by the dilation flag and one attenuation qudit. The logical bond dilation acts as the identity on every level of the attenuation qudit. The order of the two bond labels agrees with the vectorized joining contraction.

Append one padding qudit when \(N=1\), and none otherwise, to the physical register, logical auxiliaries, and reusable scratch pool. The physical basis inclusion initializes all three groups to zero.

For interval matrices \(A_{ab}:\mathbb C^s\to \mathbb C^r\), put

The first map is linear in \(P\). For prefix rows \(F_{ab}\), their density-weighted Gram is \(P_F(\rho )=\mathcal T_\rho (I)\), where the left virtual space is one dimensional.

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 [ 16 ] .

Let \(\mathcal U\) have bond dimension \(D\). The double-layer algebra \(\mathcal A\subseteq M_{D^2}(\mathbb C)\) is the not necessarily unital algebra generated by the double-layer letters \(W^{ij}\). It contains every \(W^{ij}\) and, as a linear combination of them, the normalized diagonal \(E\).

Write \(\pi _{m,n}:\{ 0,\ldots ,m{-}1\} \times \{ 0,\ldots ,n{-}1\} \to \{ 0,\ldots ,mn{-}1\} \) for the standard product coordinate bijection. For natural numbers \(d,e\), the canonical doubled product-coordinate equivalence \(s_{d,e}:\{ 0,\ldots ,(de)^2{-}1\} \to \{ 0,\ldots ,d^2e^2{-}1\} \) and its inverse satisfy

For a positive definite \(r\times r\) matrix \(P\), with \(r\geq 1\), put \(Q=P^{-1}/r\).

Concatenate a spectator basis encoding with the two bond registers and their dilation and attenuation flags. Each flag is an entire physical qudit, with the two distinguished values zero and one. For a positive cut metric \(P\) of dimension \(r\), let \(M_P\) be its normalized bond-reset operator. For joint child columns \(V\), define the normalized parent by \(r(I\otimes M_P)V\) and encode its output with both flags zero. Define the joining cost by the actual dilation and attenuation costs, including their routing, followed by the two selected-zero reflections and the amplification rounds.

Encode the label space of every internal cut in its common bond register, and encode a one-dimensional endpoint label space in the unique configuration of a zero-site register. In each case, the distinguished label zero is represented by physical zero.

For \(r\geq 0\), define

For a projection \(P\), put \(R_P=I-2P\). For two maps \(G,H\) with common input and output spaces, put

Set \(t_r=r\sin \theta (r)\).

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 \(\mathcal W\), let \(E'(\mathcal W)\) be the matrix obtained by exchanging the two entries of its column pair, \(E'(\mathcal W)_{p,q(c,a)}=E(\mathcal W)_{p,q(a,c)}\), and define

This is the contraction shown in Figure IV_index4.png of [ 16 , 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 [ 16 , 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 [ 16 , lines 2300–2306 ] is the star-algebra map

No stabilization in \(N\) is asserted.

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

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 \(F_a:\mathbb C^r\to \mathbb C^B\) and \(G_c:\mathbb C^E\to \mathbb C^r\). Define \(U_{(a,c),(b,e)}=(F_aG_c)_{be}\) and

The configurations of the interval between the two endpoint cuts are canonically identified with physical configurations of the complete chain. Both endpoint bond spaces have their normalized one-dimensional bases.

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 \(\mathcal V\).

For an injection of logical basis vectors into a larger basis, let \(J\) be its inclusion matrix. Appending an all-zero temporary register gives the zero-workspace inclusion as a special case.

Given an injective selection of qudit flags, include arbitrary configurations on the remaining sites by assigning zero to each selected flag. Denote its basis-inclusion matrix by \(E\).

Enumerate an interval’s logical support as its physical packet. Its input encoding records the physical input configuration and sets every auxiliary to zero. Its output encoding records its physical output, right bond label, and left bond label, using prescribed injective encodings in the two boundary registers and setting every strict-interior auxiliary to zero.

The auxiliaries initialized at an interval input are the auxiliary sites in its logical support, including both boundary bond registers. Enumerate any finite logical support by a finite site injection, using the fixed consecutive ordering of logical sites.

The support of an interval consists of its physical sites, all strictly interior bond and amplification sites, the inward-facing bond register at its left boundary, and the inward-facing bond register at its right boundary. Endpoint bond registers have width zero. The interior is initialized when every strictly interior auxiliary site is in its designated initial configuration. A leaf packet has one physical site and its two available boundary bond registers.

Tensor the two weighted interval columns with the identity on the remaining outside configurations, and move those configurations before the joining labels.

Decompose a complete logical configuration into its two disjoint child packets and its remaining sites. Concatenate prescribed input or output basis encodings on the child packets with a prescribed spectator encoding on the remaining sites.

View the two joining flags as sites outside both child packets. Initialize exactly these flags and retain arbitrary configurations on all other outside sites.

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 \(Q\).

Choose a basis of each \(E_k\) indexed by \(r_k\) coordinates. The site matrix for a physical letter is the matrix of its slice \(E_{k+1}\to E_k\). Evaluation at a prefix configuration is a linear functional on \(E_k\). Padding the site matrices gives their products in a common matrix space, while their rectangular forms give an open-boundary chain of bond dimensions \(r_k\).

Let \(F_k\) evaluate the chosen cut basis on each prefix configuration, and let \(G_k\) contain the coordinates of each coefficient column in that basis.

For cuts \(j\leq k\), an interval configuration assigns physical letters to the sites \(j,\ldots ,k-1\). Fixing these letters restricts the column space at cut \(k\) to the column space at cut \(j\). In the chosen cut bases, denote this restriction by \(A_{j,k}\). Separating every physical letter into output and input letters gives the corresponding operator interval.

Identify the physical configurations of a single-site interval with its local physical alphabet. Reindex its weighted interval isometry so that the output coordinates are, in order, its physical output, its right bond label, and its left bond label.

For a chain of at least two sites, choose one qudit in the common workspace. Embed each leaf packet in its physical site and its two boundary bond registers, and place its workspace qudit in the chosen common site.

Separate the output and input configurations in the prefix and suffix matrices of the paired coefficient tensor. The resulting factors are operator-valued prefix rows and suffix columns.

For a coefficient tensor \(\psi \) on a finite chain, let \(C_k\) be its coefficient matrix with rows indexed by configurations before cut \(k\) and columns indexed by the remaining configurations. Put \(E_k=\operatorname {span}\{ \text{columns of }C_k\} \) and \(r_k=\operatorname {rank}C_k\). Fixing the letter at site \(k\) defines a coordinate slice from functions on the longer prefix to functions on the shorter prefix. The last-cut vector is \(\psi \) regarded as a function on the full prefix.

Let \(w\) be the vectorization of the normalized balanced joining contraction. The bond reset sends this row functional to a specified pair of bond basis states. Its one-flag dilation is the two-copy dilation of that reset matrix.

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 [ 16 , Section III, equation (13) ] .

For a positive metric \(P\) on an \(r\)-dimensional bond, put \(S=\sqrt P\), \(T=\sqrt{P^{-1}/r}\) and \(C=r^{-1}T^{-1}S^{-1}\). Given weighted coefficients \(X_{ab}\) and \(Y_{cd}\) on adjacent intervals, their contracted coefficient is \(X_{ab}CY_{cd}\), vectorized over the two remaining virtual bonds.

Pair each output letter with its input letter, giving a coefficient tensor of local dimension \(d^2\). Splitting physical configurations at a cut and separating their output-input pairs are bijections. The resulting coefficient cut rank is the consecutive operator Schmidt rank.

For complex numbers \(c,s,x,y\), define

When the parameters are real, \(x,y\) represent the expectations of the two involutions in exterior density operators.

For a square matrix \(A\), define

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

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

For \(U=FG\), its cut coefficient matrix groups output and input indices on each side: \(W_{(a,b),(e,f)}=(F_aG_e)_{bf}\). For a three-part factorization \(FAG\), the first and second cut matrices are those of \(F(AG)\) and \((FA)G\), respectively.

Let \(0\leq k\leq N\). Number the sites before cut \(k\) by \(0,\ldots ,k-1\), and number the sites after the cut by \(0,\ldots ,N-k-1\). The corresponding bijections to the actual sites are \(i\mapsto i\) and \(j\mapsto k+j\). They induce bijections between the numbered configurations and the configurations on the two subsets of physical sites.

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

Place the real binary attenuation rotation on levels zero and one of a qudit, and let it be the identity on the other levels.

Given an injective full basis encoding with a prescribed suffix depending only on its bond label, fix one bond label and restrict the full encoding to the prefix registers. This defines the prefix basis inclusion.

Let \(\mathcal S_F=\{ P_F(\rho ):\rho \succeq 0, \operatorname {Tr}\rho =1\} \), and let \(\mathcal C_F\) be its real affine hull. The interval transfer is also an affine map over the real scalars.

Define

The columns of \(C_L,C_R\) are the left and right bond vectors, respectively. The matrix \(K\) gives the coefficients of their bilinear contraction.

For a square complex matrix \(P\) and a complex number \(z\), define

The displayed components are stacked vertically. For the interior map, their order is the lexicographic order of the two bond labels.

Set

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 diagonal positive definite matrix \(\rho \) with \(\operatorname{tr}(\rho )=1\), and \((\Phi \rvert =\operatorname {vec}(\mathbb {1}_D)^{\mathsf T}\) such that

The one-dimensional case is supplied directly: if \(D=1\), take \(J=1\) and \(\rho =\mathbb {1}_1\) using Theorem 18.8.7. If \(D{\gt}1\), chosen canonical-form-II data with full support give a positive trace-one ambient fixed point with \(J=D^2-1\) by Theorem 18.8.5. Diagonality of this ambient fixed point is an additional hypothesis: the theorem gives a diagonal matrix in the unique block coordinates, but conjugation into the ambient coordinates need not preserve diagonality. For \(D{\gt}1\), this states the fixed-pair convention in [ 16 , equations labelled Erightleft and II_UTransfer, and proof of Proposition III.2(ii) ] . The \(D=1\) branch is the direct one-dimensional specialization. When \(\mathcal U\) is an MPU, Theorem 18.8.17 gives a canonical-form-II presentation of this same tensor whose ambient fixed matrix is the supplied \(\rho \); it makes no assertion about continuous choices along a path.

Let \(d,D{\gt}0\), and let \(U\) be an MPU tensor. Choose a reduced canonical-form-II representative \(U_{\mathrm{red}}\) with \(U_{\mathrm{red}}^{(N)}=U^{(N)}\) for every \(N{\gt}0\), and set

The definition does not identify the source-cut ranks of \(U\) itself with those of \(U_{\mathrm{red}}\).

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 [ 16 , Proposition III.3(ii) ] .

For \(a\) data qudits, use \(a\) temporary qudits initialized in zero. At step \(j\), increment temporary coordinate \(j\) modulo \(d\) when data coordinate \(j\) is zero and either \(j=0\) or the preceding temporary coordinate is one. This is a reversible permutation on the full register. Put \(c_d=38(d^3)^6+1\); a step has gate budget \(4ac_d\) after routing.

The right rank is

Assume \(d{\gt}0\), and let \(T\) be the right-shift tensor. Starting from the displayed source factors for \(T\) with identity weight, rescale the first cut by

and leave \(X_2,Y_2,Z_2\) unchanged. Then these are supplied source factors for the trace-normalized scalar weight \(\rho _d=d^{-1}\mathbb {1}_d\) prescribed by CFII. In particular,

This gives the source-factor metric normalization of [ 16 , equations Erightleft, X1X2b, and YZ=1 ] for the right shift.

A prefix with scalar virtual bonds has physical matrix \(U_{ab}=F_{ab}(1,1)\). The empty prefix has one input, one output, and scalar coefficient one.

An injective choice of \(a\) sites in an \(n\)-site logical register defines the diagonal reflection with phase \(-1\) precisely when those sites are all zero. A common pool of \(A\geq a\) initialized scratch sites supplies its reversible conjunction circuit. Denote the zero-register gate bound by \(K_d(a)\).

For a periodic chain of length \(N\), the configuration space consists of triples \(((\sigma _1,\sigma _2),a)\) of two physical configurations and one \(d\)-dimensional ancilla configuration.

Define

The sitewise physical swap is

Let \(R\) cyclically rotate configurations on the chain. The crossed swap is

The sitewise swap of the second species with the ancilla is

Let \(C\) be the permutation obtained by applying \(S^{1,a}_{+}\) first, then \(S^{1,2}\), and finally \(S^{2,a}\).

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.

Canonical form II requires the positive right fixed point \(\rho \) to have trace one. If the virtual weight on the bond space of a single shift is chosen to be a scalar identity, this normalization forces

whenever \(d{\gt}0\). The same formula is defined at \(d=0\), although it then has trace zero. The tensors \(U_2\) and \(U_3\) have bond dimension \(d^2\), so the corresponding scalar trace-one matrix is

These are the scalar identity choices compatible with the source weight in [ 16 , equation labelled Y1Y1X1X1 ] . No transfer-map fixed-point equation is asserted here.

The supplied source-factor witnesses below instead use \(\rho _{\mathrm{aux}}=\mathbb {1}_d\) for each constituent shift and \(\rho _{\mathrm{aux}}=\mathbb {1}_{d^2}\) after tensoring. The paper gates computed from them carry reciprocal factors \(d\) and \(d^{-1}\) because these identity weights differ from \(\rho _d\) and \(\rho _{d^2}\) when \(d{\gt}1\).

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 [ 16 , Definition III.2 ] .

For a one-site operator \(U\), define its two-site padding by applying \(U\) to the first site and the identity to the second.

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

Let \(\mathcal U\) be simple and have a reduced full-support canonical-form-II presentation with positive diagonal fixed matrix \(\rho \). The two-site tensor \(\mathcal U_2\) has the supplied standard form with the source gates \(u,v\) and the source half factors \(X_1,X_2\). In particular,

The full-support restriction is recorded in [ 58 ] .

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 [ 16 , Section III ] .

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

Thus \(u:\mathbb {C}^{d^2}\to \mathbb {C}^{\ell r}\) has source order \(\ell \times r\), while \(v:\mathbb {C}^{r\ell }\to \mathbb {C}^{d^2}\) has source order \(r\times \ell \). These are the literal triangle contractions in [ 16 , Section III, equations uuvv–vdagger ] .

Put

with source-bond order \(r\times \ell \). Thus \(Y\) maps \((\mathbb {C}^D\otimes \mathbb {C}^d)\otimes (\mathbb {C}^d\otimes \mathbb {C}^D)\) 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 [ 16 , Theorem III.8, Section III.B, equations (31)–(32) ] .

The row space of \(M_1\) has product order \((\text{upper physical},\text{right auxiliary})\). For \(\rho \in M_{D}(\mathbb {C})\), define

This is the source weight in [ 16 , 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.

Let \(d,D,\ell ,r\) be positive integers and let \(W\) be a two-site tensor of physical dimension \(d^2\) and bond dimension \(D\). A standard form with supplied gates consists of unitary maps

and half factors \(X_1,X_2\) such that

This is the open contraction underlying [ 16 , Definition III.9 ] . No unitary periodic operator is assumed for \(W\).

Encode a child packet together with a basis encoding on its remaining logical sites. Taking the identity basis encoding on those sites allows every outside logical configuration.

On every positive-length chain, the adjunction, transposition, and conjugation actions are \(U\mapsto U^\dagger \), \(U\mapsto U^{\mathsf T}\), and \(U\mapsto \overline U\), respectively. For a site consisting of two \(d\)-dimensional spins, let \(S_N=\mathbb S^{\otimes N}\) be their sitewise exchange. The swap-combined actions are \(U\mapsto S_NU^\dagger S_N\) and \(U\mapsto S_NU^{\mathsf T}S_N\). These are the symmetries of [ 16 , Sections IV.A–IV.C ] .

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.

For four finite index sets, the source-cut shuffle sends

It is applied to the row and column products with the orders appropriate to \(M_1\) and \(M_2\).

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 [ 16 , Proposition III.3(ii) ] .

For a prefix \(F\), middle interval \(A\), and suffix \(G\), define the longer prefix \(FA\) and longer suffix \(AG\) by ordinary contraction of their shared virtual indices. The resulting physical operator has coefficients \(F_{ab}A_{cd}G_{ef}\).

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 \(\operatorname {ran}(PQ)\).

Let \(N{\gt}0\). Write each physical output and input label as an ordered pair \(i_x=(i_{x,1},i_{x,2})\) and \(j_x=(j_{x,1},j_{x,2})\), and each intermediate label as \(h_x=(h_{x,1},h_{x,2})\in \{ 0,\ldots ,\ell -1\} \times \{ 0,\ldots ,r-1\} \). Indices \(x\) are taken modulo \(N\). Define the unshifted and shifted layer matrices by

The cyclic relabeling of pairs is \(h\mapsto \bigl((h_{x,2},h_{x+1,1})\bigr)_x\), with inverse \(q\mapsto \bigl((q_{x-1,2},q_{x,1})\bigr)_x\). The alphabet of the two-site tensor is identified with ordered pairs \((i_{x,1},i_{x,2})\). This is the ordering in [ 16 , Equation StandardForm ] .

Let \(W\) have supplied two-site standard-form data. Fix virtual indices \(\alpha ,\gamma \), exterior output half-sites \(e_-,e_+\), and exterior input pairs \(j_0,j_2\). For two neighboring output pairs \(a=(a_0,a_1)\), \(b=(b_0,b_1)\) and a middle input pair \(j\), set

For an observable \(A\) on the middle input pair, let \(C=uAu^{\dagger }\) and define the matrix on the two neighboring output pairs by

The link coordinates are ordered as \((r,\ell ,r,\ell )\). For arbitrary virtual matrices \(P,Q\) and complete-block word pairs \((\sigma ^{L},\tau ^{L})\) and \((\sigma ^{R},\tau ^{R})\), set

where \(W^{\sigma ,\tau }\) denotes word evaluation. These are finite contractions of [ 16 , Equations uuvv and StandardForm ] .

Put \(q=\lceil \log _dD\rceil \), let \(g\) be the complete register size, and let \(A\) be the common workspace size. A common packet budget is

Reserve an additional \(16g^2\) for coordinate permutations, and use \(M=M_0+16g^2\) as a common leaf and node overhead. The common reflection budget is \(F=1+K_d(A)\, 2g\), where \(K_d(a)=8(38d^{18}+1)a^2+4a\). Define the midpoint-tree count using leaf cost \(M\), joining cost \(M\), and two reflection costs \(F\). Finally put

Given an isometry \(V\), success projection \(P\) and real success amplitude \(0{\lt}p{\lt}1\), put

For a real coefficient \(t\), put \(s=\sqrt{1-t^2}\) and define

The two row blocks represent the two values of a new binary flag.

For square bond weights \(L,R\), define an interval map whose amplitudes, including the two virtual output indices \(u,v\), are \((LA_{ab}R)_{uv}\).

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 and let compact decompositions of the two source cuts be given (any; the chosen ones of Definition 18.4.48 are one value). Define

These are the factors in [ 16 , Section III ] .

Extend a permutation of the logical sites by the identity on the appended workspace sites, transporting the disjoint sum to the consecutive site coordinates.

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 2.8.4, it is the complex conjugate of the Hilbert-space inner product from Definition 2.8.2.

A tensor \(A\) is normal [ 17 ] if there exists a positive integer \(L_0\) such that \(A\) is \(L_0\)-block injective.

A tensor \(A\) of physical dimension \(d\) and bond dimension \(D\) is a normal tensor if, after the spectral-radius normalization of  [ 13 ] , (i) \(A\) admits no nontrivial invariant orthogonal projection and (ii) the associated CPM \(\mathcal{E}_A(X)=\sum _i A^iX(A^i)^\dagger \) has spectral radius \(r(\mathcal{E}_A)=1\) and peripheral spectrum \(\sigma _\partial (\mathcal{E}_A)=\{ 1\} \).

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 [ 16 ] . 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 corresponds to the algebraic-local change of site grouping in [ 16 , lines 2308 and 2313–2320 ] .

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 included into \(\widehat{\operatorname {Hull}_L(\Delta )}_L\) before inverting the finite blocking.

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 [ 16 ] . No norm completion is taken.

The normed star algebra of local observables satisfies the \(C^*\)-norm identity.

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 definition is independent of the finite region and chosen representative, giving the norm completed in the Appendix of [ 16 ] 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 [ 16 , lines 2308 and 2313–2320 ] .

Let \(d,L{\gt}0\) and let \(\mathcal B_L:\mathcal A_{d^L}\to \mathcal A_d\) be the site-blocking isomorphism. A continuous functional \(\omega \) on the blocked chain defines the original-lattice functional \(T_L\omega =\omega \circ \mathcal B_L^{-1}\). This is the state transport associated with the regrouping of sites in [ 225 , lines 825–836 ] .

Multiplication defines a unital star-algebra homomorphism

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

A compatible family of finite-region states consists of complex-linear functionals \(f_\Lambda \colon \mathcal A_\Lambda \to \mathbb {C}\), indexed by finite regions \(\Lambda \Subset \mathbb {Z}\), such that, for every inclusion \(\Lambda \subseteq \Gamma \) and every \(X\in \mathcal A_\Lambda \),

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.

Compatibility determines a complex-linear functional \(f_{\mathrm{loc}}\colon \mathcal A_{\mathrm{loc}}\to \mathbb {C}\) by \(f_{\mathrm{loc}}(\iota _\Lambda (X))=f_\Lambda (X)\). For \(d{\gt}0\), this functional is continuous in the operator-norm topology.

For \(d{\gt}0\), let \(\omega \colon \mathcal A\to \mathbb {C}\) be the continuous linear extension of \(f_{\mathrm{loc}}\) to the operator-norm 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, matching the finite-region site grouping in [ 16 , lines 2308 and 2313–2320 ] .

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

Let \(d_1,d_2{\gt}0\) and let \(e:\mathcal A_{d_1}\to \mathcal A_{d_2}\) be a star-algebra isomorphism. The induced continuous complex-linear equivalence between functional spaces is \(T_e\omega =\omega \circ e^{-1}\); thus \((T_e\omega )(X)=\omega (e^{-1}X)\).

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 [ 16 ] ; 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 [ 203 , lines 1251–1266 ] . The same four-site localization occurs in [ 16 , 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 completes the change of site grouping used in [ 16 , lines 2308 and 2313–2320 ] .

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, and the singleton \(\{ 0\} \) is an identity. A singleton displacement recovers translation,

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 [ 16 ] .

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 [ 16 , lines 2292–2298 ] is exactly the displayed composition. Formula (1087) is equation RR2x in [ 203 , lines 1261–1266 ] . Cirac et al. use the homogeneous even factor in [ 16 , lines 2322–2328 ] ; the odd factor and unified family are supplied by the GNVW construction.

While GNVW allow the cell size \(d(y)\) to depend on \(y\), the present definition treats a homogeneous chain with fixed size \(d\); see [ 64 ] .

Let \(d{\gt}0\), and let \(\mathcal A_d\) denote the quasi-local observable algebra. Its state space consists of all continuous complex-linear functionals \(\omega \) such that \(\| \omega \| =1\), \(\omega (I)=1\), and \(\omega (X^*X)\ge 0\) for every \(X\in \mathcal A_d\). A state is pure if it is an extreme point of this set, viewed as a real convex set. Translation invariance is not required of these functionals. These are the state and purity conventions in [ 225 , Section 1 and Section 3 ] .

For a star-algebra automorphism \(\omega \) of the quasi-local algebra propagating within \(\{ -1,0,1\} \), define

Each \(\widehat{\mathcal R}_y\) is contained in \(G\). This is the algebraic join associated with the algebra generated by all \(\mathcal R_y\) in [ 203 , lines 1276–1279 ] . No claim that \(G\) is proper or that it fails to be norm closed is made here.

Define \(Q_{2x}=L_x\) and \(Q_{2x+1}=P_x\).

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

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 [ 16 , line 2298 ] .

For pairwise disjoint finite regions \(\Lambda ,\Gamma ,\Delta \), successive restriction in left–middle–right order gives

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 \(N {\gt} 0\).

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, for some common bond dimension \(D'\) as in Definition 18.9.67, there is a continuous path \(\mathcal W(p)\) of MPU tensors of bond dimension \(D'\) from \(\mathcal U_{\uparrow D'}\) to \(\mathcal V_{\uparrow D'}\), not necessarily in canonical form, such that

for every \(p\in [0,1]\) and every applicable chain length \(N\). When \(\mathcal U\) and \(\mathcal V\) have one bond dimension \(D\), such a path of bond dimension \(D\) from \(\mathcal U\) to \(\mathcal V\) witnesses the relation with \(D'=D\).

Let \(\mathcal U\) and \(\mathcal V\) be canonical-form MPU tensors with physical dimensions \(d_a\) and \(d_b\) and virtual bond dimensions \(D_a\) and \(D_b\). For a tensor \(\mathcal U\) of bond dimension \(D\) and \(D'\geq D\), write \(\mathcal U_{\uparrow D'}\) for the tensor of bond dimension \(D'\) whose letters are those of \(\mathcal U\) in the upper left \(D\times D\) corner and zero on the adjoined directions; adjoining unused bond directions changes no periodic operator of positive length. 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 are \(D'\geq \max (D_a,D_b)\) and a continuous path \(\mathcal W(p)\), \(p\in [0,1]\), of MPU tensors of bond dimension \(D'\) such that

Only the canonical-form requirement is relaxed along the path [ 16 , lines 708–714 ] . When \(D_a=D_b=D\), a path of MPU tensors of bond dimension \(D\) from \(\mathcal U\) to \(\mathcal V\) witnesses strict equivalence with \(D'=D\).

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  [ 233 , Section II ] .

For a time-reversal-invariant MPU in standard form, the sign \(\sigma \) of Proposition 18.9.1.19 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 18.9.2.15. If both are even, the four pairs \((u_\pm ,v_\pm )\) realize the four Case I classes. Multiplying by the matrix \(R\) of Corollary 18.9.1.17 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 18.9.2.16. If \(r\) and \(\ell \) are both even, the four pairs \((u_\pm ,v_\pm )\) represent the four Case I classes of Theorem 18.9.2.22. With

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

The literal normalized tensor printed below is not an MPU, by Theorem 18.9.3.3. The classification below evaluates the supplied tensor data under [ 180 ] . 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 of Chen, Liu and Wen  [ 11 ] (see also [ 16 , lines 1906–1943 ] ) 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 literal normalized tensor printed below is not an MPU, by Theorem 18.9.3.3. The following is the source classification claim, not an assertion of unitarity for that literal tensor; its normalization and comparison with the Z-decorated tensor are discussed in [ 180 ] . The CZX MPU of Chen, Liu and Wen  [ 11 ] (see also [ 16 , lines 1906–1943 ] ) is defined by

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

be the Hadamard matrix. The corresponding tensor is

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

Let \(\mathcal U\) and \(\mathcal V\) be simple tensors written in 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 standard forms are related by the local unitary gauges of [ 16 , 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 \(y\). This records the source’s unblocked all-length wording. Its converse has the odd-length parity obstruction of Theorem 18.9.38; the blocked-family interpretation is distinguished in [ 157 ] . The correct form of this theorem is Theorem 18.9.33 together with Theorems 18.9.34 and 18.9.35: equality of the operators at every length is equivalent to a unitary conjugation of the letters, the gate relations follow from it, and the gate relations give equality of the operators on rings of even length only.

Let \(\mathcal U\) be a tensor in canonical form generating matrix product unitaries. The source index is independent of the chosen simple blocking. More precisely, if \(k\geq k_0\geq 1\) and \(\mathcal U_{k_0}\) is simple, then \(\mathcal U_k\) is simple and

This is [ 16 , Proposition IV.2 ] for a tensor in canonical form, the hypothesis under which the source defines the index.

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 is invariant 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 [ 16 , 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 \(p\in [0,1]\).

If \(w = (J_1, \ldots , J_m)\) is a word in the blocked alphabet, where each \(J_k\) is an \(L\)-tuple in \(\{ 0, \ldots , d{-}1\} ^L\), then concatenating the block words gives a word \(\widetilde{w}\) of length \(mL\) in the original alphabet and \((A^{[L]})^w = A^{\widetilde{w}}\).

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 \(A\) and \(B\) be left-canonical irreducible tensors of common positive bond dimension satisfying \(B^i=\zeta XA^iX^{-1}\) for an invertible \(X\) and a nonzero scalar \(\zeta \). There is a unitary matrix \(U\), with the same scalar \(\zeta \), such that \(|\zeta |=1\) and \(B^i=\zeta UA^iU^\dagger \) for every \(i\).

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

Let a chain consist of a prefix of \(p\) sites, \(m\) complete blocks of length \(L\), and a suffix of \(s\) sites. Write \(M^{\alpha ,\beta }\) for the ordered product of matrices indexed by two equally long physical words, and write \(\widetilde\sigma ,\widetilde\tau \) for the concatenations of blocked words \(\sigma ,\tau \). Then, for matching row and column configurations,

This identity retains both residual products, including when the chain length is not divisible by \(L\). It extends the closed-chain blocking identity used in [ 13 , Appendix C.4, lines 1952–2017 ] .

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,

For words \(u_1,v_1\) of equal length and arbitrary words \(u_2,v_2\), word evaluation factors as \(M^{u_1u_2,\, v_1v_2}=M^{u_1,v_1}M^{u_2,v_2}\).

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

This is the unitarity equation displayed in [ 16 , Section III, equation (12) ] .

Under the hypotheses of Lemma 18.9.2.29, 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 the standard form determined by its supplied source datum, 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 18.9.1.20 is

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

Let \(\mathcal U\) be an MPU tensor with physical dimension \(d{\gt}0\). Let \(W\) be its double layer, put \(E=d^{-1}\sum _iW^{ii}\), and let \(\rho {\gt}0\) be diagonal. Put \((\Phi \rvert =\operatorname {vec}(\mathbb {1}_D)^{\mathsf T}\) and suppose that for some \(K\geq 0\),

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

Thus the compact-SVD paper gate is an isometry. This is the supplied-fixed-pair version of [ 16 , 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, corresponding to [ 16 , 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 [ 16 , Definition II.5 and equation (9) ] ; preservation of the MPU property under blocking is used in the proof of [ 16 , Proposition III.3 ] .

Let \(\mathcal U_1,\mathcal U_2\) be MPU tensors with positive physical and bond dimensions, which may differ between the tensors. There is a length \(k{\gt}0\) for which both direct length-\(k\) blockings are simple. The same conclusion holds for three such tensors \(\mathcal U_1,\mathcal U_2, \mathcal U_3\), with one common length \(k\). This is a finite-family consequence of [ 16 , Proposition III.3 ] , not an additional hypothesis on the tensors.

Let \(I\) be a finite set and let \(H\in \operatorname {Mat}_I(\mathbb C)\) be Hermitian with \(H^2=\mathbb {1}\) and \(\operatorname {tr}H=0\). Then

The matrix \(M_{i_0}\) from definition 18.4.1 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 \(W_{\mathcal U}\) and \(W_{\mathcal V}\) be the double layers of \(\mathcal U\) and \(\mathcal V\). Identify the double-layer bond space of \(\mathcal U\boxtimes \mathcal V\) with the tensor product of the two double-layer bond spaces by \(((\alpha ,\gamma ),(\alpha ',\gamma '))\mapsto ((\alpha ,\alpha '),(\gamma ,\gamma '))\). Under this identification,

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 [ 16 , Theorem IV.6(ii) ] .

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

This is the unrestricted tensor-product statement in [ 16 , 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 [ 16 , Theorem IV.6(ii) ] with the normalization in [ 16 , Section III, equation (13) ] ; it is not a separately stated result of that proof.

Let \(U\) be any MPO tensor, let \(k,p,m,s\in \mathbb N\), and write \(W=(U^{[k]})_2\). For a word \(\sigma \) of \(m\) blocks of length \(2k\), let \(\eta \sigma \) relabel each block as two consecutive words of length \(k\); use the same convention for \(\tau \). If \(i^-,j^-\) are words of length \(p\) and \(i^+,j^+\) are words of length \(s\), then

Both residual products remain present, including when \(m=0\); no unitarity, canonical form, or divisibility of the total length is assumed.

Let \(\mathcal U\) be a matrix product operator tensor of physical dimension \(d\) and bond dimension \(D\). For arbitrary matrices \(A\in \mathbb C^{E\times D}\) and \(B\in \mathbb C^{D\times E}\), put \(\mathcal V^{ij}=A\mathcal U^{ij}B\). Then

In particular, a rectangular reduction cannot increase either source rank. No equality of ranks is asserted, and no unitarity, simplicity, or canonical-form assumption is needed. If \(\mathcal V\) is a rectangular reduction of \(\mathcal U\), then for every common blocking length \(L\),

These identities follow directly from the source cuts of [ 16 , Section III, definition of \(r,\ell \) ] and permit upper bounds for composition to pass to a reduced representative.

Let \(u,v\in \operatorname {End}(\mathbb {C}^d\otimes \mathbb {C}^d)\) and let \(x,y\in U(d)\). Set

The four-site trace is unchanged:

In particular, the trace expression in (18.9.2.28) is invariant under 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.

Let \(x\) be a unitary matrix with \(x^T=-x\). There are a symmetric unitary matrix \(S\) and a real skew-symmetric unitary matrix \(\widetilde\Lambda \) with \(\det \widetilde\Lambda =1\) such that

This is the skew branch of [ 16 , Lemma lemma:conjclass-normalform-continuous ] .

Let \(K\) and \(I\) be finite sets, let \(E_k\in \mathbb R\) for \(k\in K\), and let \(P_k\) be Hermitian projectors on \(\mathbb C^I\). Suppose that the family \(\{ P_k,P_k^T\} _{k\in K}\) consists of mutually orthogonal projectors and sums to the identity. Define

Then the matrices

and

are unitary, \(S^T=S\), \(\overline{\widetilde\Lambda }=\widetilde\Lambda \), \(\widetilde\Lambda ^T=-\widetilde\Lambda \), \(\det \widetilde\Lambda =1\), and

This is the paired-projector construction in the skew branch of [ 16 , Lemma lemma:conjclass-normalform-continuous ] .

Let \(x\) be a unitary matrix with \(x^T=-x\). There are a finite set \(K\), Hermitian projectors \(P_k\) and numbers \(E_k\in [0,\pi )\), for \(k\in K\), such that the family \(\{ P_k,P_k^T\} _{k\in K}\) consists of mutually orthogonal projectors summing to the identity, and

This is the spectral decomposition in the skew branch of [ 16 , Lemma lemma:conjclass-normalform-continuous ] .

Strict equivalence under a symmetry implies strict equivalence.

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

then

Let \(x\) be a symmetric unitary matrix. There is a symmetric unitary matrix \(S\) such that

This is the symmetric branch of [ 16 , Lemma lemma:conjclass-normalform-continuous ] .

Let \(Y=\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\) and \(R=\frac{1-i}{2}[\mathbb {1}_4+i(Y\otimes Y)]\), with the two tensor factors in that order. Then

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

Let \(x\) be a normal complex matrix with finite spectrum \(\sigma (x)\), and for \(\mu \in \mathbb C\) let \(P_\mu \) be the spectral projector of \(x\) at \(\mu \), that is, the indicator function of the point \(\mu \) applied to \(x\). Then each \(P_\mu \) is Hermitian,

If moreover \(x^T=-x\), then

This is the pairing of \(P_E\) with \(P_E^T\) in the skew branch of [ 16 , Lemma lemma:conjclass-normalform-continuous ] .

Let square complex matrices \(E,F\) satisfy \(E^\dagger E+F^\dagger F=EE^\dagger +FF^\dagger =I\). There are unitaries \(L,R\) and diagonal matrices \(D_E,D_F\) with \(E=LD_ER\) and \(F=LD_FR\).

Let \(U\) be a bipartite unitary of operator Schmidt rank two. After possibly interchanging the factors, there are unitaries \(A_0,A_1\) and matrices \(B_0,B_1\) such that \(U=A_0\otimes B_0+A_1\otimes B_1\). The matrices \(B_\pm =B_0\pm B_1\) are unitary, and \(R=B_+^\dagger B_-\) is a Hermitian involution. The \(A_s\) belong to the first Schmidt space of \(U\) and the \(B_\pm \) to the second.

Replace the open word in \(F_{\alpha ,\gamma }\) by its virtual trace with arbitrary matrices \(P,Q\) at the left and right ends. The resulting matrix \(F_{P,Q}\) obeys

The same conclusion holds when \(P\) and \(Q\) are the matrix products of arbitrary complete-block prefix and suffix words, respectively. Their physical row and column words remain explicit and may have different lengths on the two sides of the three-block window.

For every supplied two-site standard form and every observable \(A\) on the middle input pair,

for every choice of \(\alpha ,\gamma ,e_-,e_+,j_0,j_2\). In particular, neither virtual end is closed in this identity, and \(B(A)\) acts only on the four output half-sites forming the two neighboring pairs.

Let \(W=(U^{[k]})_2\) have supplied two-site standard-form data. For arbitrary original-site prefix and suffix word pairs, put \(P=U^{i^-,j^-}\) and \(Q=U^{i^+,j^+}\). The three-block coefficient contracted against these end matrices satisfies

Thus the local observable on the two neighboring output pairs is independent of both residual words.

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 \(\operatorname {Hull}_L(\Delta )\). The forward composite is the identity,

This holds for every on-site dimension \(d\).

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 [ 16 , 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 [ 16 , 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 [ 16 , 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 \(L{\gt}0\), \(a\in \mathbb Z\) and \(N\geq 0\), the expansion of the blocked interval \(I(a,N)\) is

Negative starts and the empty interval are included.

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; if every complementary Weyl conjugation fixes \(X\), then

Let \(T_x=(L_x\cup P_x)\cup P_{x+1}\). The three regions \(L_x,P_x,P_{x+1}\) are pairwise disjoint, and

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

For \(x\ne y\) the pairs \(E_x\) and \(E_y\) are disjoint, and every finite region \(\Lambda \) satisfies

where \(X_\Lambda =\{ \lfloor i/2\rfloor \mid i\in \Lambda \} \).

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 [ 16 , 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 [ 16 ] .

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

Translation by two sites advances each of the three two-site regions by one index:

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 [ 16 ] , 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 [ 16 ] .

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 [ 16 , lines 2308 and 2313–2320 ] .

Let \(\Gamma \) and \(\Delta \) be disjoint finite regions and let \(A\in \mathcal A_{\Gamma \cup \Delta }\), written in the ordered \(\Gamma \times \Delta \) coordinates. For configurations \(r,s\in \mathcal C_\Delta \) and \(i,j\in \mathcal C_\Gamma \), the canonical quasi-local copies of the two coefficient families are

where a hat denotes the canonical image in the quasi-local algebra. Both right-hand sides are formed inside the quasi-local algebra alone.

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 \(\mathcal N\),

Let \(X\in M_{I\times J}(\mathbb C)\) and write \(e_{rs}\) for the matrix units of either factor. Then

Every coefficient of the expansion used at [ 203 , lines 1188–1191 ] and [ 237 , lines 1137–1169 ] is therefore a sum of products formed inside the ambient algebra.

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

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