The left endpoint space is one-dimensional, so there is one virtual path. Its product contains no factors and therefore equals \(1\).

No virtual path can have a coordinate in the empty \(k\)th bond space, so the sum in (1) is empty.

The empty matrix product is the \(D\times D\) identity, whose trace is \(D\).

The coordinatewise inclusions identify the smaller Cartesian product with the subset of the larger product on which every coordinate is below its original bound. Split the larger sum into this subset and its complement.

Encode the site and physical indices as one enlarged physical index and apply the cyclic trace expansion for a closed matrix product.

Expand the left-hand side by (4). Extend the dependent sum in (1) to the common dimension \(D\) by Theorem 23.1.10. Every extended path outside the rectangular bond ranges has a zero matrix entry. For the remaining paths, the one-dimensional endpoint spaces identify the last virtual coordinate with the first, and the products agree term by term.

Apply positive-length coefficient preservation and then the vanishing of the open-chain contraction.

Write \(D_i=\operatorname {diag}(A_0^i,\ldots ,A_{N-1}^i)\) and omit the scalar normalization temporarily. Commuting the cyclic permutation through a block diagonal gives

Repeated use of (8), followed by \(P^N=I\), yields

Taking the trace of (9) sums the traces of its diagonal blocks. Finally, \((N^{-1/N})^N=N^{-1}\) gives (7).

The cyclic rotation of the padded square chain satisfies

where \((\tau _s\sigma )_j=\sigma _{j+s}\). To prove (10), expand both square-chain traces as cyclic virtual-path sums and reindex the virtual path by the same rotation. The inverse shift appears when the rotated matrix product is restored to the distinguished open cut. Translation invariance makes every term in the cyclic average equal to \(c_A(\sigma )\), so the factor \(1/N\) cancels the sum of \(N\) identical terms.

Reflexivity uses the identity gauge on every bond. Symmetry inverts each bond gauge, and transitivity multiplies the gauges pointwise along the chain.

A square matrix with a one-sided inverse is invertible.

The coefficients satisfy \(\mu _0=\lambda _0 x\) and the recurrence \(\mu _{j+1}=x(\lambda _{j+1}+\mu _j)\), and (15) states \(\mu _{n-1}=1\). Applying \(T\) termwise and using the chain relation,

with the convention \(\mu _{-1}=0\), where the last equality substitutes (14). By the recurrence each coefficient equals \(\mu _j/x\), so the right-hand side is \(\tfrac 1x S(Y)\). Since \(\mu _{n-1}=1\) and \(Y_0,\ldots ,Y_{n-1}\) are linearly independent, \(Y\neq 0\).

Evaluating both sides of (17) at a pair of doubled indices reduces the identity factor to a Kronecker delta, which collapses the double sum to the slice products \(X_{ab}C\) and \(BX_{ab}\). For the consequence, a nonzero \(W\) has a nonzero slice.

Both solutions express \(B_i\otimes \mathbb {1}\) and \(C_i\otimes \mathbb {1}\) through the same window matrix at site \(j+1\); substituting one expression into the other and cancelling the paired gauges with (12) gives (19). Invertibility follows from Lemma 23.2.2.

The canonical representation solves the system by the freedom in the open-boundary representation. For the converse, assume that the solution tensor \(B\) represents the given length-\(N\) state. Comparing its open-boundary representation with the canonical one supplies intertwiners at all \(D^4+1\) cuts around the \(D^4\)-site window, including both endpoint cuts, together with the \(D^4\) adjacent relations [ 132 ] . Linear dependence in \(M_{D^2}(\mathbb {C})\) then provides the hypothesis of Lemma 23.2.3. The internal construction in Lemma 23.2.5 gives only \(D^4-1\) intertwiners and \(D^4-2\) relations, so it does not force this dependence for an arbitrary algebraic solution.

Once the full cut chain is supplied, the combination lemma and the slice extraction of Lemma 23.2.4 give a nonzero matrix \(R\) and a scalar \(x\neq 0\) with \(RB_i=\tfrac 1x A_iR\). The two normalizations (13), the invariance of the positive diagonal fixed point, and the uniqueness of the transfer-map fixed point under injectivity give \(|x|=1\) and \(RR^{\dagger }=c\mathbb {1}\) for some \(c{\gt}0\). Thus \(U=c^{-1/2}R\) is unitary and \(A_i=xUB_iU^{\dagger }\) for every \(i\). Since \(A\) and \(B\) represent the same nonzero length-\(N\) state, their coefficients satisfy \(V^{(N)}(A)=x^NV^{(N)}(B)=V^{(N)}(B)\), and hence \(x^N=1\). The tensor \(\widehat B_i=xB_i\) therefore represents the same state and satisfies \(A_i=U\widehat B_iU^{\dagger }\).

A vanishing combination \(\sum _{\alpha ,\beta }c_{\alpha ,\beta } \lvert \Phi _{\alpha ,\beta }\rangle =0\) says that the linear functional \(M\mapsto \sum _{\alpha ,\beta }c_{\alpha ,\beta }M_{\alpha \beta }\) annihilates every product of \(L\) matrices of \(A\). These products span \(M_D(\mathbb {C})\), so the functional vanishes on the matrix units and \(c=0\). The right cut vectors are treated in the same way with the transposed functional. For the propagation, every matrix \(M\) equals \(\sum _iA_i(A_i^\dagger M)\), and each \(A_i^\dagger M\) is a combination of products of \(L'\) matrices, so \(M\) is a combination of products of \(L'+1\) matrices.

Both decompositions (23) of the same coefficients have \(D^2\) terms. The dual functionals of the linearly independent right cut vectors of \(B\) read off a matrix \(G\) with \(\Phi ^{B}_{\mu }=\sum _\nu G_{\mu \nu }\Phi ^{C}_{\nu }\); linear independence of the left cut vectors of \(B\) shows that no nonzero row vector annihilates \(G\), so \(G\) is invertible, and \(W=(G^{-1})^{\mathsf T}\) satisfies (24). For the chain relation, extend the left words by one site: the two expressions of \(\Phi ^{C}\) at length \(k+1\) through (24) at the two cuts are combinations of the independent family \(\Phi ^{B}\) of length \(k\), and comparing coefficients gives (25) entrywise.

Consider the \(D^4+1\) cuts after \(L_0,\ldots ,L_0+D^4\) sites; each leaves at least \(L_0\) sites on both sides, so by Lemma 23.3.2 the cut vectors of \(B\) are linearly independent there, and Lemma 23.3.3 supplies invertible matrices \(W_0,\ldots ,W_{D^4}\) on the doubled bond space with \(W_k(C_i\otimes \mathbb {1})=(B_i\otimes \mathbb {1})W_{k+1}\). Since \(M_{D^2}(\mathbb {C})\) has dimension \(D^4\), there is a first index \(n\ge 1\) such that \(W_0,\ldots ,W_{n-1}\) are linearly independent and \(W_n=\sum _{k{\lt}n}\lambda _kW_k\); some \(\lambda _k\) is nonzero because \(W_n\) is invertible, so the polynomial equation (15) has positive degree and a nonzero root \(x\). Lemma 23.2.3, applied to right multiplication by \(C_i\otimes \mathbb {1}\) and left multiplication by \(B_i\otimes \mathbb {1}\), gives a nonzero combination \(W\) of \(W_0,\ldots ,W_{n-1}\) with \(W(C_i\otimes \mathbb {1})=(\tfrac 1xB_i\otimes \mathbb {1})W\) for every \(i\), and Lemma 23.2.4 extracts a nonzero multiplicity slice \(R\) satisfying (26).

The range of \(R\) is invariant under every \(B_i\), hence under every product of \(L_0\) matrices of \(B\); block injectivity leaves no nontrivial invariant subspace, so the range is everything and \(R\) is invertible. Block injectivity also makes the transfer maps of \(B\) and of \(C\) irreducible, the latter because \(C_i=\tfrac 1xR^{-1}B_iR\). The conjugate-transposed families are therefore left-canonical irreducible tensors related by the gauge \((R^\dagger )^{-1}\) and the scalar \(\overline{x}\), and Lemma 9.4.1.2 replaces the gauge by a unitary and gives \(|x|=1\). Taking adjoints returns (27). This is the fixed-point argument of [ 228 , Theorem 7 ] , which derives \(|x|=1\) from the dual fixed point of \(B\) and \(RR^\dagger =\mathbb {1}\) from the single-block property; both follow from condition C1 through the Perron–Frobenius theory of irreducible maps. The coefficient relation follows from (27) because conjugation by \(U\) does not change traces.

The weights are positive reals of modulus at most one, and the block conditions are those of Definition 23.3.6.

The block-diagonal form of the products follows from (28) by induction on the length, and (29) from the first block condition. If \(\alpha \) lies in one block and \(\beta \) in another, the entry \((\alpha ,\beta )\) of every product vanishes while the matrix unit \(\lvert \alpha \rangle \langle \beta \rvert \) has entry one there, so the products of any length do not span \(M_D(\mathbb {C})\). When \(D\ge 1\) there is at least one block, and block injectivity leaves exactly one; then (29) is the scalar \(\lambda ^2\) of that block.

For \(\lambda _B=\lambda _C=1\), Theorem 23.3.4 gives \(R\neq 0\) and \(x\neq 0\) with \(RC_i=\tfrac 1xB_iR\), and Lemma 23.3.5 turns \(R\) into a unitary \(U\) with \(|x|=1\) and \(B_i=xUC_iU^\dagger \). The coefficients then satisfy \(V^{(N)}(B)=x^NV^{(N)}(C)=x^NV^{(N)}(B)\), and since the state is nonzero, \(x^N=1\); take \(\xi =x\). For general weights, \(\lambda _B^{-1}B\) and \(\lambda _B^{-1}C\) represent the same state and the first is unital, so Theorem 23.3.4 gives \(R\) and \(x\) for this pair. The same \(R\) satisfies \(R\, (\lambda _C^{-1}C_i)=y^{-1}(\lambda _B^{-1}B_i)R\) with \(y=x\lambda _C/\lambda _B\), and Lemma 23.3.5 applied to the two unital families gives \(|y|=1\) and \(B_i=xUC_iU^\dagger \). As before \(x^N=1\), so \(|x|=1\), and \(|y|=1\) forces \(\lambda _B=\lambda _C\).

The state is nonzero, so \(D\ge 1\), and Lemma 23.3.8 leaves \(B\) with a single block: \(\sum _iB_iB_i^\dagger =\lambda _B^2\, \mathbb {1}\) with \(0{\lt}\lambda _B\le 1\). The tensors \(\lambda _B^{-1}B\) and \(\lambda _B^{-1}C\) represent the same state and the first is unital and \(L_0\)-block injective, so Theorem 23.3.4 gives \(R\neq 0\) and \(x\neq 0\) with \(R\, (\lambda _B^{-1}C_i)=\tfrac 1x(\lambda _B^{-1}B_i)R\), and \(R\) is invertible by Lemma 23.3.5. Hence \(C_i=\tfrac 1xR^{-1}B_iR\) is conjugate to a nonzero multiple of \(B_i\), so \(C\) is \(L_0\)-block injective, and Lemma 23.3.8 leaves \(C\) with a single block as well: \(\sum _iC_iC_i^\dagger =\lambda _C^2\, \mathbb {1}\) with \(0{\lt}\lambda _C\le 1\). Lemma 23.3.9 now gives \(U\) and \(\xi \). Rephasing \(C\) by \(\xi \) multiplies every length-\(N\) coefficient by \(\xi ^N=1\), and \(\xi \overline{\xi }=|\xi |^2=1\) makes \((U,\xi )\) a unit-modulus gauge-phase equivalence.

Surjectivity of \(\Phi _A\) yields a linear section by choice.

Take \(c=\Psi _A(X)\). Then \(\Phi _A(c)=\Phi _A(\Psi _A(X))=X\).

By definition, \(O_A(X)_{ij}=(\Psi _A(A^iX))_j\). Applying \(\Phi _A\) gives \(\sum _jO_A(X)_{ij}A^j=\Phi _A(\Psi _A(A^iX))=A^iX\), where the last step uses \(\Phi _A\circ \Psi _A=\operatorname{id}_{M_{D}(\mathbb {C})}\).

Applying Lemma 23.4.1.2 to \(X\) yields \(A^iX=\sum _{j=0}^{d-1}O_A(X)_{ij}A^j\). Multiplying by \(Y\) on the right and applying the linear map \(\Psi _A\) gives

Taking the \(k\)th component of both sides gives

Since this holds for all \(i\) and \(k\), the matrices are equal.

The span of the blocked matrices contains all products \(A_0^{i_0}\cdots A_{m-1}^{i_{m-1}}\). Since each \(\{ A_k^i\} _i\) spans \(M_{D}(\mathbb {C})\), these products span all of \(M_{D}(\mathbb {C})\).

The \(\mathcal{V}(A)=\mathcal{V}(B)\) hypothesis gives, for every \(X\in M_{D}(\mathbb {C})\), equality of the virtual-insertion trace coefficients

for some \(Y\) depending on \(X\). Since all three site tensors on each side are injective, the physical realisation maps \(O_A,O_B\) are algebra homomorphisms by Theorem 23.4.1.3. Setting \(T(X)=Y\) defines a \(\mathbb {C}\)-linear multiplicative map, hence a nonzero \(\mathbb {C}\)-algebra endomorphism of \(M_{D}(\mathbb {C})\), which is bijective by Theorem 3.2.1. Skolem–Noether (Theorem  [ 216 , “Skolem–Noether” ] ) gives \(Z\in \mathrm{GL}_D(\mathbb {C})\) with \(T(X)=Z^{-1}XZ\).

By cyclicity and non-degeneracy of the trace pairing, the conditions in (31) and (32) give, for all physical indices \(i,j\),

Write \(\mathbb {1}_D=\sum _jc_jA_2^j\) using the decomposition map \(\Psi _{A_2}\), and define \(Z:=\sum _jc_jB_2^j\in M_{D}(\mathbb {C})\). This is the point where the two-site comparison collapses to a single boundary matrix:

The first product identity (33) gives \(A_1^i=ZB_1^i\), and the second (34) gives \(A_1^i=B_1^iZ\). Hence \(Z\) commutes with every \(B_1^i\); since \(\{ B_1^i\} \) spans \(M_{D}(\mathbb {C})\), \(Z\) is central. By Lemma 2.1.5, \(Z=\lambda \mathbb {1}_D\) for some \(\lambda \in \mathbb {C}^{\times }\) (nonzero since the tensors are nonzero). Thus \(A_1^i=\lambda B_1^i\), and substituting into (33) forces \(A_2^j=\lambda ^{-1}B_2^j\).

Choose one site of \(\mathbf{A}\); its injectivity implies injectivity of the combined tensor by Lemma 23.5.1.1. Apply the single-block Fundamental Theorem (Theorem 3.2.4) to the two combined tensors. This yields one invertible matrix \(X\) such that every combined matrix of \(\mathbf{B}\) equals the corresponding combined matrix of \(\mathbf{A}\) conjugated by \(X\). Reading this identity back at each site gives \(B_k^i = X A_k^i X^{-1}\) for all \(k\) and \(i\), so the constant choice \(Z_k = X\) is a cyclic gauge.

Combining the site and physical indices does not change which local matrix is read at the component \((k,i)\), so both sides evaluate to \(\zeta A_k^i\) there.

If the combined tensor of \(\mathbf{B}\) differs from that of \(\mathbf{A}\) by a gauge and a nonzero scalar \(\zeta \), rescale each site of \(\mathbf{B}\) by \(\zeta ^{-1}\). Lemma 23.6.2 removes this common scalar from the combined tensor, so Theorem 23.6.1 applies to the rescaled chain. Multiplying the resulting sitewise relations by \(\zeta \) gives (35).

Write \(1 = \sum _i c_i A^i\) (by injectivity) and set \(S = \sum _i c_i B^i\).

1. Show that the length-\(N_0\) word span of \(B\) is all of \(M_{D}(\mathbb {C})\) via the linear relation between the length-\(N_0\) word-evaluation maps of \(A\) and \(B\) induced by trace agreement at length \(N_0 + 1\), together with a finrank transfer argument.

2. Derive \(S = 1\) by trace nondegeneracy: for every \(M \in M_{D}(\mathbb {C})\), \(\operatorname{tr}(M(S - 1)) = 0\).

3. Downward induction on \(k = N_0 - |w|\): decompose \(\operatorname{tr}(w_A)\) and \(\operatorname{tr}(w_B)\) using \(1 = \sum c_i A^i\) and \(S = 1\) respectively, then apply the inductive hypothesis at length \(|w| + 1\).

Apply Theorem 23.6.1.2 to obtain equality of the MPV families for all system sizes, then invoke the single-block Fundamental Theorem (Theorem 3.2.4).

Bundle the constant chains into one tensor. Injectivity of \(A\) implies injectivity of the combined tensor by Lemma 23.5.1.1, and the combined-tensor MPV equality is exactly the hypothesis needed for the single-block Fundamental Theorem. The resulting gauge matrix is independent of the site because every site carries the same local tensor.

Apply Theorem 23.7.1.1 to obtain \(B^i = X A^i X^{-1}\). Set \(Z = X^{-1}\) and \(\lambda = 1\). Then \(\lambda ^n = 1\), and (36) is exactly the same gauge relation rewritten in the form used by the corollary.

By Lemma 6.5.3, the local blocked tensor \(A^{[L]}\) is injective, and therefore every site of the constant blocked chain is injective.

By Lemma 23.7.2.2, the blocked chain built from \(A^{[L]}\) is injective. Apply Theorem 23.6.1 to the two constant blocked chains.

Under the order isomorphism between two-sided ideals of \(\prod _{j=0}^{r-1} R_j\) and products of two-sided ideal lattices, \(I_k\) corresponds to the tuple with \(\top \) in the \(k\)-th coordinate and \(\bot \) elsewhere. This tuple is an atom, hence so is \(I_k\).

By Theorem 23.8.1.1, the atoms in each two-sided ideal lattice are exactly its block ideals. The induced order isomorphism therefore assigns a unique target block ideal to every source block ideal. Applying its inverse shows that this assignment is bijective and gives \(\sigma \).

Theorem 23.8.1.2 gives the permutation \(\pi \). Theorem  [ 216 , “Summand matching and dimension preservation” ] identifies the matrix sizes \(D_{\pi (k)} = D_k\). For each \(k\), Theorem  [ 216 , “Paired component maps are bijections” ] gives a bijective multiplicative map from \(M_{D_k}(\mathbb {C})\) to \(M_{D_{\pi (k)}}(\mathbb {C})\) extracted from the elements supported on one factor of \(T\); after reindexing by the dimension equality, this is a \(\mathbb {C}\)-algebra automorphism of \(M_{D_k}(\mathbb {C})\). Skolem–Noether (Theorem  [ 216 , “Skolem–Noether” ] ) makes each such automorphism inner.

This is a standard result in the theory of semisimple algebras; see  [ 17 , Section IV.A ] for the MPS context.

Since \(\{ A_k^i\} \) spans \(M_{D_k}(\mathbb {C})\), it suffices to check on spanning elements, where the result follows from the MPV identity.

A nonzero multiplicative linear endomorphism of \(M_{D_k}(\mathbb {C})\) is automatically bijective: injectivity follows from the simplicity of \(M_{D_k}(\mathbb {C})\) (the kernel is a proper ideal, hence zero), and surjectivity from finite dimension. Non-vanishing of \(T_k\) is established via the trace pairing: if \(T_k = 0\), then all \(B_k^i = 0\), contradicting the non-vanishing of the trace pairing for injective tensors.

By Lemmas 23.8.2.2 and 23.8.2.3, \(T_k\) is a bijective multiplicative linear map. Positivity makes this algebra automorphism preserve adjoints. Unitary Skolem–Noether therefore implements it by conjugation with a unitary matrix.

Apply Theorem 23.8.1.3 to the product-algebra automorphism of Definition 23.8.2.5. The theorem writes the nonzero input block as \(\pi (k)\) in output component \(k\); for the statement above we set \(\sigma := \pi ^{-1}\), so an element supported in block \(k\) lands in block \(\sigma (k)\).

Choose \(N\) supported on a single block \(k\) (all other components zero). The hypothesis reduces to \(\operatorname{tr}(M_k N_k) = 0\) for all \(N_k\), so \(M_k = 0\) by nondegeneracy of the matrix trace pairing on \(M_{D_k}(\mathbb {C})\). Since \(k\) was arbitrary, \(M = 0\).

Reduces to per-block injectivity of the Gram map \(M_k \mapsto (i \mapsto \operatorname{tr}(M_k \cdot A_k^i))\), which holds by injectivity of \(A_k\) (the matrices \(\{ A_k^i\} \) span \(M_{D_k}(\mathbb {C})\), so the trace pairing against them separates points).

Per-block, the weight identity \(\mu _A(k) = \mu _B(k) \zeta _k\) rewrites the scaled relation as \(\mu _B(k)\, B_k^i = X_k\, (\mu _A(k)\, A_k^i)\, X_k^{-1}\). Each weighted block then satisfies an ordinary conjugation, and the block-diagonal conjugation formula assembles the per-block \(X_k\) into the flattened global gauge \(\bigoplus _k X_k\) relating \(\bigoplus _k \mu _A(k) A_k\) to \(\bigoplus _k \mu _B(k) B_k\).

Apply the single-block Fundamental Theorem (Theorem 3.2.4) to each block independently.

By the MPV decomposition (Theorem 2.6.9), each MPV is \(\sum _k \mu _k^N V^{(N)}(A_k)_\sigma \). Per-block equality of MPV coefficients gives global equality.

Apply the per-block single-block Fundamental Theorem (Lemma 23.9.1.2) to obtain blockwise gauge equivalences from injectivity and equality of MPV families, then combine them into a global gauge for the block-diagonal tensors.

Per-block gauge equivalence follows from applying the single-block Fundamental Theorem to each block, using injectivity of \(A_k\) and \(\mathcal{V}(A_k) = \mathcal{V}(B_k)\). With \(X=\bigoplus _k X_k\), (37) follows by multiplying block-diagonal matrices block by block.

Extract the gauge matrix from the per-block part of Lemma 23.9.1.7 for each block.

Apply Theorem 23.8.2.6 to the product-algebra automorphism built from the per-block linear extensions.

The sum of per-block MPV contributions is invariant under rearrangement of the summands.

Substituting the equality of dimensions makes the two tensor families identical.

Expanding both sides by Theorem 2.6.9 gives \(\sum _k \mu _k^N\, V^{(N)}(A_k)_\sigma \) and \(\sum _k \mu _{e(k)}^N\, V^{(N)}(A_{e(k)})_\sigma \). These sums are equal by reindexing the finite sum along the bijection \(e\).

The equality holds by definition; \(T_C\) is defined as the weighted block-diagonal tensor formed from the weights and block tensors of \(C\).

Rewrite \(T_C\) as a weighted block-diagonal tensor via Lemma 23.9.4.1, then apply the multi-block global gauge-equivalence lemma (Lemma 23.9.1.5).

The forward direction applies the single-block Fundamental Theorem to each block. The reverse direction holds because gauge equivalence preserves all traces of word evaluations.

Theorem 23.7.1.1 gives a single gauge matrix \(X\) with \(B^i = X A^i X^{-1}\) for all \(i\). Since gauge equivalence preserves injectivity, \(B\) is injective as well.