19 Matrix Product Unitaries
This chapter develops the algebraic prerequisites for matrix product unitaries. The first result supplies the corrected nil-matrix bound used in the blocking argument of [ CPGSV17 ] .
Let a Lie algebra \(L\) act nilpotently on a finite-dimensional vector space \(V\) over a field \(K\). Write \(C_r(L,V)\) for the \(r\)th term of the lower central series of this action. Then
As long as \(C_r(L,V)\) is nonzero, nilpotency prevents two consecutive terms from being equal: equality would make every later term equal, in contradiction with eventual vanishing. Thus each nonzero step strictly lowers dimension, and at most \(\dim _K(V)\) such steps are possible.
Let \(V\) be a finite-dimensional vector space over a field \(K\), and let \(A\subseteq \operatorname {End}_K(V)\) be a not necessarily unital associative subalgebra. If every element of \(A\) is nilpotent, then every ordered product of \(\dim _K(V)\) elements of \(A\) is zero. The empty product in dimension zero is also zero because the identity endomorphism of the zero space is zero.
Regard \(A\) as a Lie algebra under the commutator bracket. Engel’s theorem makes \(V\) a nilpotent Lie module, so the dimension bound gives
For every nonnegative integer \(r\), every ordered family \(a_1,\ldots ,a_r\) in \(A\), and every vector \(v\) in \(V\), induction on \(r\) gives
Taking \(r=\dim _K(V)\) proves that the product annihilates every vector and is therefore zero.
Let \(A\) be a not necessarily unital associative subalgebra of \(n\times n\) matrices over a field. If every matrix in \(A\) is nilpotent, then every ordered product of \(n\) matrices from \(A\) vanishes.
Transport the matrix algebra to the endomorphisms of the coordinate space through the standard algebra equivalence. Nilpotency, membership, and the ordered product are preserved, so the finite-dimensional nil-matrix theorem applies; injectivity of the equivalence returns the zero matrix.
The length \(n\) is a weak bound, not a strict bound below \(n\); the latter is false for strictly upper-triangular matrices. This corrects the intermediate strict inequality in [ CPGSV17 ] without changing its final strict \(D^4\) estimate.
19.1 The matrix product unitary condition
Let \(\mathcal U=\{ U^{ij}\} _{i,j=0}^{d-1}\) be a matrix product operator tensor of bond dimension \(D\), and write \(U^{(N)}\) for the periodic operator obtained by closing a chain of \(N\) copies of \(\mathcal U\).
Let \(I\) and \(J\) be finite sets, let \(e\colon I\to J\) be a bijection, and let \(A\) be a unitary matrix indexed by \(I\). The matrix indexed by \(J\) whose \((j,k)\) entry is
is unitary.
Simultaneous reindexing preserves the identity and matrix multiplication, and it commutes with the conjugate transpose. Reindex the equation \(AA^\dagger =\mathbb {1}\).
The tensor \(\mathcal U\) generates matrix product unitaries if \(U^{(N)}\) is unitary for every integer \(N{\gt}1\). This is the matrix product unitary condition in [ CPGSV17 , Section III, equation (10) ] .
If \(\mathcal U\) generates matrix product unitaries and \(N{\gt}1\), then
Expand the unitary-group condition for \(U^{(N)}\).
If \(\mathcal U\) generates matrix product unitaries and \(N{\gt}1\), then
This is the unitarity equation displayed in [ CPGSV17 , Section III, equation (12) ] .
Use the equivalent left-multiplied characterization of a unitary matrix.
Let \(\mathcal U\) and \(\mathcal V\) be matrix product unitary tensors with the same physical dimension. Contracting the output index of \(\mathcal U\) with the input index of \(\mathcal V\) gives a matrix product unitary tensor whose periodic operator is \(U^{(N)}V^{(N)}\), in that order. This is the concatenation tensor \(\mathcal W\) in the proof of [ CPGSV17 , Theorem IV.6(ii) ] .
For every \(N{\gt}1\), both \(U^{(N)}\) and \(V^{(N)}\) are unitary. The defining contraction of the product tensor gives the periodic operator \(U^{(N)}V^{(N)}\), which is unitary because the unitary matrices form a group under multiplication.
Suppose that \(\mathcal U\) generates matrix product unitaries. For every positive integer \(L\), the tensor \(\mathcal U^{[L]}\) obtained by blocking \(L\) adjacent sites also generates matrix product unitaries. This is the blocking operation of [ CPGSV17 , Definition II.5 and equation (9) ] ; preservation of the MPU property under blocking is used in the proof of [ CPGSV17 , Proposition III.3 ] .
At system size \(N\), closing the blocked tensor gives \(U^{(NL)}\), up to the canonical bijection between \(N\) blocked configurations and \(NL\) unblocked configurations. If \(N{\gt}1\) and \(L{\gt}0\), then \(NL{\gt}1\), so \(U^{(NL)}\) is unitary. Simultaneous reindexing preserves unitarity.
19.2 Simple matrix product unitary tensors
For an MPO tensor \(\mathcal U\), let \(\mathcal W\) be the double-layer tensor obtained by placing the physical adjoint above \(\mathcal U\) and contracting the adjacent physical legs:
The bond space of \(\mathcal W\) is the product of the two original bond spaces. The physical adjoint exchanges and conjugates the physical indices without reversing the order of the virtual chain, exactly as in the barred upper layer of Figures II_Simple1.png and II_Simple2.png of [ CPGSV17 ] .
For every chain length \(N\), closing the double layer gives
The operator family is multiplicative under the local product tensor, and the physical adjoint generates the conjugate transpose without reflecting the chain.
The physical adjoint is involutive. If \(\mathcal U\) generates matrix product unitaries, then so does \(\mathcal U^\sharp \), and for every chain length \(N\),
Applying the physical adjoint twice conjugates every entry twice and exchanges the physical indices twice. Closing a physical-adjoint tensor exchanges the two physical configurations and conjugates the resulting scalar. Hence its periodic operator is the conjugate transpose. The first MPU unitarity equation for \(U^{(N)}\) is therefore the second unitarity equation for \((U^\sharp )^{(N)}\).
Order the left doubled bond as \((\alpha ,\gamma )\) and the right doubled bond as \((\beta ,\delta )\). Then the double layer whose lower tensor is \(\mathcal U^\sharp \) has entries
Expand the double layer and then apply the physical adjoint to its upper factor. The two exchanges of physical indices leave the contracted index \(j\) in the second physical position of both factors. The product-index convention keeps \((\alpha ,\gamma )\) and \((\beta ,\delta )\) in the displayed order.
The tensor \(\mathcal U\) is simple if there are vectors \(a,b\) in the double-layer bond space such that, for all physical indices \(i,j,k,\ell \),
Thus (11) closes one double-layer letter to the physical identity, while (12) inserts the rank-one operator \(|b)(a|\) between adjacent letters. These are precisely the two diagrams in [ CPGSV17 , Definition III.2 ] .
If \(\mathcal U\) is simple and \(N{\gt}1\), then
Apply (12) sequentially between neighboring letters. The inserted rank-one operators split the periodic contraction into the product of the one-letter contractions \((a|W^{\sigma _x\tau _x}|b)\). Identity (11) turns this product into \(\prod _x\delta _{\sigma _x,\tau _x}\), which is the matrix element of the physical identity.
Every simple tensor generates matrix product unitaries. This is [ CPGSV17 , Proposition III.3(i) ] .
For every \(N{\gt}1\), the closed-double-layer identity and the sequential contraction give
Hence \(U^{(N)}\) is unitary.
19.3 Double-layer blocking and mixed physical traces
For every blocking length \(L\), physical adjunction commutes with blocking, and consequently
The virtual order is unchanged on both sides.
Physical adjunction conjugates each matrix entry without reversing the virtual word. The first identity follows entrywise. The second follows because physical blocking commutes with the local operator product.
Assume \(d{\gt}0\), and let \(E\) be the normalized transfer matrix of \(\mathcal U\). After blocking \(L\) sites, the normalized physical diagonal of the double layer is exactly \(E^L\). More precisely, if
is the standard product encoding, then its entries are
Thus the row and column reindexings have the same explicit orientation. If \(\mathcal U\) is an MPU and \(D{\gt}1\), then at the stabilization exponent \(J\) this diagonal is the same reindexing of the rank-one projector \(|\rho )(\Phi |\).
On one site, expanding the diagonal double-layer contraction gives the transfer matrix after the stated product reindexing. The factor \(d^{-1}\) is the square of the local factor \(d^{-1/2}\). Summing blocked diagonal letters over words and distributing the ordered product shows that normalized diagonals turn blocking into powers. Simultaneous reindexing preserves multiplication, giving the displayed formula. Transfer stabilization gives the final rank-one specialization.
Let \(\mathcal W\) be an MPO tensor of physical dimension \(d\), and let
be its normalized physical diagonal. Here and in the two diagonal-tail identities below, scalar inverses are the totalized complex-field inverses, so \(0^{-1}=0\). For every \(K\geq 0\),
At \(K=0\), the word product is the identity matrix. This is the algebraic diagonal-tail expansion used in the source-factor networks of [ CPGSV17 , Theorem III.8, Section III.B, equations (31)–(32) ] .
Expand the \(K\)th power of the finite sum defining \(R\). Distributivity indexes the resulting terms by physical words \(\tau \), and each term is the ordered diagonal word displayed above.
If \(d{\gt}0\), the identity matrix extends to a basis of \(M_{d}(\mathbb {C})\) whose other elements are traceless.
The trace functional is nonzero on the identity. Extend the identity by a basis of the kernel of the trace.
Assume \(d{\gt}0\). For every double-layer tensor \(\mathcal W\), there is a basis \(\{ \sigma _0,\sigma _\alpha \} \) of the physical matrix algebra and virtual matrices \(E,S_\alpha \) such that \(\sigma _0=\mathbb {1}\) and, for every nonzero index \(\alpha \), \(\operatorname{tr}(\sigma _\alpha )=0\), while
Here \(E\) is the normalized physical diagonal. Equivalently, contraction against a physical matrix \(X\) separates as
where the second physical argument is traceless. This is equation WIsom in [ CPGSV17 , Proposition III.3(ii) ] .
The trace functional is nonzero on the identity. Extend the identity to a basis adapted to the kernel of the trace; all remaining basis vectors are therefore traceless. The bilinear pairing \((X,Y)\mapsto \operatorname{tr}(XY)\) is nondegenerate. Its dual basis gives the virtual coefficients by physical contraction. The dual vector paired with the identity is \(d^{-1}\mathbb {1}\), so its virtual coefficient is precisely \(E\). Expanding matrix units in the dual basis yields the decomposition. Subtracting the trace component gives the equivalent residual formula.
Suppose a length-\(N\) closed tensor equals the physical identity. For arbitrary physical matrices \(X_0,\ldots ,X_{N-1}\),
In particular this applies to the double layer of an MPU whenever \(N{\gt}1\).
Expand each contraction and distribute the ordered product. The closed tensor coefficient is a Kronecker delta between the two physical words, so only equal words survive. The remaining independent diagonal sums factor into the product of the physical traces.
Let \(\mathcal U\) be an MPU with physical dimension \(d{\gt}0\). Every closed product of positive length that contains at least one residual coefficient has zero trace. In particular, this includes a single residual coefficient. This is the mixed-trace assertion preceding equation ESE=0 in [ CPGSV17 , Proposition III.3(ii) ] .
At lengths greater than one, represent identity factors by \(d^{-1}\mathbb {1}\) and residual factors by traceless physical arguments. The multilinear contraction identity then vanishes. For a single residual coefficient \(A\), repeat the same coefficient at every length \(q{\gt}1\) to obtain \(\operatorname{tr}(A^q)=0\). Thus the one-letter case is recovered after the longer words: the shifted zero-trace-power theorem makes \(A\) nilpotent, and therefore \(\operatorname{tr}(A)=0\).
Let \(\mathcal U\) be an MPU with physical dimension \(d{\gt}0\), and let \(N{\gt}1\). Every closed product of \(N\) identity and residual coefficients that contains at least one residual coefficient has zero trace. If the normalized diagonal is rank one, then every nonempty residual word \(P\) satisfies
The same conclusion holds after blocking \(J{\gt}0\) sites whenever the \(J\)th normalized transfer power is \(|\rho )(\Phi |\), with both virtual indices reindexed by \(q^{-1}:\{ 0,\ldots ,D^2{-}1\} \longrightarrow \{ 0,\ldots ,D{-}1\} \times \{ 0,\ldots ,D{-}1\} \).
Represent identity factors by \(d^{-1}\mathbb {1}\) and residual factors by their traceless physical arguments. Since \(N{\gt}1\), the multilinear contraction identity applies, and one trace factor vanishes. For a nonempty residual word, adjoin \(E\) as one extra factor; its closed trace has length at least two and therefore vanishes. Cyclicity of trace and
give \(EPE=0\). The blocked statement follows from the transfer-power formula with the same product-index reindexing in both matrix indices.
Let \(\mathcal U\) have bond dimension \(D\). Contract the double layer against arbitrary physical matrices, subtract the normalized diagonal coefficient, and let \(\mathcal S\) be the resulting set of residual matrices in \(M_{D^2}(\mathbb C)\). The residual algebra \(\mathfrak S\) is the not necessarily unital algebra generated by \(\mathcal S\). This is the algebra generated by the matrices \(S'_\alpha \) in [ CPGSV17 , Proposition III.3(ii) ] .
If \(\mathcal U\) is an MPU with positive physical dimension, then every nonempty product of residual generators has zero trace. Consequently, for every \(A\in \mathfrak S\) and every integer \(N{\gt}1\),
and every element of \(\mathfrak S\) is nilpotent.
A nonempty product in the multiplicative semigroup generated by \(\mathcal S\) is a nonempty residual word, so its trace vanishes by the positive-length mixed residual identity. The generated nonunital algebra is the linear span of this semigroup. Linearity of the trace therefore gives \(\operatorname{tr}(A)=0\) for every \(A\in \mathfrak S\). Since every positive power of \(A\) remains in \(\mathfrak S\), the same argument gives \(\operatorname{tr}(A^N)=0\) for \(N{\gt}1\). The shifted zero-trace-power theorem now implies that \(A\) is nilpotent.
Let \(\mathcal U\) be an MPU of bond dimension \(D\) and positive physical dimension. Every ordered product of exactly \(D^2\) elements of \(\mathfrak S\) vanishes. In particular, for arbitrary residual generators,
This is equation Sprime=0 in [ CPGSV17 , Proposition III.3(ii) ] , with the corrected exact length \(D^2\) in place of the unsupported strict intermediate bound.
The algebra \(\mathfrak S\) acts on a space of dimension \(D^2\), and every element is nilpotent. The finite-dimensional nil-matrix theorem therefore annihilates every ordered product of \(D^2\) elements. Each residual generator belongs to \(\mathfrak S\), giving the stated specialization.
Let \(E\) be the normalized diagonal after the first blocking and let \(\mathfrak S\) be its residual nonunital algebra. Define
This is the basis-independent sum of the three forms in equation sprimeforms of [ CPGSV17 , Proposition III.3(ii) ] .
Suppose \(E^2=E\), \(EAE=0\) for every \(A\in \mathfrak S\), and every ordered product of \(N{\gt}0\) elements of \(\mathfrak S\) vanishes. For scalars \(c_k\) and elements \(S_k\in \mathfrak S\),
Products are taken in increasing site order and need not commute.
Expand the product as a sum over binary choices of \(E\) or \(S_k\), retaining the original order. The all-\(E\) choice is the subtracted term because \(E^q=E\) for \(q{\gt}0\). The all-residual choice vanishes by the length-\(N\) hypothesis. In every remaining word, contiguous copies of \(E\) collapse. If two resulting copies of \(E\) are separated by a nonempty residual word, the factor \(EAE\) makes the word zero. Thus each surviving word has one contiguous projector block and belongs to one of the three summands of \(\mathcal T\).
Let the first-block tensor be an MPU of bond dimension \(D{\gt}0\). Assume its normalized double-layer diagonal \(E\) is a rank-one idempotent. After a second blocking of exactly \(D^2\) sites, every residual coefficient belongs to \(\mathcal T\).
The conclusion is membership in the sum of the three subspaces. It does not assign one exclusive type to a coefficient in an arbitrary physical basis.
First extend \(EPE=0\) from nonempty residual words \(P\) to every element of \(\mathfrak S\) by writing the generated algebra as the linear span of the multiplicative semigroup of residual generators. For a blocked matrix unit, each local double-layer entry is
The ordered expansion theorem applies with \(N=D^2\). Its all-\(E\) coefficient is the blocked Kronecker delta and is exactly removed by the residual slice. Hence every blocked matrix-unit residual lies in \(\mathcal T\). Expanding an arbitrary physical matrix in matrix units and using linearity proves the result.
Suppose \(E^2=E\) and \(EAE=0\) for every \(A\in \mathfrak S\). For all \(X,Y\in \mathcal T\),
It is enough by bilinearity and linearity of trace to consider the generators \(EA\), \(AE\), and \(AEB\) of the three summands. Representative products reduce as
The remaining generator pairs follow by the same use of associativity, \(E^2=E\), and \(EAE=0\) for \(A\in \mathfrak S\).
Let \(\mathcal V\) be an MPU of positive physical and bond dimensions whose normalized double-layer diagonal is
Then blocking \(\mathcal V\) by exactly \(D^2\) sites produces a simple tensor with witnesses \(a=\Phi \) and \(b=\rho \).
The pairing normalization makes \(E\) idempotent. Write every blocked double-layer coefficient as \(\delta _{ij}E+R_{ij}\), where \(R_{ij}\in \mathcal T\). The trace identity for \(R_{ij}E\) gives the diagonal contraction, and \(R_{ij}R_{k\ell }=R_{ij}ER_{k\ell }\), together with \(E^2=E\), gives the rank-one insertion identity after expanding both coefficients.
For a bijection \(e\) between two physical index sets, the reindexed tensor and its action on length-\(N\) configurations are
For ket and bra words \(i,j\), physical reindexing gives
Induct simultaneously along the two words. The empty pair gives the identity matrix, mismatched lengths give zero, and the nonempty case applies the defining equality to the first letters and the induction hypothesis to the tails.
At every length \(N\), the sitewise configuration bijection identifies
Apply the word identity to every pair of configurations and take the virtual trace. This is simultaneous row and column reindexing by \(e_*\).
For every physical-index bijection \(e\), taking the double layer commutes with physical reindexing:
This follows entrywise from the definitions of physical reindexing and the double layer.
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 \(mn\):
Expand the two blocked words. The canonical regrouping concatenates their \(n\) words of length \(m\) into the direct word of length \(mn\).
Every MPU of positive physical and bond dimensions has a positive blocking length \(k\leq D^4\) for which the blocked tensor is simple. If \(D{\gt}1\), one may choose
If \(D=1\), the length-one block is simple. Simplicity is preserved under every further iterated blocking by a positive factor and under a bijective relabeling of the physical basis.
At the stabilization exponent \(J\), the normalized double-layer diagonal is the rank-one fixed projector \(|\rho )(\Phi |\). The preceding theorem makes the subsequent length-\(D^2\) block simple. If \(e\) is the canonical grouping bijection, the two blocking stages satisfy
For \(D{\gt}1\),
The one-dimensional transfer matrix is already the identity. For a further block with letters \(W_1,\ldots ,W_L\) and \(R=|\rho )(\Phi |\), repeated insertion gives
The second identity multiplies the sitewise Kronecker deltas.
Let \(\mathcal U\) be an MPU of positive physical and bond dimensions. Suppose that \(\rho \) has trace one and that, for some positive integer \(J\), the normalized transfer matrix satisfies
Put \(K=JD^2\). After the standard product reindexing of the physical coordinates, the direct length-\(K\) block satisfies the contractions (11) and (12) with the specific witnesses \(\rho \) and \(\mathbb {1}\) from this transfer-power identity.
First block \(J\) sites. The normalized diagonal of this block is the reindexed rank-one matrix \(|\rho )(\mathbb {1}|\), and the trace-one condition is exactly \((\mathbb {1}|\rho )=1\). The exact second-block contraction theorem then supplies both identities after a further \(D^2\) sites. The canonical equivalence between iterated blocking and direct length-\(JD^2\) blocking transports each double-layer coefficient without changing either virtual witness.
Let \(d{\gt}0\), let \(\mathcal U\) be an MPU, and let \(K\geq 0\). For \(p,q\in \{ 0,\ldots ,d{-}1\} \times \{ 0,\ldots ,d{-}1\} \), separate the first two output sites from a tail \(\tau \in \{ 0,\ldots ,K{-}1\} \to \{ 0,\ldots ,d{-}1\} \). Then
The retained matrix entry has the reversed order \((q,p)\), and the global contraction is the output-first equation \(U^{(K+2)}(U^{(K+2)})^\dagger =\mathbb {1}\).
Apply output-first unitarity before performing any source-factor rearrangement. For each fixed tail, the inner sum is the corresponding identity-matrix entry, hence it is \(\delta _{p,q}\). There are exactly
tails, and \(d^{-K}d^K=1\) because \(d{\gt}0\).
Let \(d{\gt}0\), let \(\mathcal U\) be an MPU, and let \(K\geq 0\). For \(p,q\in \{ 0,\ldots ,d{-}1\} \times \{ 0,\ldots ,d{-}1\} \), separate the first two input sites from a common input tail \(\tau \in \{ 0,\ldots ,K{-}1\} \to \{ 0,\ldots ,d{-}1\} \). Then
The global contraction is the input-first equation \((U^{(K+2)})^\dagger U^{(K+2)}=\mathbb {1}\). This is the input-oriented tail contraction supporting the network calculation in [ CPGSV17 , Theorem III.8, Section III.B, equations (31)–(32) ] .
Apply input-first unitarity for each fixed tail. The inner output sum is \(\delta _{p,q}\). Summing over the \(d^K\) tails and multiplying by \(d^{-K}\) leaves the same Kronecker delta.
Let \(\mathcal U\) generate an MPU, and suppose its direct length-\(k\) block is simple for some \(k{\gt}0\). Then its direct length-\(k'\) block is simple for every \(k'\geq k\), without any divisibility condition on \(k'\) by \(k\). This is the corollary following [ CPGSV17 , Proposition III.3 ] .
First suppose vectors \(a,b\) satisfy the rank-one insertion identity
Set \(x_{ij}=(a|W^{ij}|b)\). Applying this identity around the constant two-site and three-site words, and then using MPU unitarity, gives
Hence \(x_{ii}=1\), while \(x_{ij}=0\) for \(i\ne j\), so the same vectors obey the one-letter contraction \((a|W^{ij}|b)=\delta _{ij}\).
For the successor step, take four words \(I,J,K,L\) of length \(k+1\) and write \(W^{I,J}=W^{i_0j_0}\cdots W^{i_kj_k}\), with the analogous notation for \(W^{K,L}\). Apply the length-\(k\) rank-one insertion identity to the suffixes \(I_+=(i_1,\ldots ,i_k)\), \(J_+=(j_1,\ldots ,j_k)\) and the prefixes \(K_-=(k_0,\ldots ,k_{k-1})\), \(L_-=(\ell _0,\ldots ,\ell _{k-1})\). Then
Thus the length-\((k+1)\) block has the same insertion witnesses, and the two- and three-site argument gives their one-letter contraction. Induction on \(k'-k\) proves the claim.
Let \(d,D{\gt}0\). Every bond-\(D\) MPU tensor becomes simple after blocking exactly \(D^4\) sites. For each fixed \(L\), the map
is continuous. Consequently, if \(x\mapsto \mathcal W(x)\) is a continuous family of bond-\(D\) MPU tensors, then \(x\mapsto \mathcal W(x)_{D^4}\) is a continuous family of simple tensors. This is only the common-blocking step in the proof of Proposition 19.9.59; it does not choose reduced representatives or assert constancy of their source ranks.
Choose for each tensor a positive simple blocking length \(k\leq D^4\). Simplicity persists at every later direct blocking, hence at \(D^4\). For fixed blocked ket and bra words, every matrix entry of the blocked tensor is a finite sum of products of entries of the original tensor. These entries are continuous, and the finite product and sum operations preserve continuity. Applying this fixed blocking map to the original path proves the path statement.
Let \(\mathcal U\) be simple with positive physical dimension, let \(W\) be its double layer, and let \(E\) be the normalized diagonal of \(W\). Suppose that for some \(J{\gt}0\),
Then the simple insertion identity holds with these same supplied witnesses:
Choose the witnesses \(a,b\) from simplicity and put \(R=|b)(a|\). Expanding the normalized diagonal as the average of the diagonal letters, the two defining simple contractions give
Indeed, insert \(R\) on both sides of each diagonal letter; its middle sandwich is \(R W^{qq}R=R\) by the first contraction, and the normalized sum contains \(d\) equal terms. The same identity makes \(W^{ij}E^N=W^{ij}E\) for every \(N{\gt}0\). Hence
Substituting the supplied equality \(E^J=|\rho )(\Phi |\) proves the claim. No uniqueness of rank-one factorizations is used.
Let \(R\) be the normalized physical diagonal of the double layer of an MPO tensor. Suppose
Then the two boundaries of the stabilized diagonal tail are fixed:
This is the fixed-pair normalization from the canonical-form-II fixed-point equations in [ CPGSV17 , equations (6a)–(6b) ] and underlies the tail contractions in [ CPGSV17 , Theorem III.8, Section III.B, equations (31)–(32) ] .
Substitute \(R^J=\lvert \rho )(\Phi \rvert \) in each expression and use \((\Phi \rvert \rho )=1\).
Let \(W\) be the double layer of an MPO tensor, and let \(R\) be its normalized physical diagonal. If
then, for all physical indices \(i,j,k,\ell \),
This is the exact diagonal-word tail between the two retained letters in [ CPGSV17 , Theorem III.8, Section III.B, equations (31)–(32) ] .
Replace the rank-one insertion by \(R^J\), apply the diagonal-word expansion, and distribute the two retained letters across the normalized finite sum.
In the setting of the preceding theorem, suppose in addition that the same fixed pair satisfies the supplied two-letter contraction
for all physical indices. Then
Thus the supplied insertion and its stabilized transfer power occur in one aligned network, as required in the calculation of [ CPGSV17 , Theorem III.8, Section III.B, equations (31)–(32) ] .
Apply the supplied two-letter contraction and then replace its rank-one insertion by the normalized diagonal-word tail.
For every MPU of positive physical and bond dimensions there are normalized left and right fixed vectors \(\Phi \) and \(\rho \) of the transfer matrix such that the unblocked double layer satisfies
The unblocked tensor need not satisfy the rank-one insertion identity.
Let \(E^J=|\rho )(\Phi |\) be a stabilized transfer power. Decompose \(W^{ij}=\delta _{ij}E+R_{ij}\). Fixedness and normalization give
The mixed residual trace identity applied to one residual factor followed by \(J\) diagonal factors gives
Since \(E^J=|\rho )(\Phi |\), these two trace identities are exactly the displayed contraction.
19.4 Trace-power lemma
For the transfer-matrix spectrum argument of [ CPGSV17 , Proposition 1 ] we first record the algebraic fact that a matrix whose positive powers all have trace \(1\) must have characteristic polynomial \(X^{n-1}(X-1)\). The source assumes only the powers with exponent strictly greater than one; this weaker hypothesis determines the nonzero spectrum as a set.
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.
The matrix \(M_{i_0}\) from definition 19.4.1 satisfies
\(M_{i_0}^k = M_{i_0}\) for every \(k\ge 1\),
\(\operatorname{tr}(M_{i_0}^k) = 1\) for every \(k\ge 1\),
\(\chi _{M_{i_0}}(X) = X^{\, n-1}(X-1)\).
The first two items follow from \(1^k=1\) and \(0^k=0\) for \(k\ge 1\). The characteristic polynomial is computed from \(\chi _{\operatorname{diag}(d)}(X) = \prod _i (X - d_i)\).
Let \(A\) be an \(n\times n\) complex matrix with \(n\ge 1\). If \(\operatorname{tr}(A^k)=1\) for every \(k\) with \(1\le k\le n\), then
Choose any index \(i_0\) (the hypothesis \(n\ge 1\) makes this possible) and let \(B\) be the rank-one diagonal idempotent from Lemma 19.4.2. For \(1\le k\le n\) we have \(\operatorname{tr}(A^k)=1=\operatorname{tr}(B^k)\). The finite-range half of the Newton–Girard the corresponding theorem in the companion quantum-channel volume [ LTC26 , “Newton–Girard trace recursion” ] then yields \(\chi _A = \chi _B = X^{\, n-1}(X-1)\).
Let \(A\) be an \(n\times n\) complex matrix. If \(\operatorname{tr}(A^k)=1\) for every \(k\ge 1\), then
(The hypothesis already forces \(n\ge 1\) because the trace of a \(0\times 0\) matrix is \(0\), contradicting \(\operatorname{tr}(A)=1\).)
If \(n=0\) the hypothesis \(\operatorname{tr}(A)=1\) contradicts the fact that the trace of an empty matrix is \(0\); the conclusion follows vacuously. When \(n\ge 1\), the hypothesis on all positive powers implies the finite-range hypothesis of Theorem 19.4.3.
Let \(A\) be an \(n\times n\) complex matrix such that \(\operatorname{tr}(A^k)=1\) for every \(k{\gt}1\), and let \(m{\gt}1\). Then \(\operatorname{tr}((A^m)^r)=1\) for every \(r\ge 1\).
For \(r\ge 1\), one has \((A^m)^r=A^{mr}\) and \(mr{\gt}1\), so the hypothesis at exponent \(mr\) gives the result.
Let \(A\) be an \(n\times n\) complex matrix. If \(\operatorname{tr}(A^k)=1\) for every integer \(k{\gt}1\), then
Thus \(1\) occurs and every nonzero spectral value equals \(1\). This is the set-spectrum consequence of [ CPGSV17 , Proposition III.1 ] ; set equality alone does not assert algebraic multiplicity.
For every positive integer \(r\), the shifted moments give
Hence
If \(\mu \in \sigma (A)\) is nonzero, spectral mapping places \(\mu ^2\) in \(\sigma (A^2)\) and \(\mu ^3\) in \(\sigma (A^3)\), so
Conversely, spectral mapping from \(1\in \sigma (A^2)\) supplies a nonzero spectral value of \(A\), which must be \(1\).
For a complex square matrix \(A\), write \(G_1(A)\) for its maximal generalized eigenspace at the eigenvalue \(1\). Then
The commuting factorization
gives, for every nonnegative integer \(k\),
Therefore every vector annihilated by a power of \(A-\mathbb {1}\) is annihilated by the corresponding power of \(A^2-\mathbb {1}\).
For every complex square matrix \(A\),
With \(G_1\) as above, algebraic multiplicity equals the dimension of the maximal generalized eigenspace. Hence
Let \(A\) be an \(n\times n\) complex matrix. If \(\operatorname{tr}(A^k)=1\) for every integer \(k{\gt}1\), then
Thus \(1\) is the sole nonzero eigenvalue and has algebraic multiplicity one. This is the exact shifted-moment conclusion of [ CPGSV17 , Proposition III.1 ] .
Applying the all-positive trace-power theorem to \(A^2\) gives
The multiplicity inequality therefore bounds \(\operatorname {mult}_{1}(\chi _A)\) by one. The set-spectrum theorem shows that \(1\) occurs, so this multiplicity is positive and hence equals one. The same set-spectrum theorem excludes every other nonzero root. Splitting the monic characteristic polynomial now yields
Let \(A\) be an \(n\times n\) complex matrix. If \(\operatorname{tr}(A^k)=0\) for every integer \(k\geq 1\), then
The zero matrix has the same positive trace powers as \(A\) and characteristic polynomial \(X^n\). Newton–Girard therefore identifies the two characteristic polynomials. This is the characteristic-polynomial consequence of the all-positive trace condition obtained in [ CPGSV17 , proof of Proposition III.3 ] .
Let \(A\) be an \(n\times n\) complex matrix such that \(\operatorname{tr}(A^k)=0\) for every \(k{\gt}1\), and let \(m{\gt}1\). Then \(\operatorname{tr}((A^m)^r)=0\) for every \(r\geq 1\). This auxiliary result supports the generalization below to a weaker hypothesis; the source itself supplies vanishing for every positive exponent.
For \(r\geq 1\), one has \((A^m)^r=A^{mr}\) and \(mr{\gt}1\), so the hypothesis at exponent \(mr\) gives the result.
Let \(A\) be an \(n\times n\) complex matrix. If \(\operatorname{tr}(A^N)=0\) for every integer \(N{\gt}1\), then \(A\) is nilpotent and
The source obtains the all-positive trace hypothesis in [ CPGSV17 , proof of Proposition III.3 ] ; its specialization is the elementwise nilpotence assertion used before the nil-matrix theorem. The result above is a stronger generalization because it assumes vanishing only for \(N{\gt}1\). It is useful independently and requires no positivity, normality, or diagonalizability.
Every positive power of \(A^2\) has vanishing trace. Hence
If \(\mu \) is a spectral value of \(A\), spectral mapping shows that \(\mu ^2\) is a root of this characteristic polynomial. Therefore \(\mu ^2=0\), and hence \(\mu =0\). Thus every root of the split monic polynomial \(\chi _A\) is zero, so \(\chi _A(X)=X^n\). Cayley–Hamilton now proves that \(A\) is nilpotent.
The standard-form construction uses the compact singular-value decomposition in which the intermediate dimension is the rank rather than the smaller ambient dimension.
Let \(M\in M_{m,n}(\mathbb {C})\) be a complex matrix of rank \(r\). There are matrices \(V\in M_{r,m}(\mathbb {C})\) and \(U\in M_{r,n}(\mathbb {C})\) and strictly positive real numbers \(s_0,\ldots ,s_{r-1}\) such that, for \(D=\operatorname{diag}(s_0,\ldots ,s_{r-1})\),
Thus no zero singular values occur in the intermediate space. This is the compact decomposition used in the standard-form construction of [ CPGSV17 , Section III ] .
Apply the spectral theorem to \(MM^\dagger \). Its nonzero eigenvalues are positive, and their number is \(\operatorname {rank}(MM^\dagger )=\operatorname {rank}(M)=r\). For an orthonormal eigenvector \(e_j\) with eigenvalue \(\lambda _j{\gt}0\), put \(w_j=M^\dagger e_j\). Then \(\langle w_j,w_k\rangle =\lambda _j\delta _{jk}\). Hence the vectors \(\lambda _j^{-1/2}w_j\) are orthonormal. Expanding each column of \(M\) in the eigenbasis and discarding the zero-eigenvalue terms gives
which has the form in (88).
We now introduce the two matrices used for the singular-value decompositions of a Matrix Product Unitary tensor in Section III of [ CPGSV17 ] . Write \(\mathcal U^i_{j,\alpha ,\beta }\) for the entries of the tensor, where \(i,j\) are physical indices and \(\alpha ,\beta \) are auxiliary indices.
The matrix \(M_1\) is obtained by combining the left auxiliary index with the lower physical index, and the upper physical index with the right auxiliary index. Thus
Transporting the row and column indices along the standard product enumerations gives a matrix \(\widetilde M_1\) indexed by \(\{ 0,\ldots ,Dd-1\} \times \{ 0,\ldots ,dD-1\} \). Its entries are obtained by applying \(M_1\) to the inverse images of the enumerated indices, and
The matrix \(M_2\) is obtained by combining the left auxiliary index with the upper physical index, and the lower physical index with the right auxiliary index. Thus
Transporting the row and column indices along the standard product enumerations gives a matrix \(\widetilde M_2\) indexed by \(\{ 0,\ldots ,Dd-1\} \times \{ 0,\ldots ,dD-1\} \). Its entries are obtained by applying \(M_2\) to the inverse images of the enumerated indices, and
The right rank is
The left rank is
Physical adjunction exchanges the two source cuts and conjugates every entry:
The physical adjoint exchanges the upper and lower physical indices while leaving both auxiliary indices fixed. Thus the first regrouping becomes the entrywise conjugate of the second, and conversely.
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.
The normalized flattening of the physical adjoint satisfies
For every pair of physical indices \((i,j)\),
The first equality is the normalized flattening, the second is physical adjunction, and the third uses the reality of \(d^{-1/2}\).
Given canonical-form-II data for \(\widetilde{\mathcal U}\), define the transformed tuple by
These data reconstruct \(\widetilde{\mathcal U^\sharp }\).
If \(\widetilde{\mathcal U}\) is in canonical form II, then \(\widetilde{\mathcal U^\sharp }\) is in canonical form II.
Theorem 8.6.4 independently preserves normality and left-canonical normalization under entrywise conjugation, and Theorem 8.6.8 preserves these properties under the physical-pair swap. Conjugating the coisometry equation gives
Conjugating the reconstruction at the swapped physical index gives
Finally, each diagonal positive matrix \(\Lambda _k\) is real, and therefore
Physical relabeling leaves this transfer map unchanged, which proves the transformed fixed-point equation.
If canonical-form-II data for \(\widetilde{\mathcal U}\) have full support, then the data from Definition 19.4.21 also have full support.
Complex conjugation of the weights leaves the block indices and their bond dimensions unchanged, so the full-support sum is unchanged.
Physical adjunction exchanges the two source ranks:
Apply Theorem 19.4.18 and use invariance of matrix rank under entrywise conjugation.
A bijective relabeling of the physical basis simultaneously relabels the physical component of every source-cut row and column. Hence
with the right and left orientations unchanged.
Evaluate each cut after relabeling. In \(M_1\), the lower physical index lies in the row and the upper physical index lies in the column; in \(M_2\) these roles are exchanged. Applying \(e\) to both entries gives the two displayed row-and-column equivalences.
Bijective physical relabeling preserves both source ranks:
Matrix rank is unchanged by bijective row and column relabelings. Apply this fact to the two source-cut identities.
Let \(A=\{ A^i\} _{i\in \{ 0,\ldots ,d{-}1\} }\) and \(B=\{ B^k\} _{k\in \{ 0,\ldots ,e{-}1\} }\) have auxiliary dimensions \(D\) and \(E\), respectively. Write
for the standard finite-product coordinate bijection. Their independent tensor product \(A\boxtimes B\) has physical dimension \(de\), auxiliary dimension \(DE\), and local matrices determined by
For every \(N\), the same coordinates give the pointwise splitting
This is infrastructure for the tensoring clause in the proof of the MPU Index Theorem IV.6(ii) [ CPGSV17 , lines 824–845 ] , not a separately stated theorem of that paper. The notation \(A=\{ A^i\} _i\) agrees with the canonical-form block notation in [ CPGSV16 , Section 2.3, lines 214–245 ] .
If \(w\) has length \(n\) and
then
Induct on the length of \(w\). For the empty word, use \(\mathbb {1}_D\otimes \mathbb {1}_E=\mathbb {1}_{DE}\) and invariance of the identity under the coordinate bijection. For a leading letter \(\pi _{d,e}(i,k)\), equation (114) and the induction hypothesis reduce the claim to
after transporting multiplication through the same coordinate bijection.
Let \(S_N(A)=\operatorname{span}_{\mathbb {C}}\{ A^w:|w|=N\} \). The pointwise splitting of length-\(N\) words gives
In particular, if \(S_N(A)=M_{D}(\mathbb {C})\) and \(S_N(B)=M_{E}(\mathbb {C})\), then \(S_N(A\boxtimes B)=M_{DE}(\mathbb {C})\). If \(A\) and \(B\) are algebraically normal, then \(A\boxtimes B\) is algebraically normal.
This is the homogeneous-word infrastructure for the tensoring sentence in the proof of the MPU Index Theorem IV.6(ii) [ CPGSV17 , lines 824–845 ] , not a separate theorem stated there. It applies to the retained blocks in the canonical-form decomposition [ CPGSV16 , Section 2.3, lines 214–245 ] .
For every product word \(w\), let \((u,v)\) be its pointwise component words. The bijection between \(w\) and \((u,v)\) and Theorem 19.4.28 give
which proves (121) after taking linear spans. Pure tensors span \(M_{D}(\mathbb {C})\otimes M_{E}(\mathbb {C})\). The Kronecker linear equivalence, followed by simultaneous reindexing along \(\pi _{D,E}\), is an isomorphism onto \(M_{DE}(\mathbb {C})\); this proves the fixed-length consequence. If \(S_{N_A}(A)=M_{D}(\mathbb {C})\) and \(S_{N_B}(B)=M_{E}(\mathbb {C})\) for positive \(N_A,N_B\), then persistence at positive multiples gives
Applying the fixed-length consequence at \(N_A N_B\) proves algebraic normality of \(A\boxtimes B\).
Let \(A\) and \(B\) be CPSV normal tensors, and suppose that
Then \(A\boxtimes B\) is a CPSV normal tensor and is left-canonical:
This is the retained-normal-block infrastructure for the tensoring sentence in the proof of the MPU Index Theorem IV.6(ii) [ CPGSV17 , lines 824–845 ] , not a separate theorem stated there. The retained-normal-block terminology is that of the canonical-form decomposition in [ CPGSV16 , Section 2.3, lines 214–245 ] ; the two left-canonical identities are explicit infrastructure hypotheses, not an additional assertion attributed to that passage.
The adjoint and multiplication laws for Kronecker products, together with the fact that simultaneous reindexing preserves adjoints, products, and finite sums, give
proving (125). Spectral normality gives positive lengths \(N_A,N_B\) at which the two homogeneous word spans are full. Theorem 19.4.29 therefore makes \(A\boxtimes B\) algebraically normal at the common length \(N_A N_B\). The algebraically normal, left-canonical tensor is CPSV normal by Theorem 8.6.3.
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 \(\mathcal V\).
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.
If \(q\) denotes the blocked product-alphabet equivalence, then each site \(s\) satisfies
Each equality follows by composing the inverse product encoding at every site with the corresponding projection.
For every system size \(N\), the sitewise configuration bijection identifies the finite-size operator with
Grouping the two auxiliary indices makes every closed-word matrix a Kronecker product. The identity
then gives the stated finite-size operator after the sitewise configuration bijection.
The independent tensor product of two matrix product unitary tensors is a matrix product unitary tensor. This is the tensoring operation in the proof of [ CPGSV17 , Theorem IV.6(ii) ] .
The multiplication rule \((A\otimes B)(C\otimes D)=AC\otimes BD\) gives
Simultaneous reindexing preserves this unitarity equation.
For every blocking length \(L\), separating each blocked product letter by the canonical equivalence \(q\) gives the exact identity
The ordering of the two component words agrees with the independent tensor-product convention.
Evaluate both sides on blocked ket and bra letters. Word multiplication of local Kronecker products gives
The two decoding identities identify \(I_1,I_2\) and \(J_1,J_2\) with the corresponding letters of \(q(I)\) and \(q(J)\).
The source-cut shuffle sends
The analogous map applies to column indices.
The source cuts satisfy
Thus the paper’s right and left orientations are unchanged.
Evaluate both source cuts entry by entry after the displayed row and column shuffles. Each entry is the product of the corresponding entries of the two source cuts.
The right and left source-cut ranks are multiplicative:
This is the rank-multiplication ingredient for the tensoring case in the proof of [ CPGSV17 , Theorem IV.6(ii) ] .
Matrix rank is unchanged by row and column bijections and satisfies \(\operatorname {rank}(A\otimes B)=\operatorname {rank}(A) \operatorname {rank}(B)\).
Let \(\mathcal U^{[L]}\) denote the direct blocking of \(L\) sites of a tensor \(\mathcal U\) with physical dimension \(d\), and let \(r_L\) and \(\ell _L\) be its right and left source-cut ranks. Then
These are the one-site forms of the two upper bounds in the proof of [ CPGSV17 , Proposition IV.2 ] .
For the first cut, choose a compact factorization of the source matrix of \(\mathcal U^{[L]}\) through a space of dimension \(r_L\). Split each physical word of length \(L+1\) into its first letter and its remaining word of length \(L\). For suitable rectangular matrices \(A_{1,L}\) and \(B_{1,L}\), the successor source matrix factors as
The intermediate space is \(\mathbb C^d\otimes \mathbb C^{r_L}\), and hence
Interchanging the upper and lower physical indices gives matrices \(A_{2,L}\) and \(B_{2,L}\) with
This factorization passes through \(\mathbb C^d\otimes \mathbb C^{\ell _L}\), so
If \(L_0\leq L\), then
These are the two blocking upper bounds used in the proof of [ CPGSV17 , Proposition IV.2 ] .
Induct on \(L-L_0\). The case \(L=L_0\) is equality. For the right rank, the successor bound and the induction hypothesis give
The same multiplication by \(d\) gives
Let \(d{\gt}0\) and \(L_0\leq L\). Suppose the endpoint source-cut ranks satisfy the supplied product identities
Then the blocking bounds are exact:
This is the saturation step in the proof of [ CPGSV17 , Proposition IV.2 ] .
The blocking bounds give
Their upper bounds have product
Since \(d{\gt}0\), the latter product is positive, so both \(r_L\) and \(\ell _L\) are positive. Hence neither upper bound can be strict, and both equalities follow.
For \(\rho \in M_{D}(\mathbb {C})\), let
In the virtual–physical row order \((\alpha ,j)\) of \(M_1\), define
Thus the source expression \(\mathbb {1}_d\otimes \rho \) is first written in the physical–virtual order \((j,\alpha )\) and then reindexed by \(\chi \) to the virtual–physical order \((\alpha ,j)\). The two Kronecker products are not identified without this reindexing. This is the source weight in [ CPGSV17 , equation labelled Y1Y1X1X1 ] .
If \(\rho \) is positive definite, then \(W_\rho \) is positive definite.
The Kronecker product of the positive-definite matrices \(\rho \) and \(\mathbb {1}_d\) is positive definite.
Choose compact singular-value decompositions
with intermediate dimensions \(r\) and \(\ell \), respectively. For \(\rho \in M_{D}(\mathbb {C})\), define the compressed positive metric
The inverse square root \(A^{-1/2}\), rather than \(A\) itself, is the normalization factor used in the first weighted source factor. This is the source-factor construction of [ CPGSV17 , Section III ] .
If \(\rho \) is positive definite, then \(A\) is positive definite.
The map \(V_1^\dagger \) is injective because \(V_1V_1^\dagger =\mathbb {1}_r\). Positive definiteness of \(W_\rho \) is therefore preserved by this congruence.
Let \(\rho \in M_{D}(\mathbb {C})\) be positive definite. Define
These are the factors in [ CPGSV17 , Section III ] .
Since \(A\) is positive definite, its support projection is the identity and
Therefore
For the second cut, \(V_2V_2^\dagger =U_2U_2^\dagger =\mathbb {1}_\ell \) and \(D_2D_2^{-1}=\mathbb {1}_\ell \) give
The two factorizations are
These are the source decompositions in [ CPGSV17 , Section III ] .
The source-factor construction supplies \(M_1=X_1Y_1\). The second compact singular-value decomposition gives \(M_2=X_2Y_2\) directly.
The entries of the two products satisfy
These are the two graphical entry identifications in [ CPGSV17 , Section III ] .
Evaluate the two matrix factorizations at \(((\alpha ,j),(i,\beta ))\) and \(((\alpha ,i),(j,\beta ))\), respectively.
The left factors satisfy
These are [ CPGSV17 , Section III ] .
The source-factor construction supplies \(X_1^\dagger W_\rho X_1=\mathbb {1}_r\); evaluating at \((r,r')\) gives the displayed weighted sum. Since \(X_2=V_2^\dagger \), the coisometry identity \(V_2V_2^\dagger =\mathbb {1}_\ell \) gives \(X_2^\dagger X_2=\mathbb {1}_\ell \); evaluating at \((l,l')\) gives the second displayed entry sum.
For every tensor \(\mathcal U\), physical indices \(p,q\), and auxiliary indices \(a,b\),
Thus the factor indexed by \((p,a)\) is starred and the factor indexed by \((q,b)\) is unstarred. This is the second-cut rotation used in [ CPGSV17 , Theorem III.8, Section III.B, equations (31)–(32) ] . The identity requires no Matrix Product Unitary, dimension, positivity, or simple-tensor hypothesis.
Substitute the entry factorization \(M_2=X_2Y_2\) into the left-hand side and reorder the finite sums. The contraction of the two \(X_2\) factors is \(X_2^\dagger X_2=\mathbb {1}_\ell \), so the two source indices coincide and the remaining sum is the displayed Gram matrix of \(Y_2\).
The right source factors are recovered from the two source matrices by
Here \(W_\rho =\rho \otimes I_d\) in the product-index order \((\text{left virtual},\text{physical})\).
Multiply \(M_1=X_1Y_1\) on the left by \(X_1^\dagger W_\rho \) and use \(X_1^\dagger W_\rho X_1=I_r\). Similarly, multiply \(M_2=X_2Y_2\) on the left by \(X_2^\dagger \) and use \(X_2^\dagger X_2=I_\ell \).
The right factors satisfy
This is [ CPGSV17 , Section III ] .
The source-factor construction supplies \(Y_1Z_1=\mathbb {1}_r\). For the second cut,
Given source-cut decompositions \(M_1=X_1Y_1\) and \(M_2=X_2Y_2\), define
by
The source legs of \(u\) have order \(\ell \times r\), whereas those of \(v\) have order \(r\times \ell \). These are the two source tensors introduced after the source-cut decompositions in [ CPGSV17 , Section III ] . The entry formulas also fix the conjugate-transpose orientation. A supplied source-factor witness provides these decompositions. The fixed compact-SVD factors specialize the formulas to the previously defined tensors.
Identify a two-site physical configuration with an ordered pair in \(\{ 0,\ldots ,d{-}1\} \times \{ 0,\ldots ,d{-}1\} \). Then
The transpose records the periodic entry orientation. The map \(\operatorname {swap}_{r,\ell }\) is the explicit equivalence \(r\times \ell \simeq \ell \times r\); the two product orders are not silently identified. This is the bond swap, or one-site translation, in the source-tensor decomposition of [ CPGSV17 , Section III ] .
At physical pairs \((i_1,i_2)\) and \((j_1,j_2)\), expand the periodic trace as
Use the entry forms of \(M_1=X_1Y_1\) for the first letter and \(M_2=X_2Y_2\) for the second. Reordering the four scalar finite sums gives
Reindexing both physical configuration spaces and explicitly swapping \((r,l)\) to \((l,r)\) gives the matrix identity.
For \(q=(q_1,q_2)\) and \(j=(j_1,j_2)\), the untraced two-site product is
The intermediate partial contraction is
Use the three identities
Substitution and reordering of the scalar finite sums give the displayed partial contraction and the untraced two-site identity, with the virtual indices \(\alpha \) and \(\gamma \) left open. Encoding both physical pairs by the standard equivalence \(\{ 0,\ldots ,d{-}1\} \times \{ 0,\ldots ,d{-}1\} \simeq \{ 0,\ldots ,({-}1\} d^2)\) gives the corresponding formula for the concrete two-site block.
The reflected factorization of the same untraced two-site product is
Thus the physical row of \(v\) is \((j_2,j_1)\), while its source column is \((r,l)\). Both orders remain explicit in the formula for the concrete two-site block under \(\{ 0,\ldots ,d{-}1\} \times \{ 0,\ldots ,d{-}1\} \simeq \{ 0,\ldots ,({-}1\} d^2)\).
Expand the first tensor letter through \(M_2=X_2Y_2\) and the second through \(M_1=X_1Y_1\). Then
Summing over \(l\) and \(r\) gives the result. The two-site block statement follows by decoding its ket and bra indices into ordered pairs.
Separate the first two physical sites of a length-\((K+2)\) configuration. For an output pair \(q\), an input pair \(j\), and tails \(\tau ,\zeta \), the periodic entry first expands as
where \(T_{\tau ,\zeta }\) is the open \(K\)-site virtual word. Reconstructing \(Y_2\) gives
No power of \(d\) occurs in either algebraic expansion.
Reindexing each configuration into its first two coordinates and its tail gives
Insert the open two-site factorization from the preceding theorem. The recovery identity and its entrywise evaluation are
Substitution combines the last raw tensor letter with \(T_{\tau ,\zeta }\) by matrix multiplication and gives the second displayed expansion.
For
define the coefficient indexed by \(l,r,l',r'\) to be
This is the coefficient obtained from the two open-tail expansions in the normalized Gram contraction in the proof of the source-tensor isometry lemma in [ CPGSV17 , Section III ] . The external factor \(u_{(l,r),q}\) is unstarred, while \(u_{(l',r'),p}\) is starred. Thus the displayed coefficient contains the unique normalization factor \(d^{-K}\).
For a tail of length \(K\), define
The expression contains exactly one normalization factor \(d^{-K}\).
Let \(W_{\mathrm{out}}\) be the output-first double layer. Then
Expand the normalized diagonal of the blocked output layer, reindex a blocked physical index by its decoded word, and evaluate the product double layer. The physical adjoint contributes the conjugated second word, while the blocked physical dimension supplies \(d^{-K}\).
The forward kernel is
Both doubled bond indices put the unstarred component before the starred component.
Expand the matrix product over its doubled bond index. The first output-layer letter supplies the sum over \(j_2\), and the normalized diagonal power supplies the common output word and contracted input word. Reordering finite sums gives the defining forward kernel without reversing the virtual chain.
On the range of the first source cut,
Substitute \(X_1^\dagger W_\rho M_1=Y_1\) and then use \(M_1=X_1Y_1\).
On the range of the second source cut,
This does not assert \(X_2X_2^\dagger =1\) on the ambient product-index space.
Substitute \(X_2^\dagger M_2=Y_2\) and then use \(M_2=X_2Y_2\).
Assume \(d{\gt}0\), let \(\mathcal U\) be an MPU, and let \(\rho \) be positive definite. Expanding both \((K+2)\)-site periodic entries through \(X_1,u,X_2\) and the open tail gives
where \(\mathcal A_{q;\tau ,j,\zeta }\) is the full source-\(u\) expansion with the \(q\) entry unstarred. This theorem evaluates the displayed expansion directly. It neither regroups the sum through \(C^{(K)}_{lr,l'r'}\) nor identifies it with the ordinary Gram matrix \(u^\dagger u\).
Replace each source-factor sum by its periodic MPO entry. Reindex the input configuration into its first two coordinates and its length-\(K\) tail, then apply normalized output-first MPU coisometry.
Let \(\mathcal U\) be an MPO tensor, let \(\rho \) be positive definite, and let \(p,q\in \{ 0,\ldots ,d{-}1\} \times \{ 0,\ldots ,d{-}1\} \). Write \(W_{\mathrm{out}}\) for the output-first double layer of \(\mathcal U\). Then
Thus \(q\) is the unstarred ket index, \(p\) is the starred bra index, and each doubled bond is ordered as \((\text{ket},\text{bra})\). This is the first closed network in [ CPGSV17 , Eq. uUnitary and lines 545–556 ] .
Expand both source-\(u\) entries and contract their common source indices. Weighted recovery of \(Y_1\), followed by the first source-cut range identity, replaces \(Y_1^\dagger Y_1\) by \(M_1^\dagger W_\rho M_1\). Expanding \(M_1\) and the output-first double layer gives the displayed \(\rho \)-weighted letter, while the remaining \(X_2\) factors form the stated range projector.
Assume \(d{\gt}0\). For any MPO tensor \(\mathcal U\), tail length \(K\), and \(p,q\in \{ 0,\ldots ,d{-}1\} \times \{ 0,\ldots ,d{-}1\} \), let \(W_{\mathrm{out}}\) be its output-first double layer and let \(E_{\mathrm{out}}\) be the normalized physical diagonal of \(W_{\mathrm{out}}\). Then
The retained letters occur in \((q,p)\) order, so \(q\) is unstarred and \(p\) is starred; their doubled bonds are in \((\text{ket},\text{bra})\) order. This is the normalized closed output tail in [ CPGSV17 , Eq. uUnitary and lines 550–556 ] .
Close the contracted input configuration as the periodic MPO of the output-first double layer. Splitting off the two retained sites leaves a common diagonal word of length \(K\). Expanding the normalized diagonal power as the normalized sum of those words and using linearity of the trace gives the displayed closed trace.
Define the second-cut metric by \(H_2=Z_2Z_2^\dagger \). Then the range projector of the second source cut factors as
This is a factorization through the chosen right inverse; it does not assert that \(H_2\), \(X_2X_2^\dagger \), or \(M_2M_2^\dagger \) is the identity.
From \(M_2=X_2Y_2\) and \(Y_2Z_2=1\) one obtains \(M_2Z_2=X_2\). Multiply this identity by its adjoint and reassociate the three factors.
Let \(\mathcal U\) be an MPO tensor, let \(\rho \) be positive definite, and let \(p,q\in \{ 0,\ldots ,d{-}1\} \times \{ 0,\ldots ,d{-}1\} \) be the retained physical pairs. No MPU or positive-dimension hypothesis is required. With \(H_2=Z_2Z_2^\dagger \), the ordinary source-\(u\) Gram entry is
The \(q\) factors are unstarred and the \(p\) factors are starred.
Substitute the second-cut metric factorization into the closed output-letter formula and expand both matrix products. Reordering the finite sums gives the displayed eight-index expression.
Suppose a direct block of length \(K\) satisfies the two-letter contraction with specified vectors \((a|\) and \(|b)\). For an arbitrary MPO tensor, one suffix/prefix extension preserves the same insertion \(|b)(a|\) at direct length \(K+1\). For any MPU tensor, specified vectors satisfying this two-letter identity also satisfy the one-letter identity with the same vectors. Applying that implication after two extensions gives, at direct length \(K+2\),
No positivity or nonzero-dimension hypothesis is required. This is the corollary following [ CPGSV17 , Proposition III.3 ] .
Extend the supplied two-letter identity twice by overlapping suffix and prefix windows. For \(x=(a|W^{IJ}|b)\), the constant configurations of lengths two and three give \(x^2=\delta _{I,J}\) and \(x^3=\delta _{I,J}\), which force \(x=\delta _{I,J}\). Physical blocking preserves MPU unitarity, so this one-letter identity and the twice-extended two-letter identity hold simultaneously at direct length \(K+2\) with the original supplied witnesses.
The doubled-bond coordinate reflection exchanges the two virtual factors,
Independently, spatial reflection reverses a blocked word of length \(K\),
These are support coordinates for the proof of [ CPGSV17 , Lemma III.7 in Section III.B ] , rather than separate definitions printed there. The blocked-word reversal is not used in the transfer transport below; conjugate transposition there reverses only the order of the two blocked letters.
The doubled-bond equivalence and the blocked-word reversal are involutions. The latter sends the letter in position \(k\) to the letter in position \(K-1-k\) and reverses the encoded word.
Decode the doubled bond, apply the product swap, and encode it again. Regard a blocked index as a word indexed by the \(K\) ordered positions, precompose with \(k\mapsto K-1-k\), and reverse the corresponding finite list.
Let \(R(\mathcal U)\) denote the normalized physical diagonal of the double layer of an MPO tensor \(\mathcal U\), and let \(s(a,b)=(b,a)\) be the doubled-bond swap. Then
This identity requires neither the MPU property nor a positivity or positive-dimension hypothesis. It is a coordinate identity supporting the calculation in the proof of [ CPGSV17 , Lemma III.7 in Section III.B ] , not a separate theorem printed there.
Expand the two normalized diagonals. Physical adjunction exchanges the two physical indices, and exchanging the two finite physical sums leaves precisely the simultaneous doubled-bond swap on the row and column.
Let \(\mathcal U\) be an MPO tensor. Write \(E\) for the transfer matrix of its normalized flattening, \(E^\sharp \) for that of \(\mathcal U^\sharp \), and \(E_{\mathrm{out}}\) for the normalized diagonal of the double layer of \(\mathcal U^\sharp \). If \(d{\gt}0\) and
then
No MPU, positivity, trace, positive bond-dimension, or positivity of \(J\) is needed for this implication.
Suppose in addition that \(D{\gt}0\) and \(\operatorname{tr}(\rho )=1\), and set \(K=JD^2\). Then the normalized diagonal \(R_K\) of the double layer of the \(K\)-site block of \(\mathcal U\) satisfies
More generally, if \(d{\gt}0\), \(\rho {\gt}0\), and the normalized diagonal \(R_L\) of an \(L\)-site block satisfies \(R_L=|\rho )(1|\), write \(R_L^\sharp \) for the normalized diagonal of the corresponding block of \(\mathcal U^\sharp \). Then
Consequently, under the combined assumptions \(d,D{\gt}0\), \(\rho {\gt}0\), \(\operatorname{tr}(\rho )=1\), and \(E^J=|\rho )(1|\),
Here all rank-one operators use the column-stacking identification \(M_D(\mathbb C)\cong \mathbb C^{D^2}\).
The blocked normalized diagonal is the corresponding transfer power. Reindex the reflected diagonal by the doubled-bond swap to obtain the transpose weight. When \(\rho \) is positive definite, its Hermitian symmetry converts the conjugate-transposed reflected diagonal to \(|1)(\rho |\). The trace-one condition makes \(|\rho )(1|\) idempotent, so the additional positive \(D^2\) power leaves it unchanged. Conjugating once more gives the stated output-layer orientation.
Let \(\mathcal U\) be an MPU with \(d,D{\gt}0\). Let \(E\) be the transfer matrix of its normalized flattening, and suppose
where \(\rho {\gt}0\), \(\operatorname{tr}(\rho )=1\), and \(J{\gt}0\). Set \(K=JD^2\). For \(n\in \{ K,K+2\} \), let \(\widehat W_n\) be the double layer of the \(n\)-site block of \(\mathcal U^\sharp \). Then, for all blocked physical indices,
The same supplied vectors occur at both lengths; the \(K+2\) identities do not replace them by existentially chosen witnesses.
Apply the direct-block simple contractions to the reflected transfer power \(|\rho ^{\mathsf T})(1|\). Positivity identifies the conjugate-transposed vectors with \((\rho |\) and \(|1)\), while conjugate transposition reverses the order of the two blocked letters. Overlapping-window reblocking preserves these witnesses through the two additional sites. These contractions are support identities for the graphical proof of [ CPGSV17 , Lemma III.7 and its proof in Section III.B ] ; they are not asserted as a separate printed theorem of that paper.
The first theorem rewrites the ordinary source-\(u\) Gram entry, while the second independently rewrites the normalized closed output tail. They do not identify these two expressions or justify inserting the source-cut range projections through the intervening chain. That range-restricted equality remains separate. Neither theorem asserts an ambient \(X_2X_2^\dagger =\mathbb {1}\), \(M_2M_2^\dagger =\mathbb {1}\), or \(Y_2Y_2^\dagger =\mathbb {1}\) identity.
If the normalized internal tail supplies
then the remaining boundary sum is
Reorder the scalar factors so that the sum separates into the weighted \(X_1\) contraction and the unweighted \(X_2\) contraction. Evaluating
at \((r',r)\) and \((l,l')\), respectively, gives the two Kronecker deltas. This is precisely the terminal boundary simplification in the proof of the source-tensor isometry lemma in [ CPGSV17 , Section III ] .
The preceding results evaluate the fully expanded normalized metric, but do not prove that regrouping it through \(C^{(K)}_{lr,l'r'}\) gives the ordinary Gram matrix \(u^\dagger u\). The remaining independent equality is between that Gram matrix and the normalized closed \((K+2)\)-site output tail, with both external source-\(u\) tensors retained before moving the cut. No result uses \(X_2X_2^\dagger =\mathbb {1}\), \(M_2M_2^\dagger =\mathbb {1}\), or \(Y_2Y_2^\dagger =\mathbb {1}\).
Put
with source-bond order \(r\times \ell \). Thus \(Y\) maps \((\mathbb {C}^d\otimes \mathbb {C}^D)^{\otimes 2}\) to \(\mathbb {C}^r\otimes \mathbb {C}^\ell \), and \(Z\) maps in the opposite direction. The regrouping of the open indices is
where \(\operatorname {flat}:\{ 0,\ldots ,D{-}1\} \times \{ 0,\ldots ,D{-}1\} \simeq \{ 0,\ldots ,({-}1\} D^2)\) is the canonical product enumeration. The dressed source-\(v\) Gram equation is
These are the tensorized factors surrounding \(v^\dagger v\) in [ CPGSV17 , Theorem III.8, Section III.B, equations (31)–(32) ] .
Let \(p=(p_1,p_2)\) be the starred physical pair, \(q=(q_1,q_2)\) the unstarred pair, and let \(a=(a_1,a_2)\) and \(b=(b_1,b_2)\) be virtual paths. The weighted first-cut contraction is
The entry of two ordinary double-layer letters from row \((a_1,b_1)\) to column \((a_2,b_2)\) is
Set
Inserting \(|\rho '\rangle \! \langle \Phi '|\) between the two letters gives
Finally, the Gram entry of \(Y_1\otimes Y_2\) expands as
Expand the ordinary and rank-one-inserted matrix products entrywise. For the tensorized Gram entry, use \(X_1^\dagger (\rho \otimes \mathbb {1}_d)X_1=\mathbb {1}_r\) on the first cut and \(X_2^\dagger X_2=\mathbb {1}_\ell \) on the second cut, then reorder the finite sums. No ambient identity \(X_2X_2^\dagger =\mathbb {1}\) is used.
Let \(\rho {\gt}0\). For output pair \(z=(z_1,z_2)\), physical pair \(p=(p_1,p_2)\), and virtual pair \(a=(a_1,a_2)\), the entry of \(v(Y_1\otimes Y_2)\) is
The first source cut has contracted to one local tensor entry, while the two factors from the second cut remain in the same summand. No partial second-cut Gram matrix has been formed. This coupling is the source-cut ordering in the calculation of [ CPGSV17 , Theorem III.8, Section III.B, equations (31)–(32) ] .
Expand the matrix product and the tensor product, reorder the three finite sums, and contract \(X_1Y_1\) by the first source-cut entry factorization. Keep both \(Y_2\) factors together throughout.
The tensorized source factors satisfy
Multiplicativity of the Kronecker product gives
For every positive definite source weight \(\rho \), and for \(x=R^{-1}(p,a)\) and \(y=R^{-1}(q,b)\), the dressed Gram equation is equivalent to
Here \(p,q\in \{ 0,\ldots ,d{-}1\} \times \{ 0,\ldots ,d{-}1\} \), \(a,b\in \{ 0,\ldots ,({-}1\} D^2)\), the indices \(s,t\) range over \(\{ 0,\ldots ,r{-}1\} \times \{ 0,\ldots ,\ell {-}1\} \), and \(z\) ranges over \(\{ 0,\ldots ,d{-}1\} \times \{ 0,\ldots ,d{-}1\} \).
Evaluate both sides of \(Y^\dagger (v^\dagger v)Y=Y^\dagger Y\) at \((x,y)\) and expand the three matrix products. Conversely, apply the displayed equality after regrouping each pair of open source-cut indices by \(R\).
If \(Y\) has a right inverse \(Z\), so that \(YZ=\mathbb {1}\), then for every square matrix \(G\) on the row space of \(Y\),
Conjugate the first equality by \(Z^\dagger \) on the left and \(Z\) on the right. Since \(Z^\dagger Y^\dagger =(YZ)^\dagger =\mathbb {1}\), both outer factors cancel. The reverse implication follows by substituting \(G=\mathbb {1}\).
For every positive definite source weight \(\rho \),
Apply right-inverse cancellation with \(G=v^\dagger v\) and \(Z=Z_1\otimes Z_2\).
If the physical and auxiliary dimensions are \(d\) and \(D\), respectively, then
The matrix \(M_1\) has \(dD\) columns. Its rank is at most its number of columns, hence at most \(dD=Dd\).
If the physical and auxiliary dimensions are \(d\) and \(D\), respectively, then
The matrix \(M_2\) has \(dD\) columns. Its rank is at most its number of columns, hence at most \(dD=Dd\).
19.5 Full-support canonical representatives
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 \(d,D{\gt}0\), let \(U\) be an MPU tensor of physical dimension \(d\) and bond dimension \(D\), and let \(\widetilde U=d^{-1/2}U\) be its normalized doubled-index tensor. There are an integer \(D_{\mathrm{red}}{\gt}0\), a normal tensor \(A_{\mathrm{red}}\) of bond dimension \(D_{\mathrm{red}}\), and canonical-form-II data for \(A_{\mathrm{red}}\) such that, for every integer \(N{\gt}0\) and every word \(\sigma \) of length \(N\),
This organizes the dimension-bounded canonical replacement used in [ CPGSV17 , lines 257–294 and 319–326 ] . Its irreducible reduction and canonical-form-II gauge are those of [ CPGSV16 , lines 195–255 and 1058–1077 ] . The shifted transfer argument is the one used in [ CPGSV17 , lines 344–356 ] .
Remove the zero irreducible blocks of \(\widetilde U\) and write
for every \(N{\gt}0\) and every word \(\sigma \) of length \(N\). At \(N=2\) the common self-overlap is one, so the retained sum is nonempty and \(D_0=\sum _kD_k{\gt}0\). For every integer \(N{\gt}1\), overlap invariance gives
Let \(J_k:\mathbb C^{D_k}\to \mathbb C^{D_0}\) be the inclusion of the \(k\)th block. Then
Thus every nonzero transfer eigenvalue of \(B_k\) is a nonzero eigenvalue of \(E_{A_0}\) and hence equals one by (257). Perron–Frobenius theory supplies \(\rho _k{\gt}0\) and \(s_k{\gt}0\) with \(\mathcal{E}_{B_k}(\rho _k)=s_k\rho _k\). Therefore \(s_k=1\), and every peripheral eigenvalue of \(B_k\) is also one. Each \(B_k\) is consequently normal.
The direct sum now has literal canonical-form data with full support. Apply the nonsingular blockwise canonical-form-II gauge. It preserves the block dimensions and all periodic vectors, so the resulting tensor \(A_{\mathrm{red}}\) still satisfies
for every integer \(N{\gt}1\). The shifted traces force \(r=1\), spectral radius one, and primitivity. Full support identifies the unique block with the ambient bond space, so \(A_{\mathrm{red}}\) is normal.
Let \(d,D{\gt}0\) and let \(U\) be an MPU tensor. There are \(D_{\mathrm{red}}{\gt}0\) and an MPU tensor \(U_{\mathrm{red}}\) of bond dimension \(D_{\mathrm{red}}\) such that, for every integer \(N{\gt}0\),
The normalized doubled-index tensor of \(U_{\mathrm{red}}\) has canonical-form-II data whose retained block dimensions sum to \(D_{\mathrm{red}}\). No equality is asserted at length zero, and no virtual gauge is asserted between the two potentially unequal-dimensional bond spaces.
Apply Theorem 19.5.2 and write its representative as \(A_{\mathrm{red}}\). Define
Since \(d{\gt}0\), normalization recovers \(A_{\mathrm{red}}\) exactly. For every \(N{\gt}0\) and every ket–bra pair \((\sigma ,\tau )\), the doubled-index coefficient identity gives
Hence the periodic operators agree at every positive length. For \(N{\gt}1\) the right side is unitary, so \(U_{\mathrm{red}}\) is an MPU.
Suppose that canonical-form data have full support:
Then every positive physical blocking of these data also has full support.
Positive blocking supplies canonical-form data with the same block dimensions, so the full-support sum is unchanged.
A bijective relabeling of the physical alphabet preserves full support of canonical-form data.
Physical relabeling leaves the blocks and their bond dimensions unchanged, so the full-support sum is unchanged.
If a block has bond dimension \(D\), then the ambient tensor \(A\) is irreducible.
If \(k\) is the block with \(D_k=D\), the total-dimension bound gives
Since every retained block dimension is positive, these equalities force \(k\) to be the only retained block. After reindexing its bond coordinates, the ambient coisometry \(U\) is square, hence unitary, and reconstruction reads
where \(B\) is the reindexed normal block and \(\mu _k\ne 0\). Thus Theorem D.3.4 transports irreducibility of \(B\) through the nonzero rescaling and unitary conjugation.
Canonical-form data with one block and full support reconstruct an irreducible ambient tensor.
Let \(k\) be the unique block. The singleton sum and full-support identity give
Thus \(k\) has ambient bond dimension, so Theorem 19.5.6 applies.
Suppose that canonical-form data have one block and full support. If the transfer map has spectral radius one and is primitive, then the ambient tensor is a normal tensor.
Combine the preceding irreducibility conclusion with the supplied spectral-radius and primitivity properties.
19.6 Transfer-matrix identities
Let \(D\) be a positive integer and let \(T:M_D(\mathbb C)\to M_D(\mathbb C)\) be a linear map. The transfer matrix \(\widehat T\) of \(T\) under the column-stacking identification \(M_D(\mathbb C)\cong \mathbb C^{D^2}\) is defined in the companion quantum-channel volume [ LTC26 , “Transfer matrix of a matrix endomorphism” ] , the matrix-units special case of the one-family transfer matrix of the corresponding definition in the companion quantum-channel volume [ LTC26 , “Transfer matrix in one family” ] .
The corresponding generic result is proved in the companion quantum-channel volume [ LTC26 , “Functoriality of transfer matrices” ] .
The corresponding generic result is proved in the companion quantum-channel volume [ LTC26 , “Trace preservation fixes the vectorized identity on the left” ] .
The corresponding generic result is proved in the companion quantum-channel volume [ LTC26 , “Powers and traces of transfer matrices” ] .
A complex number \(\mu \) is an eigenvalue of \(T\) if and only if it is an eigenvalue of \(\widehat T\).
Column stacking is a linear bijection and intertwines \(T\) with multiplication by \(\widehat T\). It therefore carries nonzero eigenmatrices to nonzero eigenvectors and conversely.
For a matrix product tensor \(A\), write \(V^{(N)}(A)_\sigma \) for the coefficient of its length-\(N\) periodic vector at the configuration \(\sigma \).
Let \(A\) and \(B\) be matrix product tensors with the same positive bond dimension, and let \(F_{A,B}(X)=\sum _i A^iX(B^i)^\dagger \) be their mixed transfer map. Then
In particular, taking \(B=A\) identifies the trace of the \(N\)th power of the ordinary transfer matrix with the length-\(N\) self-overlap.
Replace the matrix power by the transfer matrix of the corresponding operator power, identify matrix trace with operator trace, and expand the latter over periodic words. For the ordinary self-transfer statement, specialize to \(B=A\) and use the identity \(F_{A,A}=\mathcal E_A\) from Theorem B.1.1.
For an MPO tensor \(U\) of physical dimension \(d{\gt}0\), its normalized flattening is the doubled-physical-index MPS tensor
This is the normalization used in the normal-tensor argument of [ CPGSV17 , Section III, equation (13) ] .
Let \(\mathcal U\) and \(\mathcal V\) be MPO tensors with the same positive physical dimension \(d\), and let
be their composition tensor. Under the canonical identification of its auxiliary space with \(\mathbb {C}^{D_U}\otimes \mathbb {C}^{D_V}\), the normalized flattenings satisfy
This identity combines the raw composition tensor used in the proof of [ CPGSV17 , Theorem IV.6(ii) ] with the normalization in [ CPGSV17 , Section III, equation (13) ] ; it is not a separately stated result of that proof.
Substitute \(U^{ij}=\sqrt d\, \widetilde U^{ij}\) and \(V^{jk}=\sqrt d\, \widetilde V^{jk}\) into the defining contraction for \(W^{ik}\), and then multiply the result by \(d^{-1/2}\).
Let \(\mathcal U\) be an MPO tensor of physical dimension \(d\), and let \(L\geq 0\). For the normalized-tensor interpretation assume \(d{\gt}0\), as in Definition 19.6.3; with the convention \(0^{-1}=0\), the algebraic identity below also holds for \(d=0\). Pairing the ket and bra letters at each site gives a canonical bijection
If \(\widetilde{\mathcal U}=d^{-1/2}\mathcal U\) denotes normalized flattening, then
Thus normalized flattening commutes with blocking, up to the canonical doubled-index relabeling. This is the normalized form of the blocking operation in [ CPGSV17 , Definition II.5 and equation (9) ] .
A blocked letter is a product of \(L\) original letters. The identity
identifies its normalization with the product of the \(L\) one-site normalizations. Theorem 20.14.2 identifies the two physical indexings for every \(L\geq 0\); at \(L=0\) both blocked words are empty and both local matrix products are the identity.
Suppose that an MPS tensor \(A\) is left-canonical:
Then for every \(L\geq 0\) its blocked tensor is left-canonical:
If \(e\) is a bijection of physical alphabets, then \(e^*A\) is left-canonical if and only if \(A\) is:
Let \(A\) have canonical-form-II data with retained weights \(\mu _k\), blocks \(A_k\), and positive definite diagonal fixed matrices \(\Lambda _k\). For every positive integer \(L\), the blocked tensor \(A^{[L]}\) has canonical-form-II data with weights \(\mu _k^L\), blocks \(A_k^{[L]}\), and the same matrices \(\Lambda _k\). The retained blocks remain left-canonical:
A bijective relabeling of the physical alphabet likewise preserves canonical-form-II data, including both the matrices \(\Lambda _k\) and the left-canonical identities.
Consequently, if the normalized flattening \(\widetilde{\mathcal U}\) of an MPO tensor has canonical-form-II data, then for every \(L{\gt}0\) the normalized flattening of \(\mathcal U^{[L]}\) has the data obtained by first blocking and then relabeling by \(\vartheta _L\) from Theorem 19.6.5. Theorem 19.6.6 preserves the left-canonical identity under both operations. For each retained block, Lemma D.4.8 gives
The second identity follows by iterating \(\mathcal{E}_{A_k}(\Lambda _k)=\Lambda _k\). Thus the same positive definite diagonal matrix is fixed after blocking. A physical bijection leaves the transfer map unchanged. The final construction applies these two operations to (276).
For a matrix product tensor \(A\), a complex number \(c\), and a nonnegative integer \(N\), scaling every local matrix by \(c\) gives
Consequently, if \(d{\gt}0\) and \(U\) is an MPO tensor of physical dimension \(d\), then
Each length-\(N\) coefficient is multiplied by \(c^N\). Multiplication by its complex conjugate gives the first identity. Taking \(c=1/\sqrt d\) gives the second.
Let \(U\) be an MPU tensor with positive physical and bond dimensions, and let \(E\) be the transfer matrix of its normalized flattening. For every integer \(N{\gt}1\),
This is the trace identity in the proof of the normal-tensor result [ CPGSV17 , Proposition III.1 ] .
Scaling both tensor copies by \(1/\sqrt d\) multiplies the length-\(N\) self-overlap by \(d^{-N}\). The doubled-index overlap identity and MPU unitarity give the complete chain
Here the fourth equality uses \(N{\gt}1\), so that the MPU equation applies.
19.7 Block transfer multiplicity
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
The inclusions satisfy the following identities, with the second holding for distinct \(k\) and \(\ell \):
If \(B^i=\bigoplus _k\mu _k A_k^i\) is a weighted block-diagonal tensor, then
Reindex the canonical dependent-summand inclusion along the equivalence between dependent pairs and flattened coordinates. Transporting its isometry, cross-block orthogonality, and block-diagonal intertwining identities gives the three displayed formulas.
If \(U\) is the reconstruction coisometry and \(J_k\) is the inclusion of the \(k\)th retained block, define
For every retained block and physical index \(i\),
The coisometry \(UU^\dagger =\mathbb {1}\) and \(J_k^\dagger J_k=\mathbb {1}\) give
The canonical reconstruction and block-diagonal intertwining give
For a block \(k\), define its weighted transfer eigenvalue by
If the ambient transfer matrix satisfies \(\operatorname{tr}(E^N)=1\) for every integer \(N{\gt}1\), then every block satisfies
Choose a nonzero fixed matrix \(X_k\) for the unweighted normal block and set \(Y_k=V_kX_kV_k^\dagger \). The isometry and intertwining identities give
Thus \(q_k\) lies in the ambient transfer spectrum; it is nonzero because \(\mu _k\ne 0\). The shifted trace-power spectrum is \(\{ 0,1\} \), hence \(q_k=1\).
If there is exactly one block \(k\) and the blocks fill the ambient bond space, then
The full-support sum reduces to the unique summand:
Therefore \(V_k\) is a square matrix. Since \(V_k^\dagger V_k=\mathbb {1}\), finite equal-dimensional inverse symmetry gives \(V_kV_k^\dagger =\mathbb {1}\).
Let \(A\) have literal CPSV canonical-form data with \(r\) blocks. If its transfer matrix \(E\) satisfies \(\operatorname{tr}(E^N)=1\) for every integer \(N{\gt}1\), then
This is the block multiplicity consequence used in [ CPGSV17 , Proposition III.1 ] .
Each normal block \(A_k\) has a nonzero transfer fixed vector \(X_k\). Let \(U\) be the reconstruction coisometry and let \(J_k\) be the flattened inclusion of the \(k\)th block. Define the ambient inclusion and transported matrix by
The matrix \(Y_k\) is nonzero and is an ambient transfer eigenmatrix with eigenvalue \(q_k=\mu _k\overline{\mu _k}\ne 0\). Theorem 19.4.6 gives \(q_k=1\).
For distinct \(k\) and \(\ell \), the inclusion identities give
Hence the matrices \(Y_k\), for \(1\le k\le r\), are linearly independent. Vectorizing gives
For every \(r\ge 1\), the shifted traces and Theorem 19.4.4 give
The eigenspace dimension is bounded by the algebraic multiplicity, so
Finally, if \(r\) were zero, the retained direct sum and therefore \(A\) would vanish, contradicting \(\operatorname{tr}(E^2)=1\). Consequently \(r=1\).
19.8 Normalized transfer spectrum and conditional normality
Let \(T:M_D(\mathbb C)\to M_D(\mathbb C)\) be a linear map with \(D{\gt}0\). If \(\operatorname{tr}(\widehat T^{\, N})=1\) for every integer \(N{\gt}1\), then
This is the radius-and-primitivity part of [ CPGSV17 , Proposition III.1 ] .
The shifted traces give
Column stacking identifies the eigenvalues of \(T\) and \(\widehat T\). Consequently every spectral value of \(T\) is zero or one, and one occurs, so
Restricting to eigenvalues of modulus one gives
which is primitivity.
Let \(U\) be an MPU tensor with positive physical and bond dimensions, and let \(E\) be the transfer matrix of its normalized flattening. Then
This is the transfer-matrix conclusion of [ CPGSV17 , Proposition III.1 ] . The first set equality alone does not assert algebraic multiplicity.
For every integer \(N{\gt}1\), the normalized transfer identity gives
Applying the exact shifted trace-power theorem in dimension \(D^2\) yields
Let \(U\) be an MPU tensor with physical dimension \(d{\gt}0\) and bond dimension \(D{\gt}1\), and let \(E\) be the transfer matrix of its normalized flattening. There is an integer \(J\) satisfying
and vectors \(\rho ,\Phi \) satisfying
such that
Moreover, for every integer \(k\geq J\),
Thus blocking \(J\) sites removes the entire generalized zero-eigenspace. The one-dimensional case is stated separately below.
This is the Jordan-elimination step in [ CPGSV17 , proof of Proposition III.2(ii) ] .
The exact characteristic polynomial is
Cayley–Hamilton gives \(E^{D^2}=E^{D^2-1}\). Hence every power from \(J=D^2-1\) onward is equal, and \(P=E^J\) is idempotent. Since \(J{\gt}1\), the normalized transfer trace identity gives \(\operatorname{tr}(P)=1\). The range of an idempotent has dimension equal to its trace, so \(P\) has rank one and factors as \(P=\lvert \rho )(\Phi \rvert \). Taking the factors with \((\Phi \mid \rho )=1\) gives the displayed fixed-point identities from \(EP=PE=P\).
Let \(P=E^J\) be a stabilized rank-one transfer power with normalized left and right factors. If \(\rho '\) and \(\Phi '\) are any other fixed vectors with \((\Phi '\mid \rho ')=1\), then
Applying \(P\) to \(\rho '\) and applying \(\Phi '\) to \(P\) shows that the two pairs of rank-one factors differ by reciprocal scalars. Their normalized pairings force the product of those scalars to be one.
Suppose \(d{\gt}0\), \(D{\gt}1\), and \(\mathcal U\) is an MPU whose normalized flattening is presented by chosen canonical-form-II data with blocks filling the bond space. There is a unique block \(k\). Normalize its diagonal positive definite fixed matrix \(\Lambda _k\) to have trace one, and let \(V_k\) be its inclusion into the ambient bond space. Then
where \((\Phi \rvert \) is the vectorized identity. Consequently,
The diagonal matrix is the block matrix \(\Lambda _k\) in the canonical-form-II fixed-point equations in [ CPGSV17 , equations (6a)–(6b) ] ; its ambient transport \(\rho \) need not be diagonal. This is a statement about the chosen reduced representative and the blocking step in [ CPGSV17 , proof of Proposition III.2(ii) ] ; it does not assert that every ambient MPU tensor is itself in canonical form II.
The shifted trace identities satisfy \(\operatorname{tr}(E^N)=1\) for every integer \(N{\gt}1\) and give
Let \(k\) be this block. Canonical form II supplies a diagonal \(\Lambda _0{\gt}0\) fixed by the unweighted block transfer map. Normalize it by
If \(\omega _k\) is its weight, then
where \(E_k\) is the transfer map of the weighted block \(\widetilde B_k=\omega _k B_k\). Full support makes the block inclusion \(V=V_k\) both isometric and onto:
Hence \(\rho =V\Lambda V^\dagger \) is positive definite with trace one, and the intertwining equations \(A_iV=V\widetilde B_k(i)\) give the fixedness transport
The blockwise left-canonical identity transports similarly:
Finally,
where the last equality is the uniqueness of the normalized fixed-pair factorization for the stabilized power.
Let \(d{\gt}0\), \(D{\gt}1\), and let \(\mathcal U\) be an MPU whose normalized flattening has chosen canonical-form-II data with full support. Put \(J=D^2-1\) and \(W=\mathcal U_{JD^2}\). There is a matrix \(\rho {\gt}0\) with \(\operatorname{tr}(\rho )=1\) such that
With the corresponding vectorized pair \(\rho ',\Phi '\), the same blocked tensor \(W\) satisfies the two aligned simple contractions
This theorem supplies only the fixed pair and the two simple contractions. It does not assert a source-cut factorization or source-\(v\) isometry.
If the physical dimension satisfies \(d{\gt}0\) and the bond dimension is \(D=1\), then the normalized transfer matrix satisfies
Consequently its powers stabilize at the positive exponent one.
In dimension one the characteristic polynomial identity reduces to \(\chi _E(X)=X-1\), and Cayley–Hamilton gives \(E=1\).
Let \(U\) be an MPU tensor with positive physical and bond dimensions. Suppose that the normalized flattening is equipped with CPSV canonical-form data whose blocks have full support in the ambient bond space. Then the normalized flattening is a normal tensor.
This is the full-support representative form of [ CPGSV17 , Proposition III.1 ] . The full-support hypothesis is essential for this ambient representative: adjoining a zero virtual direct summand preserves every periodic operator but introduces a nontrivial invariant projection. Thus the theorem does not assert that an arbitrary, possibly nonminimal MPU representative is normal.
Write \(\widetilde U\) for the normalized flattening, set \(T=\mathcal E_{\widetilde U}\) for its transfer map, and let \(E=\widehat T\) be the corresponding transfer matrix. Also write \(r\) for the number of blocks and \(D_k\) for the bond dimension of the block \(k\). For every integer \(N{\gt}1\), the normalized transfer identity gives
The block-multiplicity theorem and the transfer-spectrum theorem then give
Full support supplies
Since \(r=1\), the unique block has ambient dimension \(D\) and therefore makes \(\widetilde U\) irreducible. The displayed radius and peripheral-spectrum identities provide its remaining normality fields. Consequently \(\widetilde U\) is a normal tensor.
19.9 Standard form, index, and symmetry classification
Let \(\widetilde{\mathcal U}=d^{-1/2}\mathcal U\) be the normalized flattening, and let \(E\) be its transfer matrix. A reduced full-support source datum for \(\mathcal U\), with physical dimension \(d{\gt}0\) and positive bond dimension, consists of an integer \(J{\gt}0\), a positive definite matrix \(\rho \) with \(\operatorname{tr}(\rho )=1\), and \((\Phi \rvert =\operatorname {vec}(\mathbb {1}_D)^{\mathsf T}\) such that
These data are supplied in one of two ways.
If \(D=1\), take \(J=1\) and \(\rho =\mathbb {1}_1\) using Theorem 19.8.7.
If \(D{\gt}1\), choose canonical-form-II data for the normalized flattening of this same tensor, with its blocks filling the ambient bond space. Theorem 19.8.5 supplies the fixed pair with \(J=D^2-1\).
For \(D{\gt}1\), this states the fixed-pair convention in [ CPGSV17 , equations labelled Erightleft and II_UTransfer, and proof of Proposition III.2(ii) ] . The \(D=1\) branch is the direct one-dimensional specialization. This definition does not assert that an arbitrary representative with \(D{\gt}1\) admits the required canonical-form-II data and does not replace \(\mathcal U\) by another tensor.
Let \(\mathcal U\) be an MPU tensor with physical dimension \(d{\gt}0\) and bond dimension \(D{\gt}0\). Let \(\rho {\gt}0\) satisfy \(\operatorname{tr}(\rho )=1\), put \((\Phi \rvert =\operatorname {vec}(\mathbb {1}_D)^{\mathsf T}\), and suppose that for some \(J{\gt}0\) the normalized transfer matrix satisfies
Construct the source factors and \(u\) from \(\mathcal U\) using \(\rho \), and set \(K=JD^2\). For retained physical pairs \(p,q\in \{ 0,\ldots ,d-1\} ^2\), the complete source-\(u\) contraction satisfies
Here \(\tau \) ranges over input words of length \(K\), and \(\eta \) ranges over output words of length \(K+2\). This is the supplied-fixed-pair form of the range-insertion equality in [ CPGSV17 , Lemma III.7 ] .
Put \(W=\mathcal U^{[K]}\). The reflected transfer-power identity, the direct matching contractions, and supplied-witness reblocking give the aligned direct and reflected contractions at lengths \(K\) and \(K+2\), with the same fixed pair at both lengths. The block \(W\) is used only inside this contraction argument; the external source tensors are the source-\(u\) factors of \(\mathcal U\).
Expand the ordinary Gram entry while retaining both external source-\(u\) tensors and the full \(\rho \)-weighted boundary. Insert the first- and second-cut range projections, but contract them only inside the closed \((K+2)\)-site network. The reflected contractions move the second-cut metric through the overlapping windows. After every virtual and source index is closed, the terminal boundary contraction gives the normalized output-first trace displayed above. No separate identity for the second-cut metric or either ambient range projection is asserted.
Let \(\mathcal U\) be an MPU tensor equipped with a reduced full-support source datum. Let \(J\), \(\rho \), and \((\Phi \rvert \) be its stabilization exponent and normalized fixed pair, so that
Compute both source cuts, their ranks \(r\) and \(\ell \), the source operator \(v\), and the right source factors \(Y_1,Y_2\) from this same tensor \(\mathcal U\) using \(\rho \). Let \(W\) be the double layer of \(\mathcal U\). Then the supplied contraction
holds if and only if
When \(\mathcal U\) is simple, Theorem 19.3.28 supplies the displayed contraction with this same fixed pair. This is the complete equivalence in [ CPGSV17 , proof of Theorem III.8 ] .
Insert \(E^J=\lvert \rho )(\Phi \rvert \) into the complete network while retaining the source factors of \(\mathcal U\). Expand both source cuts and rotate the second cut. Contract the resulting four-letter expression as one closed network to obtain the dressed Gram equation, and reverse the same calculation for the converse.
For any MPU tensor \(\mathcal U\) under the source’s standing canonical-form-II convention, let \(r\) and \(\ell \) be the two source-cut ranks and let \(u\) be the source operator. Then
Consequently \(u\) is an isometry and \(r\ell \geq d^2\). This is the unrestricted statement of [ CPGSV17 , Lemma III.7 ] .
Let \(\mathcal U\) be an MPU tensor with physical dimension \(d{\gt}0\) and bond dimension \(D{\gt}0\). Let \(\rho {\gt}0\) satisfy \(\operatorname{tr}(\rho )=1\), put \((\Phi \rvert =\operatorname {vec}(\mathbb {1}_D)^{\mathsf T}\), and suppose that for some \(J{\gt}0\) the normalized transfer matrix satisfies
Let \(r\) and \(\ell \) be the two source-cut ranks, and construct the source factors and \(u\) from \(\mathcal U\) using \(\rho \). Then
Consequently \(u\) is an isometry and \(r\ell \geq d^2\). This is the reduced-source version of [ CPGSV17 , Lemma III.7 ] .
Set \(K=JD^2\). The complete-network identity and output-first MPU unitarity give, for all retained physical pairs \(p,q\),
Hence \(u^\dagger u=\mathbb {1}\).
Let \(\mathcal U\) be an MPU tensor under the source’s standing canonical-form-II convention. Let \(r\) and \(\ell \) be its source-cut ranks, and let \(u\) and \(v\) be its source operators. Then the following conditions are equivalent:
\(\mathcal U\) is simple;
\(r\ell =d^2\);
\(u\) is unitary;
\(v\) is unitary.
This is the unrestricted statement of [ CPGSV17 , Theorem III.8 ] .
Let \(\mathcal U\) be an MPU tensor equipped with a reduced full-support source datum, and let \(\rho \) and \((\Phi \rvert \) be its normalized fixed pair. Compute the source-cut ranks \(r\) and \(\ell \) and the source operators \(u\) and \(v\) from this same tensor \(\mathcal U\) using \(\rho \). Then the following conditions are equivalent:
\(\mathcal U\) is simple;
\(r\ell =d^2\);
\(u\) is unitary;
\(v\) is unitary.
This is the reduced-source version of [ CPGSV17 , Theorem III.8 ] .
Lemma 19.9.5 gives \(u^\dagger u=\mathbb {1}\) before any of the four conditions is assumed. Thus \(u\) is a standing isometry and \(r\ell \geq d^2\) throughout the proof.
Suppose first that \(\mathcal U\) is simple. Theorem 19.3.28, applied to the stabilized transfer power, gives the second contraction with the supplied pair \(\lvert \rho )(\Phi \rvert \). By Theorem 19.9.3,
Cancelling the dressing by the right inverses of \(Y_1\) and \(Y_2\) gives \(v^\dagger v=\mathbb {1}\), hence \(r\ell \leq d^2\). Together with the standing source-\(u\) isometry, this gives \(r\ell =d^2\).
If \(r\ell =d^2\), the standing isometry \(u\) is square and therefore unitary. If \(u\) is unitary, the exact two-site factorization of \(U^{(2)}\) and unitarity of \(U^{(2)}\) imply that \(v\) is unitary. Finally, suppose that \(v\) is unitary. Unitarity gives the dressed Gram identity, and Theorem 19.9.3 gives the supplied second contraction for \(\mathcal U\). The same-witness implication in Theorem 19.4.70 supplies the first contraction, so \(\mathcal U\) is simple.
Let \(\mathcal U\) be a simple tensor under the source’s standing canonical-form-II convention. Its standard form is the depth-two form of the two-site block \(\mathcal U_2\) determined by the source unitaries \(u\) and \(v\). This is the unrestricted definition of [ CPGSV17 , Definition III.9 ] .
Let \(\mathcal U\) be a simple tensor equipped with a reduced full-support source datum. The admissible standard form of the two-site blocking \(\mathcal U_2\) is the depth-two form determined by the source unitaries
On four consecutive blocked legs its unitary is
Given two simple tensors \(\mathcal U\) and \(\mathcal V\) with standard forms, they generate the same MPU for every \(N\) if and only if there are unitaries \(x\), \(y\), and \(z\) relating the two standard forms by the local gauge equations of [ CPGSV17 , Eqns. (SFuu) and (SFvv) ] . This is the unrestricted statement of the source’s Fundamental Theorem of MPU.
Let \(\mathcal U\) and \(\mathcal V\) be simple tensors equipped with reduced full-support source data and written in admissible standard form, with source unitaries \((u_{\mathcal U},v_{\mathcal U})\) and \((u_{\mathcal V},v_{\mathcal V})\). They generate the same MPU for every \(N\),
if and only if their admissible standard forms are related by the local unitary gauges of [ CPGSV17 , Eqns. (SFuu) and (SFvv) ] : there are unitaries \(x\), \(y\), and \(z\) on the three corresponding source and canonical-form-II legs. In particular, the source unitaries obey the local relation
while the two canonical-form-II tensors are related on their virtual leg by \(z\) and on their source legs by \(x\) and \(y\).
Let \(A\) and \(B\) be finite matrices over a field. Then
Identify the image of the Kronecker product with the tensor product of the two images and multiply their dimensions.
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 \(\mathcal U\) and \(\mathcal V\) be specified tensors. Suppose that their source ranks obey
where \(c\), \(r_{\mathcal U}\), and \(\ell _{\mathcal U}\) are positive. Then
Let \(\mathcal U\), \(\mathcal V\), and \(\mathcal W\) instead have positive source ranks and suppose that
where \(c\) is a positive integer. Then
If \(d\), \(r_{\mathcal U}\), and \(\ell _{\mathcal U}\) are positive and a specified tensor has \(r_{\mathcal U}\ell _{\mathcal U}=d^2\), then
Expand the definition. For common scaling, apply \(\log _2(cx)=\log _2c+\log _2x\) to both ranks and cancel the two copies of \(\log _2c\). For the three-tensor identity, apply the product formula twice to each rank and cancel the common term. For the equivalent formulas, apply the same product identity to \(r_{\mathcal U}\ell _{\mathcal U}=d^2\) and use \(\log _2(x/y)=\log _2x-\log _2y\).
Let \(d{\gt}0\) and \(k_0\leq k\). Suppose that the endpoint source ranks satisfy
Then the specified source-index values agree:
This statement assumes the endpoint products; it neither selects a simple blocking nor defines the public index.
Exact blocking growth gives the common positive factor \(d^{k-k_0}\):
Its logarithm occurs once in each rank term and therefore cancels from their difference.
Let \(k\) be any blocking length for which \(\mathcal U_k\) is simple, and let \(r\) and \(\ell \) be its right and left source-cut ranks. The source index is
This is the unrestricted definition of [ CPGSV17 , Definition IV.1 ] .
Let \(k\) be a blocking length for which \(\mathcal U_k\) is simple and is equipped with a reduced full-support source datum, and let \(r\) and \(\ell \) be its right and left source-cut ranks. Define
Since \(r\ell =d^{2k}\), this is equivalently
The source index is independent of the chosen simple blocking. More precisely, if \(k\geq k_0\) and both \(\mathcal U_{k_0}\) and \(\mathcal U_k\) are simple, then
This is the unrestricted statement of [ CPGSV17 , Proposition IV.2 ] .
The admissible index is independent of the chosen admissible simple blocking. More precisely, if \(k\geq k_0\), both \(\mathcal U_{k_0}\) and \(\mathcal U_k\) are simple, and each is equipped with a reduced full-support source datum, then
For MPU tensors under the source’s standing canonical-form convention,
This is the unrestricted tensor-product statement in [ CPGSV17 , Theorem IV.6(ii) ] .
Suppose there is one positive blocking length \(k\) such that \(\mathcal U_k\), \(\mathcal V_k\), and \((\mathcal U\otimes \mathcal V)_k\) are simple and each carries its own reduced full-support source datum. Define all three indices from these blocks. Then
This is the common-block reduced-source version of [ CPGSV17 , Theorem IV.6(ii) ] .
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.
Use the canonical physical reindexing to identify iterated two-site blocking with direct four-site blocking. For the right cut, factor the two-site block of each tensor through its source tensor \(u\). The diagonal cut through the two blocked factors retains one physical leg and one right source leg from each factor. Thus, for suitable rectangular matrices \(A_{1,4}\) and \(B_{1,4}\),
where the columns of \(A_{1,4}\) and the rows of \(B_{1,4}\) are indexed by
Consequently,
For the left cut, use the reflected \(v\) factorization, with its physical order \((j_2,j_1)\) and source order \((r,\ell )\). For suitable rectangular matrices \(A_{2,4}\) and \(B_{2,4}\),
where the intermediate space is
Hence
Both source-cut ranks are unchanged by the canonical physical relabeling.
Let \(\mathcal W\) generate the composition of the MPUs generated by \(\mathcal U\) and \(\mathcal U'\). Under the source’s standing canonical-form convention,
This is the unrestricted composition statement in [ CPGSV17 , Theorem IV.6(ii) ] .
Suppose there is a positive integer \(k\) such that \(\mathcal U_k\), \(\mathcal U'_k\), and the composition tensor \(\mathcal W_k\) are simple and each carries its own reduced full-support source datum. Suppose in addition that \(\mathcal U_{2k}\), \(\mathcal U'_{2k}\), and \(\mathcal W_{4k}\) are simple and each carries its own reduced full-support source datum. Define the three indices initially from \(\mathcal U_k\), \(\mathcal U'_k\), and \(\mathcal W_k\); the supplied data at \(2k\) and \(4k\) make the later comparison admissible. If \(\mathcal W\) generates the composition of the MPUs generated by \(\mathcal U\) and \(\mathcal U'\), then
This is the six-block reduced-source version of [ CPGSV17 , Theorem IV.6(ii) ] .
A tensor \(A\) of physical dimension \(d\) and bond dimension \(D\) is in MPU canonical form if there are positive block dimensions \(D_k\), weights \(\mu _k\in \mathbb C\), tensors \(A_k\) of bond dimension \(D_k\), and a coisometry
such that
Every \(A_k\) is irreducible and the spectral radius of its transfer map is one. The blocks may be periodic, so no uniqueness condition is imposed on their peripheral eigenvalues. This is the canonical form of [ CPGSV17 , lines 259–267 ] ; normal tensors form a strictly stronger class.
Fix a virtual bond dimension \(D\). Let \(\mathcal U\) and \(\mathcal V\) be canonical-form MPU tensors with physical dimensions \(d_a\) and \(d_b\). Given an equality \(h\colon d_a=d_b\), reindex \(\mathcal U\) from \(d_a\) to \(d_b\) along \(h^{-1}\). The tensors are strictly equivalent with respect to \(h\) if there is a continuous path \(\mathcal W(p)\), \(p\in [0,1]\), in the fixed-bond MPU locus such that
Only the canonical-form requirement is relaxed along the path. This is the fixed-bond interpretation of [ CPGSV17 , lines 708–714 ] ; it does not compare unequal raw bond dimensions.
Let \(x{\gt}0\). After adjoining an identity ancilla of dimension \(x\), the normalized flattening is the physical reindexing of the original flattening with a normalized diagonal ancilla alphabet. Its transfer map is exactly the original transfer map. Consequently, chosen canonical-form-II data with full support determine canonical-form-II data with full support for \(\mathcal U^{(x)}=\mathcal U\otimes \mathbb {1}_x\).
The product-index equivalence has the orientation
This result preserves the normalized transfer data. It does not identify the old and new source cuts, ranks, or chosen compact-SVD factors.
Only the \(x\) diagonal ancilla letters are nonzero, each with coefficient \(x^{-1/2}\). Hence their total transfer contribution is
Thus the transfer map and each diagonal positive fixed matrix are unchanged. The block dimensions, weights, and ambient coisometry are unchanged as well, so full support follows from the same block-dimension sum. The displayed physical reindexing gives the normalized flattening of the enlarged MPO.
Fix a virtual bond dimension \(D\). Let \(\mathcal U\) and \(\mathcal V\) be MPU tensors with physical dimensions \(d_a\) and \(d_b\). They are equivalent if there are a positive blocking length \(k\) and coprime positive integers \(p_a,p_b\) with
such that \((\mathcal U^{(p_a)})_k\) and \((\mathcal V^{(p_b)})_k\) are strictly equivalent. Here the positive-dimensional identity ancillas are attached before the common blocking, and
This is the fixed-bond interpretation of [ CPGSV17 , lines 716–724 ] ; it does not introduce a stabilization relation for unequal raw bond dimensions.
An admissible strict-equivalence datum for a path \([0,1]\ni p\mapsto \mathcal W(p)\) consists of a common positive blocking length \(m\) such that the actual blocked path tensor \(\mathcal W(p)_m\) is simple and carries a reduced full-support source datum for every \(p\). The blocked tensors and their source data vary continuously with \(p\), after fixed identifications of the finite-dimensional spaces involved. No equivalent or reduced representative may replace \(\mathcal W(p)_m\). In particular, the actual blocked endpoint tensors carry their own source data.
When the path is used under a specified symmetry \(\mathcal S\), the datum also includes continuously varying reduced full-support source data for the fixed family of comparison paths
Every tensor in this family is simple. This requirement is part of the datum independently of which comparisons a later proof uses; no preservation under conjugation, physical adjoint, transposition, or \(\mathcal S\) is assumed.
Two tensors are admissibly strictly equivalent if their strict equivalence is witnessed by such a path. They are admissibly equivalent if the blocked, ancilla-enlarged endpoints in Definition 19.9.28 are joined by an admissible strict-equivalence path. The source arguments motivating these additional path hypotheses are Proposition IV.5, Theorem IV.6, Corollary IV.7, and the lemma following the symmetry-equivalence definitions in [ CPGSV17 ] ; the datum itself is introduced here and has no source-labelled counterpart.
Let \(P\) and \(Q\) be orthogonal projections on a complex inner-product space. The compression of \(Q\) to \(\operatorname {ran}P\) is self-adjoint. If a unit vector \(x\in \operatorname {ran}P\) satisfies
then there is a unit vector \(w\perp x\) such that
Thus \(\operatorname {span}\{ x,w\} \) is the generic two-dimensional angle block. This lemma concerns one eigenvalue in \((0,1)\); it does not include the endpoint eigenspaces or an orthogonal direct-sum decomposition of the ambient space.
Set \(d=Qx-\mu x\). Symmetry and idempotence give
Since \(0{\lt}\mu {\lt}1\), the vector \(w=d/\sqrt{\mu (1-\mu )}\) is well defined and has the stated properties.
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, with compression spectral data \((x_i,\mu _i)\). Then, for every \(i\), \(\mu _i\in [0,1]\) and exactly one of the following alternatives holds:
\(\mu _i=0\) and \(Qx_i=0\);
\(\mu _i=1\) and \(Qx_i=x_i\);
\(0{\lt}\mu _i{\lt}1\), and \(x_i\) belongs to the two-dimensional angle block described in Theorem 19.9.30.
This theorem classifies the compression eigenvectors.
Apply the finite-dimensional spectral theorem to the self-adjoint compression of \(Q\) to \(\operatorname {ran}P\). For a unit compression eigenvector \(x\) with eigenvalue \(\mu \), the angle-defect identity gives
hence \(0\leq \mu \leq 1\). At \(\mu =0\) or \(1\) the defect vanishes, giving the two common one-dimensional blocks. For \(0{\lt}\mu {\lt}1\), apply Theorem 19.9.30.
For compression spectral data \((x_i,\mu _i)\), define
For all indices \(i,j\),
In particular, \(\langle d_i,d_j\rangle =0\) when \(i\ne j\), including when \(\mu _i=\mu _j\). The block \(B_i\) has dimension at most two, is invariant under both \(P\) and \(Q\), and satisfies \(B_i\perp B_j\) for distinct indices.
Since \(Px_i=x_i\), symmetry of \(P\) and the compression eigenvalue equation give
Conjugate symmetry gives \(\langle d_i,x_j\rangle =0\). For the exact defect pairing, symmetry of \(Q\) and the local action formula give
Orthogonality of the generated spans follows by linearity. The action formulas
show that both projections preserve \(B_i=\operatorname {span}\{ x_i,d_i\} \). Its dimension bound follows from its two generators.
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 \(1\).
The blocks \((B_i)_i\) form an orthogonal internal direct sum in \(B\), and
Both \(B\) and \(K\) are invariant under \(P\) and \(Q\). Moreover, \(\operatorname {ran}P\subseteq B\), so for every \(z\in K\),
Consequently the restrictions \(P|_K\) and \(Q|_K\) are orthogonal projections and commute. No commutation of \(P\) and \(Q\) on the ambient space is assumed.
Pairwise orthogonality of the blocks implies independence, while their supremum is \(B\) by definition. Hence they form an internal direct sum in \(B\). The standard orthogonal-complement decomposition gives \(E=B\mathbin {\perp \! \oplus }B^\perp \).
The local action formulas show that each \(B_i\) is invariant under both projections, hence so is \(B\). Symmetry of each projection then makes \(K=B^\perp \) invariant. The compression eigenvectors form a basis of \(\operatorname {ran}P\), and each lies in its block \(B_i\); therefore \(\operatorname {ran}P\subseteq B\). Thus \(P\) vanishes on \(K\). Since \(Qz\in K\) whenever \(z\in K\), both \(P(Qz)\) and \(Q(Pz)\) vanish. The two restricted projections therefore commute.
Let \(P\) and \(Q\) be orthogonal projections on a finite-dimensional complex inner-product space. Define \(\widetilde P\) as the orthogonal projection onto \(\operatorname {ran}(PQ)\).
The reduced projection satisfies
The first two identities describe the canonical construction, while the next three establish the properties in source lines 767–769. The final range equality is the derived input to the right-factor theorem, not item (iv) of the source.
Set \(A=QP\). The standard range identity for \(A^\dagger A\) gives
By definition, \(\operatorname {ran}(\widetilde P)=\operatorname {ran}(PQ)\), which also gives \(P\widetilde P=\widetilde P\). For all \(x\) and \(z\),
Hence \(Qx-PQx\in \operatorname {ran}(PQ)^\perp \), so orthogonal projection onto this range yields \(\widetilde P Q=PQ\). Taking adjoints gives \(Q\widetilde P=QP\). Finally,
For orthogonal projections \(P\) and \(Q\), the reduced range contains no nonzero vector annihilated by \(Q\):
If \(x=PQy\) and \(Qx=0\), then \(Px=x\) and symmetry of the two projections gives
Hence \(x=0\).
There is an invertible linear map \(Y\) such that
The equality
and the equal-ranges right-factor theorem give an invertible \(Y\) satisfying \(\widetilde P QY=\widetilde P\).
In standard Euclidean coordinates, let \(\widetilde P\) be the matrix of the reduced projection associated with two square matrices \(P\) and \(Q\).
If \(P\) and \(Q\) are orthogonal projections, then \(\widetilde P\) is an orthogonal projection and
There is also an invertible matrix \(Y\) satisfying \(\widetilde P QY=\widetilde P\).
Let \(U=(U^{ij})\) be an MPO tensor, and let \(\widehat U^{ij}=AU^{ij}B\). Then
Expand the transfer map and collect the factors \(A\), \(B\), \(B^\dagger \), and \(A^\dagger \) outside the finite sum.
Let \(\rho \succeq 0\). If
then
If the quadratic form of \(V^\dagger \rho V\) vanished at a nonzero vector \(x\), positive semidefiniteness would give \(\rho Vx=0\). The support projection would then annihilate \(Vx\), contrary to the hypothesis.
Let \(W=(W^{ij})\) be an MPO tensor. Let \(L,R\succeq 0\) satisfy
and let \(P\) and \(Q\) be their support projections. Assume, letter by letter,
Set \(T=\widetilde P\) and \(\widetilde W^{ij}=TW^{ij}T\). Then, for every \(N\geq 0\),
Moreover, if \(V^\dagger V=I\) and \(VV^\dagger =T\), then
For every nonempty word \(w\), idempotence of \(T\) and (443) give
Cyclicity of the trace, together with \(QW_w=W_w=W_wP\), therefore gives \(\operatorname {tr}(TW_wT)=\operatorname {tr}(W_w)\). Length zero is immediate. Lemma 19.9.43, with \(A=B=T\) and \(T^\dagger =T\), gives
Since \(P\) and \(Q\) are the support projections of \(L\) and \(R\), respectively,
Combining these absorptions with (443), \(T^2=T\), and cyclicity of the trace gives
Let \(x\neq 0\) and set \(u=Vx\). Since \(V^\dagger V=I\), one has \(u\neq 0\); since \(VV^\dagger =T\), one has \(Tu=u\). Equation (443) gives \(Pu=u\), and (439) gives \(Qu\neq 0\). Thus
Applying Lemma 19.9.44 to \(L\) and \(R\) yields
Let \(L,R\succeq 0\), let \(P\) and \(Q\) be their support projections, and let \(\widetilde P\) be the reduced projection of \(P\) and \(Q\). There are invertible support-inverse extensions of \(L\) and \(R\), together with an invertible matrix \(Y\), such that \(PQY=\widetilde P\).
Extend the inverses on the supports by the identity on their orthogonal complements, and choose an invertible right factor for the reduced projection.
Let \(W=(W^{ij})\) be an MPO tensor, let \(L\), \(R\), \(X\), \(Z\), \(Y\) be square matrices of the same size, let \(P\) be an orthogonal projection, let \(Q\) be a square matrix of the same size, and set \(\widetilde P\) equal to their reduced projection. Assume, letter by letter, \(W^{ij}P=W^{ij}=QW^{ij}\). Set \(\widehat W^{ij}=LW^{ij}R\) and \(\widetilde W^{ij}=\widetilde P W^{ij}\widetilde P\). If \(X,Z,Y\) satisfy \(XL=P\), \(RZ=Q\), and \(PQY=\widetilde P\), then
If in addition \(X\), \(Z\), and \(Y\) are invertible, then
If \(L,R\succeq 0\) and \(P=\operatorname {supp}(L)\), \(Q=\operatorname {supp}(R)\), such invertible matrices may be chosen as \(X=L^++(\mathbb {1}-P)\), \(Z=R^++(\mathbb {1}-Q)\), together with an invertible right factor \(Y\) of the reduced projection.
Expanding each virtual sandwich and using \(XL=P\) and \(RZ=Q\) gives (459). The absorptions \(QW^{ij}=W^{ij}=W^{ij}P\) give, letter by letter,
which is (460). Since \(\widetilde P Q=PQ\) and \(PQY=\widetilde P\),
This proves (461) and (462). Thus \(\widetilde W\) is obtained from \(\widehat W\) by left multiplication by \(X\) and right multiplication by \(ZY\). Invertibility of these factors and the source ranks of a virtual sandwich (Theorem 19.9.56) give
the first equality in each line being the invariance identity with invertible \(X\) and \(ZY\), and the second rewriting \(X\widehat W(ZY)\) through (459)–(461) and (462) to \(\widetilde W\), which gives (463) and (464). For positive semidefinite \(L\) and \(R\), their ambient support-inverse extensions supply \(X\) and \(Z\); the invertible right-factor identity for \(\widetilde P Q\) supplies \(Y\).
For finite index sets \(M,N\), matrix rank defines a lower-semicontinuous map
Consequently, if \(A\colon X\to \operatorname {Mat}_{M\times N}(\mathbb C)\) is continuous, then \(x\mapsto \operatorname {rank}A(x)\) is lower semicontinuous.
For every \(k\in \mathbb N\), the sublevel set is the closed determinantal set
These closed sublevel sets characterize lower semicontinuity. Composition with a continuous map preserves lower semicontinuity.
Let \(r,\ell \colon X\to \mathbb N\) be lower semicontinuous. Suppose that \(c{\gt}0\) and
for every \(x\in X\). Then both \(r\) and \(\ell \) are locally constant.
Fix \(x\in X\). Since \(c{\gt}0\), both \(r(x)\) and \(\ell (x)\) are positive. Lower semicontinuity gives a neighborhood \(U\) of \(x\) on which
For \(y\in U\), monotonicity of multiplication and the fixed-product identity give
Positivity permits cancellation, first of \(\ell (x)\) and then of \(r(x)\), so \(r(y)=r(x)\) and \(\ell (y)=\ell (x)\).
Under the hypotheses of Theorem 19.9.49, if \(S\subseteq X\) is preconnected, then for all \(x,y\in S\),
In particular, both functions are constant when \(X\) is preconnected and when \(X=[0,1]\).
A locally constant function has open fibers. A preconnected set cannot meet two disjoint nonempty open fibers, so each of \(r\) and \(\ell \) has one value on \(S\). The unit interval is connected, hence preconnected.
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
If \(x\mapsto A(x)\), \(x\mapsto \mathcal U(x)\), and \(x\mapsto B(x)\) are continuous families, then
is continuous.
Every matrix entry is a finite sum of products of entries of \(A(x)\), \(\mathcal U(x)\), and \(B(x)\), hence is continuous.
Let \(q:\{ 0,\ldots ,D{-}1\} \times \{ 0,\ldots ,D{-}1\} \to \{ 0,\ldots ,D^2{-}1\} \) be the standard enumeration of a pair. For an MPO tensor \(\mathcal W\) and a doubled-virtual matrix \(E\in \mathbb C^{D^2\times D^2}\), define
If \(E(\mathcal W)\) denotes the normalized physical diagonal of the physical-adjoint double layer of \(\mathcal W\), define
This is the contraction shown in Figure IV_index4.png of [ CPGSV17 , Proposition IV.5 ] .
Let \(L,R\in \mathbb C^{D\times D}\) and let \(\operatorname {vec}\) stack matrix columns, so
Then the rank-one doubled-virtual matrix satisfies
No adjoint or normalization factor occurs in the contraction result. The binary map \((\mathcal W,E)\mapsto C(\mathcal W,E)\), the map \(\mathcal W\mapsto E(\mathcal W)\), and consequently the map \(\mathcal W\mapsto \widehat{\mathcal W}_{\mathrm{ff}}\) are continuous. These assertions neither choose \(L\) and \(R\) continuously nor infer their positivity. They make no claim about reduced representatives or constancy of ranks.
Expanding the left-hand side of (478) and using column stacking gives
This proves (478). Each entry in (474) is a finite sum of products, so the binary contraction is continuous. The double-layer entries are finite sums of products of entries of \(\mathcal W\) and their complex conjugates, and the normalized physical diagonal is another finite sum. Hence \(\mathcal W\mapsto E(\mathcal W)\) is continuous. Substitution into (475) proves continuity of the factor-free sandwich.
The two raw source cuts satisfy the literal formulas
No transpose or conjugation occurs in either identity.
For \(a=1,2\), expand an entry of the right-hand side. The identity factors collapse the two physical sums, leaving
which is the corresponding entry of \(M_a(\widehat{\mathcal U})\).
If \(A\) and \(B\) are invertible, then
This assertion concerns the raw source ranks only.
The determinant identities for Kronecker products show that \(A\otimes I_d\) and \(I_d\otimes B\) are invertible. Left and right multiplication by these matrices preserves rank. Apply this observation to each source-cut formula in Theorem 19.9.55.
Let \(A\succeq 0\) have support projection \(P\), and let \(A^+\) denote its inverse on the support. Define
Define also
This construction gives explicit witnesses for the invertible matrices required in Proposition IV.5 of [ CPGSV17 ] ; the paper requires those matrices but does not state these formulas.
The matrices \(X_A\) and \(Z_A\) are mutually inverse, and
Consequently, \(X_A\) is a unit in the matrix algebra, with inverse \(Z_A\), and \(\det (X_A)\) is a unit in \(\mathbb C\).
In the notation of Proposition IV.5 of [ CPGSV17 ] , let \(L\) and \(R\) be the positive semidefinite outer factors with support projections \(P\) and \(Q\), respectively. The required witnesses are
Thus
or equivalently \(XL=P\) and \(RZ=Q\). Here the paper’s matrix \(Z\) is \(X_R\), not the inverse-extension matrix \(Z_R\) from Definition 19.9.57.
The support identities give
Their complementary forms are
and idempotence of \(P\) gives
Therefore
Likewise,
while multiplication by \(A\) gives \(X_AA=AX_A=P\).
Let \([0,1]\ni x\mapsto \mathcal W(x)\) be a continuous path of tensors generating MPUs, without requiring the path to be in canonical form. For each \(x\), take the canonical form and the two source-cut ranks \(r(x)\) and \(\ell (x)\) of its associated simple tensor. Then \(r(x)\) and \(\ell (x)\) are constant on \([0,1]\).
Let \([0,1]\ni x\mapsto \mathcal W(x)\) be a continuous path of tensors generating MPUs, without requiring the path to be in canonical form. Suppose that the path carries an admissible equivalence datum with blocking length \(m\). Then the ranks \(r(x)\) and \(\ell (x)\) of the actual blocked path tensors \(\mathcal W(x)_m\) are constant on \([0,1]\).
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.
It does not change under blocking.
It is additive under tensor products and composition.
It is constant under continuous deformations through MPU tensors.
Two MPU tensors have the same index if and only if they are equivalent.
This is the unrestricted statement of [ CPGSV17 , Theorem IV.6 ] .
Assume the following operation-specific admissibility hypotheses. Every pair of simple blockings compared for blocking independence carries reduced full-support source data. For a tensor product \(\mathcal U\otimes \mathcal V\), there is one positive length \(k\) such that \(\mathcal U_k\), \(\mathcal V_k\), and \((\mathcal U\otimes \mathcal V)_k\) are simple and carry their own data. For a composition tensor \(\mathcal W\) of \(\mathcal U\) and \(\mathcal U'\), there is one positive length \(k\) such that
are simple and carry their own data. Continuous-deformation and equivalence claims use the path data of Definition 19.9.29. Under these hypotheses, the MPU index has the following properties.
It does not change under blocking.
It is additive under tensor products and composition.
It is constant under continuous deformations through MPU tensors.
Admissibly equivalent MPU tensors have the same index. Conversely, tensors with the same index are admissibly equivalent whenever the standard-form interpolation is supplied with an admissible path datum.
These are the conclusions of [ CPGSV17 , Theorem IV.6 ] , restricted to the explicit admissibility hypotheses above.
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 \(v(p)\).
Let \(k_0\) be the common blocking length in an admissible equivalence datum for a continuous path \(\mathcal U(p)\) of MPU tensors, and suppose that this supplied length satisfies \(k_0\leq D^4\). Then there are continuous paths of unitaries \(u(p)\) and \(v(p)\) such that the MPU \(U^{(2k_0N)}(p)\) has standard form given by \(u(p)\) and \(v(p)\) for every \(p\in [0,1]\). This is the reduced-source version of [ CPGSV17 , Corollary IV.7 ] .
19.9.1 Local characterization of symmetries
An MPU in standard form is invariant under complex conjugation if and only if there are unitaries \(x\) and \(y\) such that either \(x=x^T\) and \(y=y^T\), or \(x=-x^T\) and \(y=-y^T\), and
Let a tensor and its conjugate comparison tensor both be equipped with reduced full-support source data and written in admissible standard form. The generated MPU is invariant under complex conjugation if and only if there are unitaries \(x\) and \(y\) such that either \(x=x^T\) and \(y=y^T\), or \(x=-x^T\) and \(y=-y^T\), and
This is the reduced-source version of the conjugation characterization in [ CPGSV17 , Proposition V.1 ] .
Let \(x\) be unitary and satisfy \(x=x^T\) or \(x=-x^T\). There are a symmetric unitary \(S\) and a matrix \(\widetilde\Lambda \) such that
In the symmetric case, \(\widetilde\Lambda \) is real and symmetric. In the skew-symmetric case, it is real and skew-symmetric. In both cases, \(\det (\widetilde\Lambda )=1\).
Let \(I\) be a finite set of odd cardinality and let \(A\in \operatorname {Mat}_I(\mathbb C)\) satisfy \(A^T=-A\). Then
Transpose invariance, skew symmetry, and the determinant of a negated matrix give
Since the complex numbers have characteristic zero, \(\det A=0\).
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.
If \(|I|\) were odd, Lemma 19.9.1.4 would give \(\det A=0\). Hence \(|I|\) is not odd and is therefore even.
Let \(U\in \operatorname {Mat}_d(\mathbb C)\) be unitary and satisfy \(U^T=-U\). Then \(d\) is even. This is the parity obstruction in [ CPGSV17 , Proposition V.2, Case II ] .
The determinant of a unitary matrix is a unit, so it is nonzero. Apply Lemma 19.9.1.5 to \(U\).
Let an MPU in standard form be invariant under complex conjugation. Its standard-form unitaries can be chosen in exactly one of the following two cases. In Case I,
uniquely up to local orthogonal transformations. In Case II,
uniquely up to local symplectic transformations, where \(\Sigma _x=\mathbb {1}_{x/2}\otimes \left(\begin{smallmatrix} 0 & 1 \\ -1 & 0 \end{smallmatrix}\right)\). Case II can occur only when \(r\) and \(\ell \) are even.
Let an MPU in admissible standard form be invariant under complex conjugation, and suppose every standard-form tensor compared below carries reduced full-support source data. Its standard-form unitaries can be chosen in exactly one of the following two cases. In Case I,
and the choice is unique up to local orthogonal transformations. In Case II,
where
The choice is unique up to local transformations \(S_x\) satisfying \(S_x^T\Sigma _xS_x=\Sigma _x\). Case II can occur only when \(r\) and \(\ell \) are even.
In Case II of Proposition 19.9.1.7, one can write
where \(\widetilde u\) and \(\widetilde v\) are orthogonal and
In the admissible Case II of Proposition 19.9.1.8, one can write
where \(\widetilde u\) and \(\widetilde v\) are orthogonal and
Let an MPU be in standard form, with one-site physical space \(\mathbb C^{d_0}\otimes \mathbb C^{d_0}\). It satisfies \(U^{(N)}=U^{(N)\dagger }\) for every even \(N\) if and only if there is a unitary \(x\) such that
Let an MPU be in admissible standard form. Suppose its one-site tensor \(\mathcal U\), its actual two-site block \(\mathcal U_2\), and the physical-adjoint comparison tensor \(\mathcal U_2^\sharp \) are simple and each carries its own reduced full-support source datum. Write the one-site physical space as \(\mathbb C^{d_0}\otimes \mathbb C^{d_0}\). Time-reversal symmetry then forces \(\operatorname {ind}(\mathcal U)=0\). Under these hypotheses, \(U^{(N)}=U^{(N)\dagger }\) for every even \(N\) if and only if there is a unitary \(x\) such that
Under time reversal, \(\mathcal U_2\) and \(\mathcal U_2^\sharp \) generate the same MPU. The admissible fundamental theorem identifies their source spaces, so their corresponding source-cut ranks agree. Physical adjunction exchanges the two cuts. Hence
Definition 19.9.17 therefore gives \(\operatorname {ind}(\mathcal U_2)=0\), and blocking independence gives \(\operatorname {ind}(\mathcal U)=0\). Apply the admissible fundamental theorem to the supplied standard forms of \(\mathcal U_2\) and \(\mathcal U_2^\sharp \) and follow the source proof. The converse follows by contracting the displayed local relation around every even periodic chain.
Let an MPU be in standard form, with one-site physical space \(\mathbb C^{d_0}\otimes \mathbb C^{d_0}\). It satisfies \(U^{(N)}=U^{(N)T}\) for every even \(N\) if and only if there are a unitary \(x\) and a phase \(e^{i\phi }\) such that
Let an MPU be in admissible standard form. Suppose its one-site tensor \(\mathcal U\), its actual two-site block \(\mathcal U_2\), and the transposed comparison tensor \(\mathcal U_2^{\mathsf T}\) are simple and each carries its own reduced full-support source datum. Write the one-site physical space as \(\mathbb C^{d_0}\otimes \mathbb C^{d_0}\). Transposition symmetry then forces \(\operatorname {ind}(\mathcal U)=0\). Under these hypotheses, \(U^{(N)}=U^{(N)T}\) for every even \(N\) if and only if there are a unitary \(x\) and a phase \(e^{i\phi }\) such that
Under transposition, \(\mathcal U_2\) and \(\mathcal U_2^{\mathsf T}\) generate the same MPU. The admissible fundamental theorem identifies their source spaces, so their corresponding source-cut ranks agree. Transposition exchanges the two cuts. Hence
Thus \(\operatorname {ind}(\mathcal U_2)=0\) by Definition 19.9.17, and blocking independence gives \(\operatorname {ind}(\mathcal U)=0\). Apply the admissible fundamental theorem to the supplied standard forms of \(\mathcal U_2\) and \(\mathcal U_2^{\mathsf T}\) and repeat the time-reversal argument, retaining the unrestricted phase \(e^{i\phi }\). The converse follows by contracting the local relation around every even periodic chain.
19.9.2 Equivalence under symmetries
A finite-chain operator symmetry \(\mathcal S\) specifies a set \(A\) of applicable chain lengths and, for every physical dimension \(d\) and length \(N\), a map
We suppress the subscripts on \(\mathcal S_{d,N}\) when \(d\) and \(N\) are understood. An MPO tensor \(\mathcal W\) is invariant under \(\mathcal S\) if
for every \(N\in A\).
Fix a virtual bond dimension \(D\). Two canonical-form tensors \(\mathcal U\) and \(\mathcal V\) are strictly equivalent under a symmetry \(\mathcal S\) if they have the same physical dimension and there is a continuous path \(\mathcal W(p)\) of fixed-bond MPU tensors from \(\mathcal U\) to \(\mathcal V\), not necessarily in canonical form, such that
for every \(p\in [0,1]\) and every applicable chain length \(N\).
Strict equivalence under a symmetry implies strict equivalence.
The same path lies in the larger locus obtained by forgetting invariance under \(\mathcal S\).
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 \(\mathcal S\).
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 \(D^4\).
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 \(p\in [0,1]\).
Fix a positive integer \(m\leq D^4\). The actual \(m\)-site blocked endpoint MPUs \(U_1^{(mN)}\) and \(U_2^{(mN)}\) are admissibly strictly equivalent under \(\mathcal S\) if and only if there are continuous families of unitaries \(\widetilde u(p)\) and \(\widetilde v(p)\) whose endpoints generate \(U_1^{(mN)}\) and \(U_2^{(mN)}\), whose generated MPU is invariant under \(\mathcal S\) for every \(p\in [0,1]\), and whose tensor path and fixed conjugate, physical-adjoint, transposed, and \(\mathcal S\)-image comparison paths carry continuously varying reduced full-support source data. These families arise from the actual blocked path \(\mathcal W(p)_m\) in an admissible equivalence datum of common length \(m\); the blocked path itself is the length-one admissible witness for the displayed endpoint families. When \(m{\gt}1\), this criterion does not assert strict equivalence of the unblocked families \(U_1^{(N)}\) and \(U_2^{(N)}\). This is the reduced-source form of [ CPGSV17 , Lemma following the symmetry-equivalence definitions ] .
Suppose that 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
For \(i=1,2\), suppose the original one-site endpoint tensor \(\mathcal U_i\) and its conjugate comparison tensor are simple and each carries reduced full-support source data. Suppose that either \(r\) or \(\ell \) is odd and that the standard-form interpolation determined below carries an admissible path datum of blocking length one. Then \(\mathcal U_1\) and \(\mathcal U_2\) are admissibly strictly equivalent under conjugation symmetry if and only if
If either \(r\) or \(\ell \) is odd, all conjugation-invariant MPUs with these source ranks are equivalent under conjugation symmetry.
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.
Suppose \(r\) and \(\ell \) are even. Two conjugation-invariant MPUs in the standard forms of Proposition 19.9.1.7 are strictly equivalent under conjugation if and only if they belong to the same case and
Suppose that \(r\) and \(\ell \) are both even and that the standard-form interpolation determined below carries an admissible path datum of blocking length one. Choose admissible standard forms of two conjugation-invariant MPUs as in Proposition 19.9.1.8. They are admissibly strictly equivalent under conjugation symmetry if and only if they belong to the same case of Proposition 19.9.1.8 and
Two conjugation-invariant MPUs are equivalent under conjugation symmetry if and only if they belong to the same case of Proposition 19.9.1.7.
For \(i=1,2\), suppose the original one-site endpoint tensor \(\mathcal U_i\), its conjugate comparison \(\overline{\mathcal U_i}\), the actual two-site block \(\mathcal U_{i,2}\), and its conjugate comparison \(\overline{\mathcal U_{i,2}}\) are simple and each carries reduced full-support source data. Suppose also that the resulting standard-form interpolation carries an admissible path datum of blocking length one. Then \(\mathcal U_1\) and \(\mathcal U_2\) are admissibly equivalent under conjugation symmetry if and only if they belong to the same case of Proposition 19.9.1.8.
Suppose first that either \(r\) or \(\ell \) is odd. Then both original endpoints lie in Case I. After one blocking, write \(r^{(2)}\) and \(\ell ^{(2)}\) for the blocked source ranks. Admissible blocking independence, applied to the supplied one-site and two-site endpoint data, gives
and
For each endpoint,
Blocking preserves Case I. If either \(r^{(2)}\) or \(\ell ^{(2)}\) is odd, the odd-rank theorem applied to the blocked endpoints gives admissible strict equivalence. If both blocked ranks are even, then \(d\) is even because one original source rank is odd. Hence \(r\ell =d^2\) is even. Consequently,
and
The even-rank theorem now applies to the preserved Case I endpoints. Thus the odd-source-rank branch has a single admissible equivalence class without imposing a parity condition on the blocked ranks.
If \(r\) and \(\ell \) are both even, admissible blocking independence shows that the blocked ranks remain even. Since \(r\ell \) is even, the same calculations give
and
Blocking preserves Case I and Case II. The even-rank theorem therefore shows that the blocked endpoints are admissibly strictly equivalent exactly when they belong to the same case. This is precisely admissible equivalence of the original endpoints.
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
In product coordinates, the exchange matrix has entries \(\mathbb S_{(i,j),(k,l)}=\delta _{i,l}\delta _{j,k}\). Therefore
Taking \(A=U\) and \(B=U^\dagger \) gives \(\operatorname{tr}(UU^\dagger )=\operatorname{tr}(\mathbb {1})=n\).
For a time-reversal-invariant MPU in standard form, the sign \(\sigma \) is
where \(\mathbb S_{12,34}\) exchanges the pairs of sites \(12\) and \(34\).
For a time-reversal-invariant MPU in admissible standard form, suppose its one-site tensor \(\mathcal U\), its actual two-site block \(\mathcal U_2\), and the physical-adjoint comparison tensor \(\mathcal U_2^\sharp \) are simple and each carries its own reduced full-support source datum. Then the sign \(\sigma \) in Proposition 19.9.1.12 is
where \(\mathbb S_{12,34}\) exchanges the pairs of sites \(12\) and \(34\).
Under \(u'=(x\otimes y)u\) and \(v'=v(y^\dagger \otimes x^\dagger )\), the trace formula in Lemma 19.9.2.18 has the same value for \((u',v')\) as for \((u,v)\). Thus \(\sigma \) is independent of the standard-form gauge.
Under the hypotheses of Lemma 19.9.2.19, let \(x\) and \(y\) be unitaries and change the standard-form gauge by
Then the trace formula has the same value for \((u',v')\) as for \((u,v)\). Consequently the sign \(\sigma \) is independent of the standard-form gauge.
Write \(x_i\) and \(y_i\) for the unitaries acting on site \(i\). Then \(u'_{12}=x_1y_2u_{12}\), \(u'_{34}=x_3y_4u_{34}\), and \(v'_{23}=v_{23}y_2^\dagger x_3^\dagger \). Substitution into Lemma 19.9.2.19, cancellation on the middle legs, and \(\mathbb S_{12,34}x_1y_4=x_3y_2\mathbb S_{12,34}\) give
The last equality follows from cyclicity of the trace and unitarity of \(x\) and \(y\).
For a time-reversal-invariant MPU in standard form, the sign \(\sigma =\pm 1\) is independent of the gauge and is constant under continuous deformations that preserve time-reversal symmetry.
Suppose a time-reversal-preserving path carries an admissible equivalence datum of blocking length one. At every path point, require the one-site tensor, its actual two-site block, and the physical-adjoint comparison of that two-site block to be simple and to carry continuously varying reduced full-support source data. Then the sign \(\sigma =\pm 1\) is independent of the admissible standard-form gauge and is constant along the path.
Let \(\mathcal U\) be a time-reversal-invariant MPU in standard form, and let \(\sigma ^{(k)}\) be the sign obtained from a standard form of its \(k\)-site block. Then
For every blocking length \(k\) under consideration, suppose the endpoint tensor \(\mathcal U_k\), its actual two-site block \(\mathcal U_{2k}\), and the physical-adjoint comparison tensor \(\mathcal U_{2k}^\sharp \) are simple and each carries its own reduced full-support source datum. Let \(\sigma ^{(k)}\) be the sign of the resulting standard form. Then
Gauge invariance permits the standard blocked representatives used in the source proof. Choose
and put \(\zeta =\sigma ^{(2)}/\sigma ^{(1)}\). The one- and two-site local time-reversal relations give the unwinding identity
Applying this identity successively through a block of length \(k\) gives
Since each sign is \(\pm 1\), this equals \(\sigma ^{(1)}\) for odd \(k\) and \(\sigma ^{(2)}\) for even \(k\).
Strictly equivalent time-reversal-invariant MPUs have the same \(\sigma ^{(1)}\) and \(\sigma ^{(2)}\). Equivalent time-reversal-invariant MPUs have the same \(\sigma ^{(2)}\).
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.
For a conjugation-invariant MPU, the MPS sign \(G\overline G=\pm \mathbb {1}\) agrees with Case I and Case II, respectively, of Proposition 19.9.1.7.
Let \(\mathcal U\) be conjugation invariant. Suppose \(\mathcal U\) and its conjugate comparison tensor are simple and carry their own reduced full-support source data. Let \(p\mapsto \mathcal W(p)\) be the actual standard-form interpolation from \(\mathcal U\) to its Case I or Case II representative. Suppose this interpolation carries an admissible conjugation-symmetry path datum of blocking length one: the path \(\mathcal W(p)\) and its fixed conjugate, physical-adjoint, transposed, and conjugation-image comparison paths are simple and carry continuously varying source data. Then the sign \(G\overline G=\pm \mathbb {1}\) in the MPS classification agrees with Case I and Case II, respectively, of Proposition 19.9.1.8.
For a time-reversal-invariant MPU in standard form, the MPS sign \(G\overline G=\pm \mathbb {1}\) equals the sign \(\sigma ^{(2)}\) obtained after blocking two sites.
Let an MPU in admissible standard form be invariant under time reversal. Suppose \(\mathcal U\), \(\mathcal U_2\), and \(\mathcal U_4\), together with the physical-adjoint comparison tensors \(\mathcal U_2^\sharp \) and \(\mathcal U_4^\sharp \), are simple and each carries its own reduced full-support source datum. Then the MPS sign \(G\overline G=\pm \mathbb {1}\) equals the sign \(\sigma ^{(2)}\).
19.9.3 Symmetry examples
Let \(u_+=v_+=\mathbb {1}_{r\ell }\) and \(u_-=v_-=\operatorname {diag}(-1,1,\ldots ,1)\). If at least one of \(r\) and \(\ell \) is odd, \((u_+,v_+)\) and \((u_+,v_-)\) realize the two classes of Theorem 19.9.2.9. If both are even, the four pairs \((u_\pm ,v_\pm )\) realize the four Case I classes. Multiplying by the matrix \(R\) of Corollary 19.9.1.9 realizes the four Case II classes.
Assume the standard-form tensors displayed below carry reduced full-support source data and that the comparison paths used to classify them carry admissible path data of blocking length one. Let
If at least one of \(r\) and \(\ell \) is odd, the pairs \((u_+,v_+)\) and \((u_+,v_-)\) represent the two classes of Theorem 19.9.2.10. If \(r\) and \(\ell \) are both even, the four pairs \((u_\pm ,v_\pm )\) represent the four Case I classes of Theorem 19.9.2.14. With
the four pairs \((u'_\pm ,v'_\pm )\) represent the four Case II classes.
The CZX MPU is defined by
where \(CZ=\operatorname {diag}(1,1,1,-1)\). Let
be the Hadamard matrix. The corresponding tensor is
It has \(\sigma ^{(1)}=\sigma ^{(2)}=-1\).
Suppose the CZX tensors \(\mathcal U\), \(\mathcal U_2\), and \(\mathcal U_4\), together with the physical-adjoint comparison tensors \(\mathcal U_2^\sharp \) and \(\mathcal U_4^\sharp \), are simple and each carries its own reduced full-support source datum. The CZX MPU is defined by
Its tensor is
It has \(\sigma ^{(1)}=\sigma ^{(2)}=-1\).
The standard forms
realize, respectively,
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 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.
For every \(N{\gt}0\), the tensors in Definition 19.9.3.7 generate \(T^{(N)}\) and \(T^{(N)\dagger }\), respectively. Both tensors are matrix product unitaries.
Closing the virtual indices of \(A^{ij}=\lvert i\rangle \langle j\rvert \) leaves one nonzero virtual configuration. If \(\tau _n=\sigma _{n+1}\) for every \(n\) modulo \(N\), then
Otherwise the same matrix entry is zero. Hence the output configuration is \((\tau _{N-1},\tau _0,\ldots ,\tau _{N-2})\), which fixes the permutation-matrix convention and gives the right shift. Its permutation matrix is unitary. Physical adjunction gives the conjugate transpose, hence the left shift and its unitarity.
Let \(A\) be the right-shift tensor and let \(z_{(\alpha ,i)}=\delta _{\alpha i}\) be the vectorized identity. Its raw source cuts are
For the left-shift tensor \(A^\sharp \) they are
In every physical dimension, including \(d=0\),
If \(d{\gt}0\), the remaining two ranks are
These are the raw source-cut quantities from [ CPGSV17 , Section III ] ; no choice of source factors is involved.
The entry formula \(A^{ij}_{\alpha \beta } =\delta _{i\alpha }\delta _{j\beta }\) gives \(M_1(A)=\mathbb {1}\) and \(M_2(A)=zz^{\mathsf T}\). Hence the first rank is \(d^2\). For \(d{\gt}0\), the vector \(z\) has a nonzero diagonal coordinate, so its outer product has rank one. Physical adjunction exchanges the two cuts and their ranks, which gives all four left-shift formulas without a second rank calculation.
On two \(d\)-dimensional spins per site, define
The three tensors generate the operators displayed in Definition 19.9.3.10 on every nonempty chain, and each tensor is a matrix product unitary.
The bond-one identity tensor generates the identity at every length. The finite-chain operator of an independent tensor product is the Kronecker product of the two finite-chain operators, after the sitewise identification \((\mathbb C^d\otimes \mathbb C^d)^{\otimes N} \cong (\mathbb C^d)^{\otimes N}\otimes (\mathbb C^d)^{\otimes N}\). Applying this identity to the right and left shifts gives the three displayed formulas. The identity tensor is unitary, and tensor products preserve unitarity, so all three tensors are matrix product unitaries.
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.
Assume \(d{\gt}0\). In the source coordinates of the supplied factorizations,
These are the formulas in [ CPGSV17 , lines 2009–2016 ] . The \(U_3\) equalities use the source-coordinate presentation of that equation; they are distinct from the blocked four-spin presentation below.
Write the source and physical indices as \(((a,b),(c,e))\) and \(((i,j),(k,l))\), respectively, and set \(s=d^{-1/2}\). The identity factors give
For \(U_3\), the right-shift and left-shift factors instead give
where \(d\, s^2=1\). The first display is the identity matrix, while the second interchanges the two composite indices and is therefore \(\mathbb S\) in both orientations.
Assume \(d{\gt}0\). The two open-leg factorizations of the blocked tensor have, as their respective central gates,
Here the two physical sites are decoded into the four-spin order of the paper, and the factors \(X_1,Y_2\) and \(X_2,Y_1\) remain as the two open boundaries of the respective factorizations. This is [ CPGSV17 , lines 2018–2026 ] .
Write \(B_2=\mathcal U_{2,2}\) for the actual two-site block and decode \(I=(I_1,I_2)\) and \(J=(J_1,J_2)\). If \(\pi \) is the four-spin physical identification and \(\iota _u,\iota _v\) are the two source-rank identifications, the asserted open-leg equalities are
Write \(s=d^{-1/2}\), so that \(d\, s^2=1\). In the product-rank coordinates, the entries of the two central gates are
These are respectively the entries of \(\mathbb {1}\otimes \mathbb S\otimes \mathbb {1}\) and \((\mathbb S\otimes \mathbb S)(\mathbb {1}\otimes \mathbb S\otimes \mathbb {1})\); substituting them into the exact open two-site factorizations gives the asserted equalities.
Assume \(d{\gt}0\). The two open-leg factorizations of the blocked tensor have, as their respective central gates,
Again the four-spin order is the one displayed in the paper, and the source factors at the two open boundaries are retained explicitly. This is [ CPGSV17 , lines 2028–2034 ] .
With \(B_3=\mathcal U_{3,2}\) for the actual two-site block and the corresponding identifications \(\pi ,\iota _u,\iota _v\), the exact equalities are
Again write \(s=d^{-1/2}\), so that \(d\, s^2=1\). Reversing the two supplied factors gives the entry identities
The first is the entry of \((\mathbb {1}\otimes \mathbb S\otimes \mathbb {1})(\mathbb S\otimes \mathbb S)\), and the second is the entry of \(\mathbb {1}\otimes \mathbb S\otimes \mathbb {1}\). Substitution into the two exact open two-site factorizations proves the claim.
Assume \(d{\gt}0\). For the fixed standard-form source factors, the corresponding unitaries satisfy
The preceding theorems establish these matrices for explicit supplied source-factor witnesses. It remains to identify that supplied choice with the fixed standard-form choice.
Define
The first two standard forms are
Two MPUs \(U_i^{(N)}\) and \(U_j^{(N)}\) are in the same phase under time reversal combined with the local swap, transposition combined with the local swap, or conjugation if and only if the corresponding \(\widetilde U_i^{(N)}\) and \(\widetilde U_j^{(N)}\) are in the same phase under time reversal, transposition, or conjugation, respectively.
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.
After blocking two sites, \(U_2^{(N)}\) and \(U_3^{(N)}\) are both in the trivial phase under conjugation.
Suppose the two-site blocked tensors carry reduced full-support source data and the standard-form comparison paths carry admissible path data of blocking length one. Then the actual two-site blocked tensors of \(U_2\) and \(U_3\) are both in the trivial admissible strict phase under conjugation. Consequently \(U_2\) and \(U_3\) lie in the same admissible equivalence class.
The MPUs \(\widetilde U_2^{(N)}\) and \(\widetilde U_3^{(N)}\) are in the same phase under both time reversal and transposition.
For each of time reversal and transposition, let \(p\mapsto \mathcal W_{\mathcal S}(p)\) be the actual source-constructed interpolation from \(\widetilde U_2\) to \(\widetilde U_3\). Suppose it carries an admissible symmetry-path datum of blocking length one. Thus the path \(\mathcal W_{\mathcal S}(p)\) and its fixed conjugate, physical-adjoint, transposed, and \(\mathcal S\)-image comparison paths are simple and carry continuously varying reduced full-support source data. Suppose also that the one- and two-site endpoint tensors used in the local comparison carry their own data. Then \(\widetilde U_2^{(N)}\) and \(\widetilde U_3^{(N)}\) are in the same admissible phase under both time reversal and transposition.
The MPUs \(U_1^{(N)}\) and \(U_2^{(N)}\) lie in different strict conjugation phases exactly when \(d=4k+2\) for a nonnegative integer \(k\). Otherwise they lie in the same strict phase. All these phases are in Case I, so the two MPUs are equivalent after blocking and adjoining ancillas.
For each endpoint tensor \(\mathcal A\) among \(U_1\), \(U_2\), \(\widetilde U_1\), and \(\widetilde U_2\), suppose that \(\mathcal A\), its conjugate comparison \(\overline{\mathcal A}\), its actual two-site block \(\mathcal A_2\), and the conjugate comparison \(\overline{\mathcal A_2}\) are simple and carry their own reduced full-support source data. For every strict comparison used below, let \(p\mapsto \mathcal W(p)\) be the actual source-constructed interpolation between the corresponding endpoint tensors. Suppose it carries an admissible conjugation-symmetry path datum of blocking length one: the path and its fixed conjugate, physical-adjoint, transposed, and conjugation-image comparison paths are simple and carry continuously varying source data. Require a length-one admissible path datum also for the actual blocked interpolation used in the final equivalence claim. Under these hypotheses, \(U_1^{(N)}\) and \(U_2^{(N)}\) are in different admissible strict phases under conjugation exactly when \(k\) is a nonnegative integer and
Otherwise they are in the same admissible strict phase. All these standard forms lie in Case I, so \(U_1^{(N)}\) and \(U_2^{(N)}\) are admissibly equivalent after blocking and adding ancillas.
After adding one ancilla of dimension \(d\) per site, \(\widetilde U_1^{(N)}\) and \(\widetilde U_2^{(N)}\) are strictly equivalent under both time reversal and transposition.
After adding one ancilla of dimension \(d\) per site and performing the source blocking, denote the actual enlarged blocked endpoints by \(\widehat{\mathcal U}_1\) and \(\widehat{\mathcal U}_2\). For either time reversal or transposition, let \(p\mapsto \mathcal W_{\mathcal S}(p)\) be the source-constructed interpolation from \(\widehat{\mathcal U}_1\) to \(\widehat{\mathcal U}_2\). Suppose this interpolation carries an admissible symmetry-path datum of blocking length one. Thus the path \(\mathcal W_{\mathcal S}(p)\) and its fixed conjugate, physical-adjoint, transposed, and \(\mathcal S\)-image comparison paths are simple and carry continuously varying reduced full-support source data. Suppose also that the one- and two-site enlarged endpoint tensors used in the local comparison carry their own data. Then \(\widehat{\mathcal U}_1\) and \(\widehat{\mathcal U}_2\) are admissibly strictly equivalent under both time reversal and transposition. No equivalence of the original ancilla-free tensors is asserted, because the equal ancilla dimensions used here need not satisfy the coprimality condition in Definition 19.9.2.6.
19.10 Finite-region observable algebras
Fix a nonnegative integer \(d\). For a finite region \(\Lambda \subset \mathbb Z\), the configuration space and local observable algebra are
The algebra \(\mathcal A_\Lambda \) carries its operator norm. This is the finite-region construction used in the QCA appendix of [ CPGSV17 ] ; no completion is taken here.
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 \(d{\gt}0\), the map \(\iota _{\Lambda ,\Gamma }\) is injective.
Since \(d{\gt}0\), choose a configuration \(c\) on \(\Gamma \setminus \Lambda \). For configurations \(x,y\) on \(\Lambda \),
Hence \(\iota _{\Lambda ,\Gamma }(A)=\iota _{\Lambda ,\Gamma }(B)\) implies \(A=B\).
If \(d{\gt}0\), then for every \(A\in \mathcal A_\Lambda \),
An injective star-algebra homomorphism between complex \(C^*\)-algebras is isometric.
For \(d{\gt}0\), the map \(\iota _{\Lambda ,\Gamma }\) is an operator-norm isometry.
Use injectivity and the general theorem that an injective star-algebra homomorphism between complex \(C^*\)-algebras is an isometry.
For every finite region \(\Lambda \), \(\iota _{\Lambda ,\Lambda }\) is the identity on \(\mathcal A_\Lambda \).
When the two regions agree, their complement is empty. Hence
If \(\Lambda \subseteq \Gamma \subseteq \Delta \), then
The complement decomposes as the disjoint union
Therefore
If \(\Lambda \cap \Gamma =\varnothing \), \(A\in \mathcal A_\Lambda \), and \(B\in \mathcal A_\Gamma \), then their images in the algebra of the union satisfy
This tensor-factor identity is local algebra infrastructure for the quasi-local algebra used in [ CPGSV17 , lines 2292–2300 ] ; the appendix does not state it as a separate theorem.
For configurations \(x,y\) on \(\Lambda \cup \Gamma \), split each intermediate configuration \(z\) into \((z|_\Lambda ,z|_\Gamma )\). The matrix entry of the left-hand side is
Reversing the two factors gives the same scalar product.
Let \(\Lambda \subseteq \Gamma \) be finite regions and \(A\in \mathcal A_\Gamma \). Then \(A\) acts trivially outside \(\Lambda \) if and only if it commutes with the full observable algebra on the complement:
This is a finite-dimensional tensor-factor identity. It is not a statement of Schumacher–Werner or the QCA appendix of [ CPGSV17 ] .
Splitting configurations gives \(\mathcal A_\Gamma \cong \mathcal A_\Lambda \otimes \mathcal A_{\Gamma \setminus \Lambda }\). Under this identification, the two inclusions are \(C\mapsto C\otimes \mathbf1\) and \(B\mapsto \mathbf1\otimes B\). Write an arbitrary matrix in blocks indexed by configurations on \(\Lambda \). If it commutes with every \(\mathbf1\otimes B\), each block commutes with the full matrix algebra on \(\Gamma \setminus \Lambda \) and is therefore scalar. Thus the matrix is \(C\otimes \mathbf1\) for some \(C\in \mathcal A_\Lambda \). The converse follows from
19.10.1 Bipartite support algebras
Let \(I\) and \(J\) be finite sets. For \(X\in M_{I\times J}(\mathbb C)\), the existing left coefficient slice and right operator block are the matrices
For a star-subalgebra \(\mathcal D\subseteq M_K(\mathbb C)\), write \(\mathcal D'=\{ Y\in M_K(\mathbb C)\mid [Y,Z]=0\text{ for every }Z\in \mathcal D\} \) for its commutant. The construction below is the full complex matrix-algebra specialization of the coefficient construction in [ SW04 , Section 4.3 ] and of the leastness and commutant statements in [ GNVW12 , Lemma 7 ] .
Local fix (star-closed input): The printed statement of Lemma 7 says only that \(\mathcal A\) is a subalgebra. Its coefficient-generated support and leastness clauses remain valid for arbitrary algebra input, but its commutant characterization fails unless \(\mathcal A\) is closed under the involution. For instance, the algebra \(\operatorname {span}\{ \mathbf1,E_{12}\} \subset M_2(\mathbb C)\) gives a counterexample to the printed commutant clause if involution closure is omitted. Schumacher and Werner first define the ordinary algebra generated by the coefficient support and then derive its adjoint closure from adjoint closure of the observable-algebra input [ SW04 , lines 1161–1171 ] . Accordingly, the statements below use a star-subalgebra. This is a necessary correction to the printed hypothesis. In the QCA application the input is the image of a local observable algebra under a star-automorphism [ GNVW12 , Section 7.2 ] . The MPU application likewise takes the image of a local star-algebra under a star-automorphism [ CPGSV17 , lines 2300–2306 ] . This correction, including the counterexample without involution closure and the exact application ranges, is recorded in [ con26w ] .
Scope restriction (full matrix factors): After this correction, Lemma 7 permits arbitrary finite-dimensional \(C^*\)-algebras \(\mathcal B_1\) and \(\mathcal B_2\), whereas the statements below take \(\mathcal B_1=M_I(\mathbb C)\) and \(\mathcal B_2=M_J(\mathbb C)\). Full matrix factors suffice for the MPU spin chain because every one-site algebra is isomorphic to \(M_d(\mathbb C)\) [ CPGSV17 , lines 2320–2334 ] . The general-factor result is not claimed here; see [ con26k ] .
No full-matrix structure theorem, dimension formula, Margolus gate, depth-two circuit, MPU standard form, or index comparison is asserted here.
For every \(X\in M_{I\times J}(\mathbb C)\) and \(i,j\in I\),
Compare the \((r,s)\) entries. Both sides equal \(\overline{X((j,s),(i,r))}\).
Let \(\mathcal A\subseteq M_{I\times J}(\mathbb C)\) be a star-subalgebra. Its set of left coefficients is
Let \(\mathcal A\subseteq M_{I\times J}(\mathbb C)\) be a star-subalgebra. Its set of right coefficients is
The left support algebra of \(\mathcal A\) is the star-subalgebra
generated by all left coefficients.
The right support algebra of \(\mathcal A\) is the star-subalgebra
generated by all right coefficients.
If \(X\in \mathcal A\) and \(r,s\in J\), then
The matrix \(X^{\mathrm L}_{rs}\) is one of the generators of the defining star-algebra closure.
If \(X\in \mathcal A\) and \(i,j\in I\), then
The matrix \(X^{\mathrm R}_{ij}\) is one of the generators of the defining star-algebra closure.
For every \(X\in M_{I\times J}(\mathbb C)\),
At the entry \(((i,r),(j,s))\), only the matrix unit \(E_{rs}\) contributes, and its coefficient is \(X^{\mathrm L}_{rs}(i,j)=X((i,r),(j,s))\).
For every \(X\in M_{I\times J}(\mathbb C)\),
At the entry \(((i,r),(j,s))\), only the matrix unit \(E_{ij}\) contributes, and its coefficient is \(X^{\mathrm R}_{ij}(r,s)=X((i,r),(j,s))\).
For every star-subalgebra \(\mathcal B\subseteq M_I(\mathbb C)\),
Use the universal property of the star-subalgebra generated by \(\operatorname {Coeff}_{\mathrm L}(\mathcal A)\).
For every star-subalgebra \(\mathcal C\subseteq M_J(\mathbb C)\),
Use the universal property of the star-subalgebra generated by \(\operatorname {Coeff}_{\mathrm R}(\mathcal A)\).
Let \(I=\{ 0,\ldots ,p-1\} \) and \(J=\{ 0,\ldots ,q-1\} \). For every star-subalgebra \(\mathcal B\subseteq M_p(\mathbb C)\),
where the right-hand side is the tensor-product submodule consisting of matrices whose left coefficient slices lie in \(\mathcal B\).
Membership in the tensor-product submodule is equivalent to membership of every left coefficient in \(\mathcal B\). Apply Theorem 19.10.1.10.
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.
For \(X\in M_{I\times J}(\mathbb C)\),
For \(A\in S\) and \(B\in T\), direct evaluation gives
Thus the forward implication extends by linearity. Conversely, expand the right blocks in a basis of \(T\). Applying the dual coordinate functionals to those blocks produces matrices in \(S\) from the left-slice hypothesis, and the resulting finite sum reconstructs \(X\) as a sum of Kronecker products from \(S\) and \(T\).
Let \(I=\{ 0,\ldots ,p-1\} \) and \(J=\{ 0,\ldots ,q-1\} \). For every submodule \(S\subseteq M_p(\mathbb C)\),
Thus the two-sided Kronecker span extends the existing left-oriented tensor-submodule construction.
In Theorem 19.10.1.14, take the right submodule to be the full matrix space. The right-block condition is then automatic, leaving precisely the left-slice criterion.
For every star-subalgebra \(\mathcal C\subseteq M_J(\mathbb C)\),
The coefficient criterion reduces the right-hand containment to membership of every right block in \(\mathcal C\). This is equivalent to the left-hand inclusion by Theorem 19.10.1.11.
Every bipartite star-subalgebra satisfies the literal two-sided containment
This is the full complex matrix-algebra specialization of the support-algebra containment in Schumacher–Werner Section 4.3 and of Gross–Nesme–Vogts–Werner equation (22). The latter follows after applying Lemma 7 to both tensor factors. The displayed assertion is an inclusion in an actual tensor-product span, not merely the conjunction of the two one-sided inclusions. The MPU appendix applies this construction to \(\mathcal R_{2x}\) in [ CPGSV17 , lines 2320–2334 ] .
If \(X\in \mathcal A\), every left coefficient of \(X\) lies in the left support algebra and every right coefficient lies in the right support algebra. Theorem 19.10.1.14 places \(X\) in their Kronecker-product submodule.
For \(B\in M_I(\mathbb C)\) and \(X\in M_{I\times J}(\mathbb C)\),
For \(i,j\in I\) and \(r,s\in J\), the two product entries are
Equality of the two products is therefore equivalent to \([B,X^{\mathrm L}_{rs}]=0\) for every \(r,s\in J\).
For \(C\in M_J(\mathbb C)\) and \(X\in M_{I\times J}(\mathbb C)\),
For \(i,j\in I\) and \(r,s\in J\), the two product entries are
Equality of the two products is therefore equivalent to \([C,X^{\mathrm R}_{ij}]=0\) for every \(i,j\in I\).
For \(B\in M_I(\mathbb C)\),
Equivalently,
By Theorem 19.10.1.18, commutation of \(B\otimes \mathbf1_J\) with every \(X\in \mathcal A\) is equivalent to commutation of \(B\) with every left coefficient. The coefficient set is closed under adjoints because \(\mathcal A\) is a star-subalgebra. The universal property of the generated star-subalgebra then extends this commutation relation to all of \(\operatorname {Spp}_{\mathrm L}(\mathcal A)\); the converse follows from generator membership.
For \(C\in M_J(\mathbb C)\),
Equivalently,
By Theorem 19.10.1.19, commutation of \(\mathbf1_I\otimes C\) with every \(X\in \mathcal A\) is equivalent to commutation of \(C\) with every right coefficient. Theorem 19.10.1.1 shows that the right coefficient set is closed under adjoints. The universal property of the generated star-subalgebra extends commutation to all of \(\operatorname {Spp}_{\mathrm R}(\mathcal A)\); the converse follows from generator membership.
Let \(I\), \(J\), and \(K\) be finite sets, and let
be star-subalgebras. Suppose that, for every \(X\in \mathcal A_1\) and \(Y\in \mathcal A_2\),
Then the two support algebras on the common factor commute:
This is the full complex matrix-algebra specialization of Schumacher–Werner Lemma sppcomm [ SW04 , lines 1174–1194 ] and GNVW Lemma sppcomm [ GNVW12 , Section 7.1, lines 1221–1246 ] .
Local fixes. For the generated \(C^*\)-algebras, the input must be closed under the involution; see [ con26w ] . In both printed coefficient proofs, the basis element \(e'_\nu \) belongs to the third factor, rather than the second factor as printed; see [ con26o ] .
Scope restriction (full matrix factors). The cited lemmas permit arbitrary finite-dimensional \(C^*\)-algebra factors, whereas the statement above treats full complex matrix algebras. This restriction suffices for the MPU application in [ CPGSV17 , lines 2300–2306 and 2313–2334 ] ; the general-factor statement remains open as recorded in [ con26k ] .
For \(X\in \mathcal A_1\), \(Y\in \mathcal A_2\), \(i,i'\in I\), and \(k,k'\in K\), applying Theorem [ LTC26 , “Middle-factor coefficients of commuting overlapping operators” ] to (664) gives
Applying the same identity to \(Y^\dagger \in \mathcal A_2\), with \(k\) and \(k'\) exchanged, also gives commutation with \((Y^{\mathrm L}_{kk'})^\dagger \). Hence every generator of \(\operatorname {Spp}_{\mathrm R}(\mathcal A_1)\) lies in the star-subalgebra commuting with all left coefficients of \(\mathcal A_2\). Passing to the generated star-subalgebra yields
for every \(R\in \operatorname {Spp}_{\mathrm R}(\mathcal A_1)\) and every left coefficient \(L\) of \(\mathcal A_2\).
Since \(\operatorname {Spp}_{\mathrm R}(\mathcal A_1)\) is closed under adjoints, applying the first equality in (667) to \(R^\dagger \) shows that each left coefficient commutes with both \(R\) and \(R^\dagger \). Passing to the star-subalgebra generated by the left coefficients gives (665).
For a finite region \(\Lambda \subset \mathbb Z\) and \(a\in \mathbb Z\), define
Thus
The map \(\Lambda \mapsto \Lambda +a\) is injective. If \(\Lambda \subseteq \Gamma \), then \(\Lambda +a\subseteq \Gamma +a\).
For finite regions \(\Lambda ,\mathcal N\subset \mathbb Z\), define their sumset by
Hence \(j\in \Lambda +\mathcal N\) exactly when \(j=i+a\) for some \(i\in \Lambda \) and \(a\in \mathcal N\). The sumset is monotone in both arguments, either empty argument gives the empty region, and the singleton \(\{ 0\} \) is an identity. A singleton displacement recovers translation,
Translation commutes with either argument, and sumsets are associative:
These identities support composition of propagation bounds. This is the finite region \(\Lambda +\mathcal N\) in the propagation condition of the QCA appendix of [ CPGSV17 ] .
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 \(d\) 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 \(\mathcal A_{\Lambda +a}\).
Translation bijectively relabels the configuration basis. The resulting star-algebra equivalence between the two finite matrix algebras preserves the operator norm. Applying this identity to \(A-B\) and using linearity gives
so distances are preserved.
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.
The site identities follow from
Relabelling a configuration is precomposition by the inverse site translation, so evaluation at the corresponding translated site gives
Applying this identity successively proves the configuration action laws. Finally, the local observable translation relabels both matrix indices by \(T_{a,\Lambda }^{-1}\); evaluating on translated configurations therefore gives \(A(x,y)\), and applying this entrywise identity successively proves the local observable action laws.
If \(\Lambda \subseteq \Gamma \), then for every \(A\in \mathcal A_\Lambda \),
The lattice translation identifies \(\Gamma \setminus \Lambda \) with \((\Gamma +a)\setminus (\Lambda +a)\). Hence relabelling configurations carries the identity on the first complement to the identity on the second complement, and
The finite-region local algebras form a directed system under the maps \(\iota _{\Lambda ,\Gamma }\). Its algebraic direct limit is the algebraic local algebra
For every finite region \(\Lambda \), write \(\iota _\Lambda \colon \mathcal A_\Lambda \to \mathcal A_{\mathrm{loc}}\) for the canonical inclusion. Whenever \(\Lambda \subseteq \Gamma \), these inclusions satisfy
This is the union \(\bigcup _{\Lambda \Subset \mathbb Z}\mathcal A_\Lambda \) in the Appendix of [ CPGSV17 ] . No norm completion is taken.
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 \(U^{(N)}\).
If \(\mathcal U\) is an MPU and \(N{\gt}1\), then \(U_a^{(N)}\) is unitary and defines the star-algebra automorphism
The factor order is the one in [ CPGSV17 , lines 2300–2306 ] .
Simultaneous reindexing preserves both unitarity equations for \(U^{(N)}\). Conjugation by the resulting unitary preserves multiplication, the unit, complex scalars, and the adjoint.
Let \(\Lambda \subseteq I_{a,N}\) and \(A\in \mathcal A_\Lambda \). Set \(A_L=\iota _{\Lambda ,I_{a,N}}(A)\). For \(N{\gt}1\), the finite-chain term in [ CPGSV17 , lines 2300–2306 ] is the star-algebra map
No stabilization in \(N\) is asserted.
Assume \(d{\gt}0\). If a local observable \(A\in \mathcal A_{\mathrm{loc}}\) is represented by \(A_\Lambda \in \mathcal A_\Lambda \) on a finite region \(\Lambda \), define its operator norm by
This does not depend on the finite region or representative. It is the norm whose topology is completed in the Appendix of [ CPGSV17 ] to obtain the quasi-local algebra.
If \(A_\Lambda \) and \(A_\Gamma \) represent the same local observable, then the defining relation for the inductive limit gives a finite region \(\Delta \supseteq \Lambda \cup \Gamma \) such that
Both inclusions preserve the operator norm, so \(\lVert A_\Lambda \rVert =\lVert A_\Gamma \rVert \).
For \(d{\gt}0\), the local algebra with the operator norm is a normed complex star algebra. No completeness assertion is made.
Choose finite-region representatives of the local observables and enlarge their regions to a common finite region. The norm axioms, the product inequality, and compatibility with complex scalar multiplication then follow from the corresponding properties of the finite-region matrix algebra.
For \(d{\gt}0\) and every local observable \(A\),
Choose a finite region \(\Lambda \) and a representative \(A_\Lambda \in \mathcal A_\Lambda \). Then
The normed star algebra of local observables satisfies the C-star axiom. This structure does not include completeness.
The C-star identity gives the required norm inequality in the reverse direction; submultiplicativity gives the other direction.
For \(d{\gt}0\) and every finite region \(\Lambda \), the canonical map
is an operator-norm isometry.
The norm of \(\iota _\Lambda (A)\) is defined to be the operator norm of its representative \(A\in \mathcal A_\Lambda \). Hence \(\lVert \iota _\Lambda (A)\rVert =\lVert A\rVert \); applying this equality to differences proves preservation of distance.
Let \(B\) be a complex star algebra. Suppose that for every finite region \(\Lambda \) there is a star-algebra homomorphism \(g_\Lambda \colon \mathcal A_\Lambda \to B\). Whenever \(\Lambda \subseteq \Gamma \), suppose that
Then there is a unique star-algebra homomorphism \(g\colon \mathcal A_{\mathrm{loc}}\to B\) satisfying \(g\circ \iota _\Lambda =g_\Lambda \) for every finite region \(\Lambda \).
Define \(g\) on a local observable represented by \(A\in \mathcal A_\Lambda \) as \(g_\Lambda (A)\). Compatibility makes this independent of the representing region. Since each \(g_\Lambda \) preserves addition, multiplication, the unit, scalar multiplication, and the star operation, so does \(g\). Every element of the direct limit has a finite-region representative, which also proves uniqueness.
A local observable \(A\in \mathcal A_{\mathrm{loc}}\) is supported in the finite region \(\Lambda \) if it lies in the image of \(\iota _\Lambda \colon \mathcal A_\Lambda \to \mathcal A_{\mathrm{loc}}\).
Every local observable is supported in some finite region. If \(A\) is supported in \(\Lambda \) and \(\Lambda \subseteq \Gamma \), then \(A\) is supported in \(\Gamma \). If \(A\) and \(B\) are supported in \(\Lambda \) and \(\Gamma \), respectively, then \(A+B\) and \(AB\) are supported in \(\Lambda \cup \Gamma \). The adjoint \(A^*\) is supported in \(\Lambda \), and both \(0\) and \(1\) are supported in every finite region. For positive on-site dimension, each map \(\iota _\Lambda \) is injective.
Enlarge representatives by tensoring with the identity on the added sites. Two representatives can therefore be moved to the finite union of their regions, where sums and products are formed. The canonical inclusions are unital star-algebra homomorphisms, so they preserve the adjoint, zero, and the unit. For \(d{\gt}0\), every transition map between finite-region algebras is injective; the canonical map from each member of such a directed system to its direct limit is therefore injective. Finally, every element of an algebraic direct limit has a representative in one member of the directed system.
If \(X,Y\in \mathcal A_{\mathrm{loc}}\) are supported in finite regions \(\Lambda \) and \(\Gamma \) with \(\Lambda \cap \Gamma =\varnothing \), then
In particular, for \(A\in \mathcal A_\Lambda \) and \(B\in \mathcal A_\Gamma \),
This is the algebraic-local form of the tensor-factor identity underlying the construction in [ CPGSV17 , lines 2292–2300 ] , not a named theorem of the appendix.
Choose representatives \(X=\iota _\Lambda (A)\) and \(Y=\iota _\Gamma (B)\). Embed both into \(\mathcal A_{\Lambda \cup \Gamma }\). The preceding finite-region identity gives
Assume \(d{\gt}0\). A local observable \(X\in \mathcal A_{\mathrm{loc}}\) is supported in a finite region \(\Lambda \) if and only if it commutes with every local observable having a finite support disjoint from \(\Lambda \):
This is an algebraic-local statement. It does not assert the corresponding characterization in the norm completion and is not stated by Schumacher–Werner. Their Theorem 6 gives the generalized Margolus structure, and Corollary 7 says that the inverse of a nearest-neighbor QCA exists and is a nearest-neighbor QCA.
The forward implication is commutation of disjointly supported observables. Conversely, choose a finite support \(\Delta \) of \(X\) and represent \(X\) by \(A\in \mathcal A_{\Lambda \cup \Delta }\). The hypothesis says that \(A\) commutes with every observable on \((\Lambda \cup \Delta )\setminus \Lambda \). Since \(d{\gt}0\), the canonical map from \(\mathcal A_{\Lambda \cup \Delta }\) into \(\mathcal A_{\mathrm{loc}}\) is injective, so this commutation already holds in the finite-region algebra. The finite relative-commutant theorem gives \(A=\iota _{\Lambda ,\Lambda \cup \Delta }(A_\Lambda )\) for some \(A_\Lambda \in \mathcal A_\Lambda \). Hence \(X=\iota _\Lambda (A_\Lambda )\).
For \(a\in \mathbb Z\), finite-region translations induce a star-algebra homomorphism
determined on every finite-region representative by
Naturality gives, for \(\Lambda \subseteq \Gamma \),
Hence the finite-region maps form a compatible family and induce the stated homomorphism on the algebraic direct limit.
The homomorphisms satisfy
On a representative \(A\in \mathcal A_\Lambda \), the two sides of the composition law are represented on the equal finite regions \((\Lambda +a)+b\) and \(\Lambda +(a+b)\). The finite-region action identity therefore gives
The zero law is proved in the same way. Equality on all finite-region representatives determines equality on the algebraic direct limit.
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
The homomorphism action laws give
Thus \(\tau _a\) is bijective with inverse \(\tau _{-a}\). The formula on finite-region representatives is the defining formula for the homomorphism.
The automorphisms form an additive action of \(\mathbb Z\):
The first two identities are the homomorphism action laws expressed as identities of star-algebra automorphisms. Taking \(b=-a\) in the composition law gives
which is the inverse identity.
A local observable \(A\) is supported in a finite region \(\Lambda \) if and only if \(\tau _a(A)\) is supported in \(\Lambda +a\).
If \(A=\iota _\Lambda (A_\Lambda )\), then
which proves the forward implication. Apply the same argument to \(\tau _a(A)\) with displacement \(-a\), and use \(\tau _{-a}\tau _a=\operatorname {id}\) and \((\Lambda +a)-a=\Lambda \), for the reverse implication.
If \(d{\gt}0\), then every translation preserves the operator norm and distance:
Choose \(A_\Lambda \in \mathcal A_\Lambda \) with \(A=\iota _\Lambda (A_\Lambda )\). Finite-region translation is a star-algebra equivalence of matrix algebras, hence preserves the operator norm. Therefore
Applying this equality to \(A-B\) gives distance preservation.
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.
Extend the isometric involution continuously. The algebraic identities and the C-star identity hold on the dense canonical image and hence on the completion.
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.
The involution is isometric on \(\mathcal A_{\mathrm{loc}}\), hence extends continuously to the completion. The identities \((A+B)^*=A^*+B^*\), \((AB)^*=B^*A^*\), and \((A^*)^*=A\) hold on the dense subalgebra \(j(\mathcal A_{\mathrm{loc}})\) and therefore on \(\mathcal A\). The conjugate-linear identity \((cA)^*=\overline cA^*\) extends by the same argument. Density also extends
so the completed algebra satisfies the C-star identity.
For every finite region \(\Lambda \), define
For every finite region \(\Lambda \), the map \(j_\Lambda \) is an injective isometry. If \(\Lambda \subseteq \Gamma \), then
The algebraic inclusion \(\iota _\Lambda \) and the completion map \(j\) both preserve the norm. Their composite is therefore an isometry and hence injective. Compatibility follows from \(\iota _\Gamma \circ \iota _{\Lambda ,\Gamma }=\iota _\Lambda \).
The canonical star-algebra homomorphism from a normed star algebra into its completion has dense range.
This is the defining density property of the completion.
Algebraic local observables are dense in the quasi-local algebra. Equivalently,
Density of \(j(\mathcal A_{\mathrm{loc}})\) is the defining density property of the completion. Every algebraic local observable has a representative in some finite region, so
Taking closures proves the second equality. This theorem concerns the union over all finite regions and makes no claim about the image of a fixed region.
The appendix introduces the quasi-local algebra and then invokes lattice translation in the definition of a QCA and in its covariance condition [ CPGSV17 , lines 2292–2306 ] . The following completion result and translation laws make this background structure explicit; they are not stated as separate results in the appendix.
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
Apply the completion functor to \(f\) and \(f^{-1}\). On the dense canonical images their composites satisfy
Continuity gives the two inverse identities on the completions. Multiplication, addition, scalar multiplication, and the star operation commute with the extension because the corresponding identities hold on the canonical images and both sides are continuous.
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
Algebraic-local translation and its inverse translation by \(-a\) are operator-norm isometries, hence continuous. Extend this equivalence to the norm completion. The extension identity gives
Substituting \(A=\iota _\Lambda (A_\Lambda )\) and using the finite-region translation formula gives the second identity.
For \(d{\gt}0\), the quasi-local automorphisms form an additive action of \(\mathbb Z\):
On the canonical image of the algebraic-local algebra,
Continuity extends these identities to the completion. Taking \((a,b)=(a,-a)\) and \((-a,a)\) identifies the inverse of \(\tau _a\) with \(\tau _{-a}\).
For \(d{\gt}0\), a star-algebra automorphism \(\omega \) of the quasi-local algebra \(\mathcal A\) is translation covariant if
for every \(a\in \mathbb Z\). This is the covariance condition in the definition of a one-dimensional QCA [ CPGSV17 , line 2298 ] .
For \(d{\gt}0\), a star-algebra automorphism \(\omega \) of \(\mathcal A\) is translation covariant if and only if
for every \(a\in \mathbb Z\) and every \(A\in \mathcal A\).
Two automorphisms are equal if and only if they agree on every observable. Apply this observation to the two composites in the definition.
For \(d{\gt}0\), a star-algebra automorphism \(\omega \) of \(\mathcal A\) commutes with every integer translation if and only if it commutes with unit translation:
The forward implication follows by taking the translation step to be \(1\). Conversely, use two-sided integer induction. The zero case follows from \(\tau _0=\operatorname {id}\). For the successor step, the additive action law gives \(\tau _{i+1}=\tau _1\circ \tau _i\), and
For the predecessor step, \(\tau _{-1}=\tau _1^{-1}\) transfers commutation to \(\tau _{-1}\), while \(\tau _{i-1}=\tau _{-1}\circ \tau _i\), so
For \(d{\gt}0\), the identity automorphism of \(\mathcal A\) is translation covariant. If \(\omega \) and \(\eta \) are translation covariant, then so is the composite
which first applies \(\omega \) and then \(\eta \).
The identity commutes pointwise with every translation. For the composite, translation covariance gives
For \(d{\gt}0\), every quasi-local translation preserves the operator norm and distance:
A star-algebra equivalence between complex C-star algebras preserves the C-star norm. Since \(\tau _a\) is linear,
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 \(A\in \mathcal A_\Lambda \).
These assertions follow directly from membership in the range of \(j_\Lambda \).
Let \(d{\gt}0\). If \(x\in \mathcal A\) is supported in \(\Lambda \) and \(\Lambda \subseteq \Gamma \), then \(x\) is supported in \(\Gamma \).
Write \(x=j_\Lambda (A)\). Enlarge \(A\) to \(\mathcal A_\Gamma \) by tensoring with the identity on \(\Gamma \setminus \Lambda \). Compatibility of the canonical embeddings identifies the image of this enlargement with \(x\).
Let \(d{\gt}0\). If \(x,y\in \mathcal A\) are supported in finite regions \(\Lambda \) and \(\Gamma \) with \(\Lambda \cap \Gamma =\varnothing \), then
Equivalently, for \(A\in \mathcal A_\Lambda \) and \(B\in \mathcal A_\Gamma \),
This is finite-support infrastructure for the quasi-local algebra in [ CPGSV17 , lines 2292–2300 ] ; it is not stated there as a separate theorem.
The completion map is a star-algebra homomorphism. Applying it to the algebraic-local identity gives
and finite-region support provides representatives \(x=j_\Lambda (A)\) and \(y=j_\Gamma (B)\).
The following finite-dimensional averaging fact is standard and is not a claim from [ CPGSV17 ] . It will be used to characterize exact local support by commutation with observables on the complementary tensor factor.
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 every \(M\in \operatorname {Mat}_{S\times C}(\mathbb C)\),
Split \(M\) into \(C\times C\) blocks indexed by \(S\). The Weyl average sends each block \(B\) to \((\operatorname {tr}B/n)\mathbf1_C\), so its block trace is the corresponding entry of \(\operatorname {tr}_C(M)\).
Let \(S\) be a nonempty finite set. The average is contractive in the operator norm. Moreover, if every complementary Weyl conjugation fixes \(X\), then
Unitary conjugation preserves the operator norm, and the coefficients of the finite average sum to one. Apply this contraction to \(M-X\).
Let \(d{\gt}0\), let \(\Lambda \subset \mathbb Z\) be finite, and let \(x\in \mathcal A\). Then
This is standard infrastructure for the UHF algebra used in TNLean. It is not a theorem stated in [ CPGSV17 ] or by Schumacher–Werner, quant-ph/0405174.
The forward implication follows from commutation of disjoint tensor factors. Conversely, approximate \(x\) by \(A\in \mathcal A_\Delta \), where \(\Lambda \subseteq \Delta \). Split \(\Delta =\Lambda \sqcup (\Delta \setminus \Lambda )\) and average \(A\) over Weyl unitaries on the second factor. Every such conjugation fixes \(x\) by the commutation hypothesis, so unitary invariance and convexity give
The Weyl identity writes the average as \(B\otimes \mathbf1\) for some \(B\in \mathcal A_\Lambda \). Thus elements of \(\mathcal A_\Lambda \) approximate \(x\) arbitrarily well. The embedding of \(\mathcal A_\Lambda \) is isometric, hence has closed range, and therefore \(x\in \mathcal A_\Lambda \).
The following theorem gives the scalar center of the homogeneous UHF algebra. GNVW invoke this conclusion for the quasi-local algebra at [ GNVW12 , lines 1276–1282 ] , specifically line 1279, but do not prove it there. The homogeneous construction in [ CPGSV17 , lines 2292–2298 ] likewise defines the norm completion without stating this conclusion separately. Schumacher–Werner give the corresponding homogeneous norm-completed construction in [ SW04 , lines 263–285 ] , again without a separate scalar-center theorem.
Let \(d{\gt}0\) and let \(z\in \mathcal A\). If
then there is a unique \(c\in \mathbb C\) such that
Consequently,
This statement concerns the fixed on-site dimension \(d\); it does not assert the site-dependent extension used in GNVW.
Apply Theorem 19.10.1.70 with \(\Lambda =\varnothing \). The commutation hypothesis gives \(z\in j_\varnothing (\mathcal A_\varnothing )\), so there is an \(A\in \mathcal A_\varnothing \) with \(z=j_\varnothing (A)\). Let \(e\colon \mathbb C\xrightarrow {\ \sim \ }\mathcal A_\varnothing \) be the canonical isomorphism and write \(A=e(c)\). If also \(z=j_\varnothing (e(c'))\), then injectivity of \(j_\varnothing \) and of \(e\) give
Thus \(c\) is the unique scalar for which \(z=c\mathbf1\).
Write \(Z(\mathcal A)\) for the center. Since every \(j_\Gamma (B)\) belongs to \(\mathcal A\), the two inclusions are
Hence \(Z(\mathcal A)=\mathbb C\mathbf1\).
The appendix requires that “there exists a finite subset \(\mathcal N\subset \mathbb Z\)” such that, for every finite \(\Lambda \subset \mathbb Z\),
This is the propagation condition stated at line 2298 of [ CPGSV17 ] . The definitions below isolate this forward inclusion. They do not impose translation covariance or a propagation bound for \(\omega ^{-1}\).
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 \(x\in \mathcal A\),
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 \(\omega \colon \mathcal A\to \mathcal A\) be a star-algebra automorphism, and let \(\mathcal N\subset \mathbb Z\) be finite. The following conditions are equivalent:
Thus condition (c) is precisely the range-inclusion formula from line 2298 of [ CPGSV17 ] , expressed inside the quasi-local algebra, and condition (d) is its finite-region witness form.
Every observable supported in \(\Lambda \) has the form \(j_\Lambda (A)\), so (a) and (b) are equivalent. Replacing support by membership in the range of the canonical embedding gives (c). Finally, membership in \(j_{\Lambda +\mathcal N}(\mathcal A_{\Lambda +\mathcal N})\) is equivalent to the existence of a representative \(B\) in that finite-region algebra, which gives (d).
Let \(d{\gt}0\) and let \(\omega \colon \mathcal A\to \mathcal A\) be a star-algebra automorphism. If \(\mathcal N\subseteq \mathcal M\) and \(\omega \) propagates within \(\mathcal N\), then \(\omega \) propagates within \(\mathcal M\). More explicitly, for every finite region \(\Lambda \),
Consequently, if \(\omega \) has finite forward propagation, then for every prescribed finite set \(\mathcal N\subset \mathbb Z\) there is a finite set \(\mathcal M\subset \mathbb Z\) such that
This is the neighborhood-enlargement consequence of the propagation condition in Appendix line 2298 of [ CPGSV17 ] .
Monotonicity of the sumset in its second argument gives \(\Lambda +\mathcal N\subseteq \Lambda +\mathcal M\). Monotonicity of quasi-local support then enlarges the support of \(\omega (x)\) from \(\Lambda +\mathcal N\) to \(\Lambda +\mathcal M\). For the final assertion, choose a finite propagation neighborhood \(\mathcal K\) and take \(\mathcal M=\mathcal K\cup \mathcal N\).
Let \(d{\gt}0\), let \(\omega ,\eta \colon \mathcal A\to \mathcal A\) be star-algebra automorphisms, and let \(\mathcal N,\mathcal M\subset \mathbb Z\) be finite. Suppose that \(\omega \) propagates within \(\mathcal N\) and \(\eta \) propagates within \(\mathcal M\). The composite \(\eta \circ \omega \) first applies \(\omega \) and then \(\eta \), so
Hence \(\eta \circ \omega \) propagates within \(\mathcal N+\mathcal M\). In particular, if \(\omega \) and \(\eta \) have finite forward propagation, then \(\eta \circ \omega \) has finite forward propagation. This is the composition closure of the finite-neighborhood condition in Appendix line 2298 of [ CPGSV17 ] .
Apply the two propagation bounds successively and use associativity of finite region sumsets. For the existential assertion, choose one finite propagation neighborhood for each automorphism and use their sumset.
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:
Let \(R\) be the maximum of \(|n|\) over \(n\in \mathcal N\), with value \(0\) when \(\mathcal N\) is empty. Then \(-R\leq n\leq R\) for every \(n\in \mathcal N\), which gives \(\mathcal N\subseteq [-R,R]\cap \mathbb Z\). Enlargement of propagation neighborhoods gives the fixed-neighborhood assertion. Finally, choose a finite propagation neighborhood and apply this enlargement to its symmetric interval.
Let \(d{\gt}0\), and let \(\omega \) be a star-algebra automorphism of \(\mathcal A\). A finite set \(\mathcal N\subset \mathbb Z\) is a two-sided propagation neighborhood when
and
for every finite \(\Lambda \subset \mathbb Z\). The automorphism has two-sided finite propagation when \(\omega \) and \(\omega ^{-1}\) each have a finite forward propagation neighborhood; the two neighborhoods need not initially agree.
These are derived convenience predicates. The definition in [ CPGSV17 , line 2298 ] and Schumacher–Werner, quant-ph/0405174, Definition 1, both define locality by the forward inclusion for \(\omega \) alone. In particular, these predicates do not assert that forward locality implies locality of the inverse.
For a finite set \(\mathcal N\subset \mathbb Z\), write
Reflection fixes every symmetric interval:
If \(\mathcal N\subseteq \mathcal M\), then every common two-sided bound by \(\mathcal N\) is also a common bound by \(\mathcal M\). More generally, if \(\omega \) propagates within \(\mathcal N\) and \(\omega ^{-1}\) propagates within \(-\mathcal N\), then both propagate within
Finally, any two-sided finite-propagation pair admits a common finite neighborhood and therefore a common symmetric-interval neighborhood. Thus there is an \(R\in \mathbb N\) such that
and
for every finite \(\Lambda \subset \mathbb Z\).
Negation reverses the two endpoint inequalities, so it preserves \([-R,R]\cap \mathbb Z\). Enlarge each supplied neighborhood using monotonicity. For reflected bounds, both inclusions into \(\mathcal N\cup (-\mathcal N)\) are immediate. For arbitrary finite forward and inverse neighborhoods, first take their union and then enlarge that common neighborhood to a symmetric interval.
Let \(\Gamma ,\Lambda ,\mathcal N\subset \mathbb Z\) be finite. Then
If \(g+n=\lambda \) for \(g\in \Gamma \), \(n\in \mathcal N\), and \(\lambda \in \Lambda \), then \(g=\lambda +(-n)\in \Lambda +(-\mathcal N)\), proving the forward implication. Conversely, if \(g\in \Gamma \) and \(g=\lambda +(-n)\) with \(\lambda \in \Lambda \) and \(n\in \mathcal N\), then \(\lambda =g+n\in \Gamma +\mathcal N\).
Let \(d{\gt}0\), let \(\omega \) be a star-algebra automorphism of \(\mathcal A\), and let \(\mathcal N\subset \mathbb Z\) be finite. If \(\omega \) propagates within \(\mathcal N\), then \(\omega ^{-1}\) propagates within \(-\mathcal N\). Therefore finite forward propagation of \(\omega \) implies finite forward propagation of \(\omega ^{-1}\). Moreover, if \(\omega \) is translation covariant, then so is \(\omega ^{-1}\).
This is a consequence of the forward-only locality definition, not part of that definition. Schumacher–Werner, quant-ph/0405174, derive inverse locality through Lemma 5, Theorem 6, and Corollary 7 using the generalized Margolus structure. The proof here instead uses TNLean’s exact outside-commutant infrastructure.
Let \(x\in \mathcal A_\Lambda \) and let \(y\in \mathcal A_\Gamma \), where \(\Gamma \cap (\Lambda +(-\mathcal N))=\varnothing \). Forward locality gives \(\omega (y)\in \mathcal A_{\Gamma +\mathcal N}\). The regions \(\Gamma +\mathcal N\) and \(\Lambda \) are disjoint, so \(x\) commutes with \(\omega (y)\). Applying \(\omega ^{-1}\) shows that \(\omega ^{-1}(x)\) commutes with \(y\). The exact outside-commutant characterization now gives
Choosing the reflected finite neighborhood proves the existential statement. For each lattice translation \(\tau _a\), multiplying the covariance identity on the left and right by \(\omega ^{-1}\) gives
Let \(d{\gt}0\) and let \(\omega \) be a star-algebra automorphism of \(\mathcal A\). Then
More precisely, a forward bound by a finite set \(\mathcal N\) gives an inverse bound by \(-\mathcal N\), so both maps propagate within the enlarged common neighborhood
This statement remains valid when \(\mathcal N=\varnothing \). For an arbitrary finite forward-propagation witness, there is therefore a common finite neighborhood for \(\omega \) and \(\omega ^{-1}\), and it may be enlarged to a symmetric interval \([-R,R]\cap \mathbb Z\).
This equivalence is a derived convenience theorem. It does not change the forward-only locality definition at line 2298 of [ CPGSV17 ] or in Schumacher–Werner, Definition 1.
Derived inverse locality sends a forward bound by \(\mathcal N\) to an inverse bound by \(-\mathcal N\). Enlarging both bounds to \(\mathcal N\cup (-\mathcal N)\) gives the fixed-neighborhood statement. At the existential level, forward finite propagation gives finite propagation of the inverse, hence the two-sided predicate. The converse is its first conjunct. Finally, apply the common-neighborhood and symmetric-interval enlargement results to the resulting two-sided pair.
Let \(d{\gt}0\) and let \(\omega \colon \mathcal A\to \mathcal A\) be a star-algebra automorphism. The automorphism \(\omega \) is a one-dimensional quantum cellular automaton if it is translation covariant and has finite forward propagation. Thus there exists a finite set \(\mathcal N\subset \mathbb Z\) such that, for every finite \(\Lambda \subset \mathbb Z\),
This is the definition at line 2298 of [ CPGSV17 ] . It imposes only the displayed forward propagation condition and makes no locality claim for \(\omega ^{-1}\).
For \(d{\gt}0\), a star-algebra automorphism \(\omega \) of \(\mathcal A\) is a quantum cellular automaton if and only if it is translation covariant and has finite forward propagation. These two properties therefore construct an automaton, and every quantum cellular automaton satisfies each of them.
This is the conjunction defining a quantum cellular automaton, together with its introduction and elimination rules.
For \(d{\gt}0\), the identity automorphism of \(\mathcal A\) is a quantum cellular automaton.
The identity is translation covariant. For finite forward propagation, choose the neighborhood \(\{ 0\} \). An observable supported in \(\Lambda \) remains unchanged, and \(\Lambda +\{ 0\} =\Lambda \).
Let \(d{\gt}0\). If \(\omega \) and \(\eta \) are quantum cellular automata on \(\mathcal A\), then \(\eta \circ \omega \) is a quantum cellular automaton. This composite first applies \(\omega \) and then \(\eta \). No propagation condition for either inverse is required.
Both defining properties are closed under composition in the stated order. Translation covariance of \(\omega \) and \(\eta \) gives translation covariance of \(\eta \circ \omega \). If \(\mathcal N\) and \(\mathcal M\) are forward propagation neighborhoods for \(\omega \) and \(\eta \), respectively, then \(\mathcal N+\mathcal M\) is a forward propagation neighborhood for \(\eta \circ \omega \). Hence the composite is a quantum cellular automaton.
Let \(d{\gt}0\). If \(\omega \) is a quantum cellular automaton on \(\mathcal A\), then \(\omega ^{-1}\) is also a quantum cellular automaton.
Translation covariance and finite forward propagation are both preserved by inversion. These are the two defining properties of a quantum cellular automaton.
Let \(L{\gt}0\) be a block length and let \(\Lambda \subset \mathbb Z\) be finite. The expansion of \(\Lambda \) is
Every \(z\in \widehat\Lambda _L\) has unique block coordinates \((x,r)\) with \(x\in \Lambda \) and \(0\leq r{\lt}L\). Increasing \(r\) gives the consecutive-site order within the block. This is the finite-region site grouping used in [ CPGSV17 , lines 2308 and 2313–2320 ] ; it does not assert the QCA-to-MPU converse stated at line 2308.
An integer \(z\) belongs to \(\widehat\Lambda _L\) exactly when its quotient by \(L\) belongs to \(\Lambda \). The inverse block-coordinate map sends \((x,r)\) to \(Lx+r\), and \(\Lambda \subseteq \Gamma \) implies \(\widehat\Lambda _L\subseteq \widehat\Gamma _L\).
Apply Euclidean division by the positive integer \(L\). Its quotient and residue are unique, the residue lies in \(\{ 0,\ldots ,L-1\} \), and taking the inverse coordinates gives \(Lx+r\). Monotonicity follows by retaining the same quotient and residue.
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 \(Lx+r\).
Evaluation at \(Lx+r\) reads the \(r\)th letter of the blocked spin at \(x\). Conversely, encoding collects these letters in increasing residue order. At the configuration level, decoding commutes with restriction from \(\Gamma \) to \(\Lambda \subseteq \Gamma \).
The two evaluation formulas are the inverse laws for the ordered word encoding. Restriction preserves the quotient block coordinate and residue, so both sides have the same value at every expanded site.
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.
Matrix entries are transported by applying the inverse configuration equivalence to both indices. The blocking equivalence preserves the operator norm and is an isometry.
The entry formula is matrix reindexing. A star-algebra equivalence between finite matrix algebras preserves the C-star norm and hence distances.
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 \(L{\gt}0\). 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:
The quotient coordinate of each \(z\in \Delta \) belongs to the image defining the hull, which proves containment after expansion. Monotonicity follows by retaining the same witness \(z\) in the larger region. Finally, every expanded site has quotient in \(\Lambda \), while for each \(x\in \Lambda \) the site \(Lx\) belongs to \(\widehat\Lambda _L\) and has quotient \(x\). Hence \(\operatorname {Hull}_L(\widehat\Lambda _L)=\Lambda \).
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 configurations \(x,y\) on \(\widehat\Gamma _L\), let \(\widetilde x\) and \(\widetilde y\) be their block encodings on \(\Gamma \). The two matrix entries in the commuting square are
Restriction commutes with block encoding, and decoding identifies \(\Gamma \setminus \Lambda \) with the original sites in \(\widehat\Gamma _L\setminus \widehat\Lambda _L\). Hence the two indicator conditions are equivalent and the entries agree.
Let \(L{\gt}0\). The compatible finite blocking maps induce a star-algebra homomorphism
In the reverse direction there is a star-algebra homomorphism
A representative on an arbitrary original region \(\Delta \) is first included into \(\widehat{\operatorname {Hull}_L(\Delta )}_L\) and then finite blocking is inverted. Thus the reverse construction does not require \(\Delta \) itself to be an expanded region.
For \(\Lambda \subseteq \Gamma \), the finite commuting square gives
so the forward finite-region maps form a compatible family. For the reverse family, block-hull monotonicity sends \(\Delta \subseteq \Theta \) to \(\operatorname {Hull}_L(\Delta )\subseteq \operatorname {Hull}_L(\Theta )\). Transitivity of the canonical inclusions and the inverse commuting square then give
The universal property of the algebraic direct limit therefore induces both star-algebra homomorphisms.
Let \(L{\gt}0\). The forward map applies finite blocking to a representative on \(\Lambda \) and regards the result as supported on \(\widehat\Lambda _L\). The reverse map enlarges a representative on \(\Delta \) to the expansion of its block hull, applies the inverse finite blocking equivalence, and regards the result as supported on \(\operatorname {Hull}_L(\Delta )\). Their composites have the orientations
The first identity holds for every on-site dimension \(d\); the second also assumes \(d{\gt}0\).
These are the evaluation formulas for the two compatible-family lifts. Forward compatibility is the commuting square. Reverse compatibility follows by monotonicity of block hulls, composition of local inclusions, and the same square applied in the inverse direction. Forward blocking after the reverse map reduces to the canonical inclusion of \(\Delta \) into its expanded block hull. The other composite is the identity by injectivity of the forward map.
For \(d{\gt}0\) and \(L{\gt}0\), forward blocking gives a star-algebra equivalence
This formalizes only the algebraic-local change of site grouping used at lines 2308 and 2313–2320 of [ CPGSV17 ] . It does not assert the QCA-to-MPU converse at line 2308, extend blocking to the quasi-local completion, intertwine translations, provide a propagation estimate or a circuit representation, or transport a quantum cellular automaton.
The identity \(\mathcal B_L\mathcal U_L=\operatorname {id}\) makes \(\mathcal B_L\) surjective. For injectivity, represent two blocked observables on finite regions \(\Lambda \) and \(\Gamma \), enlarge both to \(\Omega =\Lambda \cup \Gamma \), and suppose their images under \(\mathcal B_L\) agree. Injectivity of the canonical map from the finite algebra on \(\widehat\Omega _L\), followed by injectivity of \(\mathcal B_{L,\Omega }\) and the finite commuting square, gives equality of the two representatives in the algebra on \(\Omega \). Thus \(\mathcal B_L\) is bijective, with inverse \(\mathcal U_L\), and defines the stated star-algebra equivalence.
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.
The first two formulas are the evaluation identities for the forward and reverse maps. For an expanded region, the block-hull identity removes the enlargement and leaves the inverse finite blocking equivalence. On a finite representative, forward norm preservation reduces to norm preservation by finite blocking and by the canonical local inclusion; distance preservation follows by applying the norm identity to differences. The inverse isometry follows from the forward isometry and the right-inverse identity, rather than repeating the norm calculation.
For \(d{\gt}0\) and \(L{\gt}0\), site blocking extends uniquely by continuity to a star-algebra equivalence
This is only the completion of the change of site grouping used at [ CPGSV17 , lines 2308 and 2313–2320 ] . This slice does not define scaled translations or carry-aware neighborhoods, prove propagation estimates, construct blocked automorphisms, transport the property of being a quantum cellular automaton, realize a circuit, or reconstruct an MPU from a quantum cellular automaton.
The algebraic blocking equivalence and its inverse are isometries, hence continuous. Extending both maps to the operator-norm completions gives mutually inverse star-algebra homomorphisms.
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,
The first identity is the defining completion map, and the two dense formulas are its canonical-image formulas for the algebraic equivalence and its inverse. Composing them with the finite-region canonical maps gives the two finite formulas. The exact expanded-region formula uses \(\operatorname {Hull}_L(\widehat\Lambda _L)=\Lambda \). A star-algebra equivalence of C-star algebras preserves the norm; applying this to \(X-Y\) gives distance preservation.
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:
Choose a finite-region representative and apply the corresponding forward or inverse finite evaluation formula. For the reverse implication in the exact statement, apply inverse blocking and simplify the inverse composite and the block hull of the expanded region.
Let \(L{\gt}0\), let \(a\in \mathbb Z\), and let \(\Lambda \subset \mathbb Z\) be a finite blocked region. Then
This is the translation rule for the site grouping used in [ CPGSV17 , lines 2308 and 2313–2320 ] .
Write each original site uniquely as \(xL+r\), with \(0\leq r{\lt}L\). Translation of the blocked coordinate by \(a\) sends this site to \((x+a)L+r=xL+r+aL\). The argument is unchanged when \(x\) or \(a\) is negative.
Let \(L{\gt}0\) and let \(\mathcal N\subset \mathbb Z\) be a finite original-lattice neighborhood. Write \(\widehat{\{ 0\} }_L=\{ 0,\ldots ,L-1\} \) for the sites in the reference block. Define the induced blocked-lattice neighborhood by
The full reference block is included because the block reached after a displacement can depend on the starting residue. This is the finite-region geometry implicit in the grouping of sites in [ CPGSV17 , lines 2308 and 2313–2320 ] .
Let \(L{\gt}0\), let \(\mathcal N\subset \mathbb Z\) be finite, and let \(b\in \mathbb Z\). Then
The quotient is the Euclidean quotient by the positive integer \(L\). Consequently these formulas include negative displacements and both positive and negative block carries. In general, \(\operatorname {Hull}_L(\mathcal N)\) alone omits some carries caused by the starting residue. This makes explicit the boundary geometry implicit in [ CPGSV17 , lines 2313–2320 ] .
An element of \(\widehat{\{ 0\} }_L+\mathcal N\) has the form \(r+n\) with \(0\leq r{\lt}L\) and \(n\in \mathcal N\). Taking its block coordinate gives the first equivalence. Euclidean division writes \(r+n=bL+s\) with the unique remainder \(0\leq s{\lt}L\), which gives the second equivalence without any sign restriction on \(n\).
Let \(L{\gt}0\). For every finite blocked region \(\Lambda \) and every finite original-lattice neighborhood \(\mathcal N\),
This is the exact finite-region carry identity associated with the site grouping in [ CPGSV17 , lines 2313–2320 ] .
Write an expanded site as \(xL+r\), where \(x\in \Lambda \) and \(0\leq r{\lt}L\). For \(n\in \mathcal N\), Euclidean division gives
The second summand ranges over exactly \(\mathcal C_L(\mathcal N)\) by Lemma 19.10.1.106. This proves both inclusions, including for negative \(x\) and \(n\).
Let \(L{\gt}0\). For every finite blocked region \(\Lambda \) and every finite original-lattice neighborhood \(\mathcal N\),
Every finite original region is contained in the expansion of its block hull. Apply this containment to \(\widehat\Lambda _L+\mathcal N\) and use Lemma 19.10.1.107 to identify that hull.
Let \(d,L{\gt}0\). If \(\mathcal B_L\) denotes the algebraic-local blocking equivalence, then for every \(a\in \mathbb Z\),
This is the translation rule induced by the blocking of sites used in [ CPGSV17 , lines 2308 and 2313–2320 ] .
On a representative supported in a finite blocked region \(\Lambda \), both sides apply the same matrix reindexing. Their supports agree because
Equality on all finite-region representatives proves the identity on the algebraic direct limit.
Let \(d,L{\gt}0\). If \(\overline{\mathcal B}_L\) denotes the quasi-local blocking equivalence, then for every \(a\in \mathbb Z\),
This is the completion of the translation rule induced by the site grouping in [ CPGSV17 , lines 2308 and 2313–2320 ] .
The two sides are continuous maps on the quasi-local algebra. On the dense algebraic-local subalgebra their values agree by Lemma 19.10.1.109. They therefore agree on the completion.
Line 2308 of [ CPGSV17 ] invokes a converse representation theorem: after blocking finitely many sites, the restriction of every one-dimensional QCA to the local algebra is represented by a depth-two circuit of nearest-neighbor unitaries, and hence by an MPU in standard form. The results below concern the preceding change of grouping. A given QCA remains a QCA under any prescribed finite blocking. This one-way transport alone does not produce the circuit or MPU representation asserted by the converse theorem.
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
For every \(X\in \mathcal A^{(d^L)}\),
This is the evaluation formula for the defining conjugation.
The inverse of the blocked automorphism is the blocked inverse:
Evaluate both sides on a blocked-chain observable. The blocking equivalence and its inverse cancel on the two sides of \(\omega ^{-1}\).
Let \(\mathcal N\subset \mathbb Z\) be finite. If \(\omega \) propagates within \(\mathcal N\), then \(\omega ^{[L]}\) propagates within the carry-aware blocked neighborhood \(\mathcal C_L(\mathcal N)\). Explicitly, for every finite blocked region \(\Lambda \) and every \(X\in \mathcal A^{(d^L)}\) supported in \(\Lambda \), the observable \(\omega ^{[L]}(X)\) is supported in \(\Lambda +\mathcal C_L(\mathcal N)\).
Forward blocking sends support in \(\Lambda \) to support in \(\widehat\Lambda _L\). Apply the propagation bound for \(\omega \), then apply inverse blocking. The resulting support is
by the exact carry-aware block-hull identity.
If \(\omega \) has finite forward propagation, then \(\omega ^{[L]}\) has finite forward propagation.
Choose a finite propagation neighborhood \(\mathcal N\) for \(\omega \). Lemma 19.10.1.114 gives the finite neighborhood \(\mathcal C_L(\mathcal N)\) for \(\omega ^{[L]}\).
The blocked automorphism is translation covariant if and only if the original automorphism commutes with translation by one whole block:
Thus blocked unit-translation covariance records commutation of \(\omega \) with translation by \(L\), not necessarily commutation with original unit translation. For \(L\neq 1\), the latter condition can be strictly stronger.
By Theorem 19.10.1.61, translation covariance on the blocked chain is equivalent to commutation with blocked unit translation. Under \(\overline{\mathcal B}_L\), blocked unit translation corresponds to original translation by \(L\). Conjugating the commutation identity by the blocking equivalence proves both implications.
If \(\omega \) is translation covariant, then \(\omega ^{[L]}\) is translation covariant.
Translation covariance of \(\omega \) gives commutation with \(\overline\tau ^{(d)}_L\). Apply Theorem 19.10.1.116.
Let \(d,L{\gt}0\). If \(\omega \) is a one-dimensional QCA on \(\mathcal A^{(d)}\), then
is a one-dimensional QCA on \(\mathcal A^{(d^L)}\).
The two defining properties of a one-dimensional QCA are finite forward propagation and translation covariance. They are preserved by Lemma 19.10.1.115 and Lemma 19.10.1.117, respectively.
Line 2308 of [ CPGSV17 ] invokes the external Schumacher–Werner structure theorem to obtain support algebras, a depth-two nearest-neighbor circuit, and an MPU in standard form. The next statements establish only the elementary normalization of a finite propagation neighborhood to \([-1,1]\cap \mathbb Z\) by regrouping sites. They do not prove any part of that external structure theorem beyond this change of grouping.
Let \(L{\gt}0\) and let \(\mathcal N\subset \mathbb Z\) be finite. If \(|n|{\lt}L\) for every \(n\in \mathcal N\), then
If \(\mathcal N=\varnothing \), membership in \(\mathcal C_L(\mathcal N)\) is impossible. In general, a point \(b\in \mathcal C_L(\mathcal N)\) supplies residues \(0\leq r,s{\lt}L\) and \(n\in \mathcal N\) such that
Since \(-L{\lt}n{\lt}L\), the equality is incompatible with \(b\leq -2\) and with \(b\geq 2\). Thus \(-1\leq b\leq 1\), including for negative displacements.
Let \(L{\gt}0\) and let \(\mathcal N\subset \mathbb Z\) be finite. If
then
Every \(|n|\) is bounded above by the finite supremum, so the strict supremum bound supplies the pointwise hypothesis. When \(\mathcal N=\varnothing \), the supremum is \(0\) and the conclusion is vacuous.
For every finite \(\mathcal N\subset \mathbb Z\), define
Then \(L_{\mathcal N}{\gt}0\) and
With the finite-supremum convention, \(L_{\varnothing }=1\).
The finite supremum is strictly smaller than its successor, which is positive even when \(\mathcal N\) is empty.
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
Blocking first gives propagation within \(\mathcal C_L(\mathcal N)\). The carry bound places this neighborhood inside \([-1,1]\cap \mathbb Z\), and enlargement of propagation neighborhoods gives the claim.
Let \(d{\gt}0\). If \(\omega \) has finite forward propagation, then there exists a positive integer \(L\) such that, 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
Choose a finite propagation neighborhood \(\mathcal N\) and take \(L=1+\sup _{n\in \mathcal N}|n|\). This \(L\) is positive, including when \(\mathcal N\) is empty. Blocking gives propagation within \(\mathcal C_L(\mathcal N)\); the canonical carry inclusion and propagation monotonicity then give the nearest-neighbor bound.
Let \(d{\gt}0\) and let \(\omega \) be a one-dimensional QCA on \(\mathcal A^{(d)}\). There exists a positive integer \(L\) such that \(\omega ^{[L]}\) is a QCA on \(\mathcal A^{(d^L)}\) and, 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
Apply the preceding theorem to the finite propagation of \(\omega \), using the canonical positive block length. At the same length, site blocking preserves both finite propagation and translation covariance, hence the QCA property.
Lines 2292–2298 of [ CPGSV17 ] specify the finite local algebras, their canonical inclusions, translation covariance, and the locality inclusion
The following statements record the finite-dimensional maps determined by this inclusion. They are consequences and coordinate formulations of the cited locality relation, not separately named theorems of the paper. The left/right order below agrees with the adjacent-pair convention in [ GNVW12 ] .
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
The finite-region restriction is injective, and for every \(A\in \mathcal A_\Lambda \) one has
The defining equality follows from the locality inclusion and the injectivity of \(\iota _{\Lambda +\mathcal N}\). If two local observables have the same image under \(\omega _{\Lambda ,\mathcal N}\), their canonical images become equal after applying \(\omega \). Injectivity of \(\omega \) and of \(\iota _\Lambda \) then gives equality of the two local observables.
If \(\Lambda \subseteq \Gamma \), then for every \(A\in \mathcal A_\Lambda \),
Apply the canonical inclusion into the quasi-local algebra to both sides. Both expressions become \(\omega (\iota _\Lambda (A))\); injectivity of the target inclusion gives the equality.
Suppose in addition that \(\omega \) is translation covariant. For \(a\in \mathbb Z\) and \(A\in \mathcal A_\Lambda \),
Here \((\Lambda +a)+\mathcal N=(\Lambda +\mathcal N)+a\).
The left-hand side is \(\omega (\tau _a(\iota _\Lambda (A)))\). Translation covariance moves \(\tau _a\) past \(\omega \), and the finite-region translation formula gives the right-hand side.
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 \(A\in \mathcal A_\Gamma \) and \(B\in \mathcal A_\Delta \),
Evaluate both matrices on pairs of configurations. For the first local inclusion, restriction to \(\Gamma \) gives the matrix coefficient of \(A\), while equality of the complementary \(\Delta \)-configurations gives the matrix coefficient of the identity. This is \(A\otimes \mathbf1\). Interchanging the two regions gives \(\mathbf1\otimes B\).
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.
For \(x\in \mathbb Z\), set
The order \(L_x,P_x\) is the left–right factor order in equation RR2x of [ GNVW12 , lines 1251–1266 ] . The same four-site localization occurs in [ CPGSV17 , lines 2313–2318 ] .
For the neighbourhood \(\mathcal N=\{ -1,0,1\} \),
Substitute (869). Adding \(-1,0,1\) to the two points of \(E_x\) gives
The largest point of \(L_x\) is \(2x\), while the smallest point of \(P_x\) is \(2x+1\), so the two sets are disjoint. Finally, \(L_{x+1}=\{ 2x+1,2x+2\} =P_x\).
Let \(\omega \) be a star-algebra automorphism of the homogeneous quasi-local algebra which propagates within \(\{ -1,0,1\} \). Write the finite image of \(\mathcal A_{E_x}\) in the ordered \(L_x\times P_x\) coordinates as
Define
After the canonical finite-region embeddings, these form one family of star-subalgebras of the quasi-local algebra. Here the matrix embedding is the composition
Under this identification, the canonical local inclusion in [ CPGSV17 , lines 2292–2298 ] is exactly the displayed composition. Formula (874) is equation RR2x in [ GNVW12 , lines 1261–1266 ] . Cirac et al. use the homogeneous even factor in [ CPGSV17 , lines 2322–2328 ] ; the odd factor and unified family are supplied by the GNVW construction.
Scope restriction (homogeneous chain): GNVW allow the cell size \(d(y)\) to depend on \(y\), whereas the present definition fixes one positive size \(d\). The site-dependent statement is not claimed here; see [ con26l ] .
For every \(x\in \mathbb Z\), the finite image satisfies
This is the homogeneous, full-matrix instance used in the two-sided containment at [ GNVW12 , lines 1276–1278 ] ; see [ con26l ] . Cirac et al. state only the weaker inclusion in the left support algebra tensored with the unrestricted right physical factor at [ CPGSV17 , lines 2322–2328 ] .
Apply Theorem 19.10.1.17 to \(S_x\) in the ordered coordinates of (872). Its left support is \(\mathcal R_{2x}\) and its right support is \(\mathcal R_{2x+1}\) by (874), which gives (876).
If \(\widehat{\mathcal R}_y\) denotes the canonical quasi-local copy of \(\mathcal R_y\), then for every \(x\in \mathbb Z\),
More generally, the canonical image of any matrix on a finite region \(\Lambda \) is supported in \(\Lambda \). This is the homogeneous specialization of the localization accompanying equation RR2x in [ GNVW12 , lines 1261–1274 ] ; see [ con26l ] .
An element of \(\widehat{\mathcal R}_{2x}\) is the canonical image of a matrix in \(\mathcal R_{2x}\subseteq \mathcal A_{L_x}\) and is therefore supported in \(L_x\). The same argument places \(\widehat{\mathcal R}_{2x+1}\) in \(\mathcal A_{P_x}\).
Equation (A1) in lines 2300–2306 of [ CPGSV17 ] defines the local action associated with an MPU by eventual finite-chain conjugation. In lines 2300–2306, eventual constancy is used to obtain a norm-preserving star-algebra automorphism of \(\mathcal A_{\mathrm{loc}}\), together with a translation formula and the support bound \(\mathcal N=[-4D^4,4D^4]\cap \mathbb Z\), and then to extend this automorphism to \(\mathcal A\). The results below isolate the direct-limit and completion argument. They begin with compatible forward and inverse finite-local actions and assume a forward norm formula, a support bound, and a unit-translation formula. The inverse norm formula follows from forward isometry and mutual invertibility. The numerical radius above belongs to the MPU-specific stabilization argument: the generic result below neither proves that radius nor constructs the finite-local actions from an MPU. It also does not provide the blocked circuit or MPU standard form discussed at line 2308.
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
Apply the star-algebra universal property to each compatible family.
For every finite region \(\Lambda \) and \(A\in \mathcal A_\Lambda \),
These are the evaluation identities supplied by the universal property.
Compatible finite-local automorphism data determine a star-algebra equivalence
whose forward and inverse maps are \(f\) and \(g\), respectively.
Induct on finite-region representatives in the algebraic direct limit. The two inverse assumptions give
Thus \(f\) and \(g\) are mutually inverse.
For every \(A\in \mathcal A_{\mathrm{loc}}\),
Both identities follow from the defining forward and inverse maps of \(F\).
Assume \(d{\gt}0\). The forward local norm hypothesis is
for every finite region \(\Lambda \) and every \(A\in \mathcal A_\Lambda \).
Under the forward local norm hypothesis, every \(A\in \mathcal A_{\mathrm{loc}}\) satisfies
Induct on finite-region representatives. For \(A=\iota _\Lambda (A_\Lambda )\),
Under the forward local norm hypothesis, for all \(A,B\in \mathcal A_{\mathrm{loc}}\),
Apply the forward norm identity to \(A-B\) to prove that \(F\) is an isometry. Since \(F\circ F^{-1}\) is the identity, the right-inverse law then makes \(F^{-1}\) an isometry as well.
Under the forward local norm hypothesis, every \(A\in \mathcal A_{\mathrm{loc}}\) satisfies
Apply the inverse isometry to the distance from \(A\) to \(0\) and use \(F^{-1}(0)=0\).
Under the forward local norm hypothesis, the algebraic equivalence extends to a star-algebra equivalence
of the quasi-local completion.
The forward norm formula makes \(F\) an isometry, and the inverse law makes \(F^{-1}\) an isometry. Apply the completion extension for star-algebra equivalences to these two continuous maps.
Let \(j\colon \mathcal A_{\mathrm{loc}}\to \mathcal A\) be the canonical completion map. For every \(A\in \mathcal A_{\mathrm{loc}}\),
These are the canonical-image formulas for the continuous extension of a star-algebra equivalence and for its inverse.
For every finite region \(\Lambda \) and \(A\in \mathcal A_\Lambda \),
Substitute \(j_\Lambda =j\circ \iota _\Lambda \) into the dense evaluation formulas and use \(F(\iota _\Lambda (A))=f_\Lambda (A)\) and \(F^{-1}(\iota _\Lambda (A))=g_\Lambda (A)\).
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 \(A\in \mathcal A_\Lambda \).
If the forward family has support bound \(\mathcal N\), then the completed automorphism \(\overline F\) propagates within \(\mathcal N\).
Let \(X=j_\Lambda (A)\) be supported in \(\Lambda \). Choose \(B\in \mathcal A_{\Lambda +\mathcal N}\) such that \(f_\Lambda (A)=\iota _{\Lambda +\mathcal N}(B)\). Then
so the image is supported in \(\Lambda +\mathcal N\).
The forward family commutes with unit translation if, for every finite region \(\Lambda \) and every \(A\in \mathcal A_\Lambda \),
If the finite-local unit-translation formula holds, then every \(A\in \mathcal A_{\mathrm{loc}}\) satisfies
Induct on finite-region representatives and write \(A=\iota _\Lambda (A_\Lambda )\). Then
If the finite-local unit-translation formula holds, then the completed automorphism \(\overline F\) is translation covariant.
By Theorem 19.10.1.61, it suffices to prove that \(\overline F\) commutes with translation by one site. Induct on the completion. On every \(A\in \mathcal A_{\mathrm{loc}}\),
The equality set is closed because both composites are continuous, so the identity extends from the canonical dense subalgebra to the completion.
Let \(d{\gt}0\). Suppose the compatible forward family preserves the local operator norm, has a finite support bound, and satisfies the finite-local unit-translation formula. Then the completed automorphism \(\overline F\) is a one-dimensional QCA.
The support hypothesis supplies a finite forward propagation neighborhood, while the unit-translation formula supplies translation covariance. These are exactly the two defining properties of a one-dimensional QCA.