We argue by induction on the common length of \(u_1\) and \(v_1\). For the empty prefix, \(M^{u_2,v_2}=\mathbb {1}_D M^{u_2,v_2}\). If \(u_1=iu\) and \(v_1=jv\), then

using associativity of matrix multiplication.

Lemma 16.1.4 gives \(M^{ua,\, vb}=M^{u,v}M^{ab}\) and \(M^{au,\, bv}=M^{ab}M^{u,v}\). Hence, by cyclicity of the trace,

The trace of \(\prod _n c_{t_n}M^{s_nt_n}\) is \(\bigl(\prod _nc_{t_n}\bigr)\operatorname {tr}\prod _nM^{s_nt_n}\). Every word of the doubled-index tensor is multiplied by the product of the scalars of its letters, which is nonzero, so the span of the words of each length is unchanged.

Induction on the common length of \(u,v\), expanding one letter at a time and using that block-diagonal sums multiply blockwise.

The trace of a block-diagonal sum is the sum of the traces of the two blocks; apply Lemma 16.1.9 and take traces entrywise.

The entry is a closed bond trace, \(\rho ^{(N)}(M)_{\sigma ,\tau } =\operatorname{tr}(M^{\sigma _1\tau _1}\cdots M^{\sigma _N\tau _N})\). Shifting both configurations by \(r\) rotates the matrix product by one factor, and the trace is invariant under that cyclic rotation:

so \(\rho ^{(N)}(M)_{r\sigma ,\, r\tau } =\rho ^{(N)}(M)_{\sigma ,\tau }\).

At the level of word evaluations, the conjugate transpose of a product reverses the factors, so

Taking traces gives the entrywise identity

For an MPDO, \(\rho ^{(N)}(M)\) is Hermitian, so

positive semidefiniteness is preserved under the simultaneous relabeling of rows and columns by the reflection.

After identifying the single physical index of the doubled tensor with a pair \((i,j)\), for every \(X\in M_{D}(\mathbb {C})\),

By Lemma 16.2.2, \(\mathcal{E}_M\) is the transfer map of the doubled-index MPS tensor. The corresponding MPS transfer map is positive, so \(\mathcal{E}_M(X)\ge 0\) whenever \(X\ge 0\).

By the mixed-product property \((A\otimes B)(C\otimes D)=(AC)\otimes (BD)\), the \(N\)-site product decomposes as a sum of Kronecker products after transporting along the bond-space identification \(e\):

where \(P_\kappa =A^{(\sigma _1,\kappa _1)}\cdots A^{(\sigma _N,\kappa _N)}\) and \(Q_\kappa =A^{(\tau _1,\kappa _1)}\cdots A^{(\tau _N,\kappa _N)}\). Since the trace is invariant under the reindexing by \(e\), we apply \(\operatorname{tr}(X\otimes Y)=\operatorname{tr}(X)\operatorname{tr}(Y)\) to obtain

where \(\psi _\kappa (\sigma )=\operatorname{tr}(P_\kappa )\). Hence \(\rho ^{(N)}(M) =\sum _\kappa |\psi _\kappa \rangle \! \langle \psi _\kappa |\ge 0\).

Each copy similarity cancels inside every positive-length closed matrix-product trace.

This follows entrywise from the exchange of the ket and bra indices and conjugate transposition of the virtual matrix.

If the common injective prefix has length \(L\) and the selector has length \(S\), concatenation gives simultaneous words of length \(L+S\). For every tuple \((X_j)_j\), the resulting coefficients satisfy \(\sum _{|w|=L+S}c_w A_j^w=X_j\). Hence the simultaneous word tuples span the product algebra, and nondegeneracy of the trace pairing gives finite-length trace separation.

For every matrix \(X\), block injectivity gives coefficients such that \(X=\sum _{|w|=L}c_wA_0^w\). Since the family has only the block \(A_0\), this is precisely the required simultaneous word-tuple spanning identity.

At a fixed tail length \(L\), expand the first-site equality at system size \(L+1\) and group all copies lying over the same representative. For every length-\(L\) word \(w\) this gives

Representative word-tuple separation therefore forces each matrix \(c_j^{(L+1)}(YA_j-ZA_j)\) to vanish. A nonzero coefficient may then be cancelled.

For the eventual assertion, choose a threshold \(L_0\) and put \(M=L_0+1\). For a fixed representative \(j\), apply Corollary 8.5.2 to the nonzero numbers \(\mu _{j,q}^{M}\). It yields \(r\geq 1\) such that \(c_j^{(Mr)}=\sum _q(\mu _{j,q}^{M})^r\neq 0\). Since \(Mr-1\geq L_0\), the fixed-length argument applies with \(L=Mr-1\). Finally, equality of the positive-length MPV families gives the first-site hypothesis for the sector-decomposition tensor.

Expand both matrices as direct sums. Their product is the direct sum of \(\mu _{j,q}^{|w|+1}C_j^sA_j^w\). Taking the trace and summing first over \(q\) gives the stated coefficient \(c_j^{(|w|+1)}\).

Fix \(s\) and a tail length \(L\), and put \(\Delta _j=c_j^{(L+1)}(C_j^s-E_j^s)\). Lemma 16.4.13 gives \(\sum _{j=0}^{g-1}\operatorname{tr}(\Delta _jA_j^w)=0\) for every length-\(L\) word \(w\). Representative word-tuple separation implies \(\Delta _j=0\) for every \(j\). If \(c_{j_0}^{(L+1)}\neq 0\), cancellation gives \(C_{j_0}^s=E_{j_0}^s\). Since \(s\) was arbitrary, \(C_{j_0}=E_{j_0}\).

Under eventual separation, apply Corollary 8.5.2 to the nonzero weights above each fixed representative, exactly as in Theorem 16.4.11. This selects a sufficiently large \(L\) with \(c_j^{(L+1)}\neq 0\) and proves \(C_j=E_j\).

Reindex the finite sum over listed coordinates by the class-copy pairs \((j,q)\) and use the bond-dimension identifications to replace each block by its unit-phase representative. The marked block supplies one factor \(\lambda _{j,q}\) and the word supplies \(|w|\) further factors. Summing first over \(q\) gives (20).

Theorem 16.4.15 rewrites (21) into marked-trace agreement for \(P_{\mathrm{rep}}\). Its representatives have eventual word-tuple separation, and every copy weight is nonzero. Theorem 16.4.14 therefore gives \(C_j=E_j\).

First-site agreement for \(T_{\mathrm{grp}}\) also holds for \(P_{\mathrm{rep}}\) because their matrix product vectors agree at every positive length. Applying Theorem 16.4.11 gives (22), namely

Induction on \(w\) in (24) gives \(CA^w=V^\dagger EB^wV\). Cyclicity of trace and \(VV^\dagger =\mathbb {1}\) give (25). Linearity of the physical mark gives the grouped marked tensor.

Apply (25) to both sides of (26), remove the block gauge by cyclicity of trace, and use Theorem 16.4.16 on the representatives. The resulting equality of the full grouped marked tensors, followed by (24), gives \(F_A=G_A\).

For the forward implication, split the first block into its first letter and its remaining \(L\) letters. The insertion identity (28) reduces the blocked MPV equality to the original first-site equality, with the remaining letters of the first block prepended to the flattened tail.

For the converse, put \(\Delta ^i=(YA)^i-(ZA)^i\). Equality after blocking gives \(\Delta ^i A^w=0\) for every word \(w\) of length \(L\). Since the length-\(L\) words span \(M_{D}(\mathbb {C})\), the same equality holds with \(A^w\) replaced by the identity, and therefore \(\Delta ^i=0\) for every physical index \(i\).

Injectivity of \(A_j\) implies that its homogeneous words span \(M_{D_j}(\mathbb {C})\) at every positive length. Apply the three-block separation argument with source length \(1\). For each ordered pair of distinct representatives it gives a pairwise block-separating equation at length \(6\). Multiplying the equations for the \(g-1\) representatives different from \(A_j\) gives coefficients \(c_w^{(j)}\) satisfying

If \(L\geq 6(g-1)+1\), put \(m=L-6(g-1){\gt}0\). The length-\(m\) words of each \(A_j\) span \(M_{D_j}(\mathbb {C})\). Concatenating these words with the selectors in (32) shows that the simultaneous length-\(L\) word tuples span \(\prod _jM_{D_j}(\mathbb {C})\). This proves eventual representative word-tuple separation. The representative-grouped Lemma L gives the two first-site conclusions.

For the unblocked canonical form, propagate \(p\)-block injectivity to the lengths \(2p-1\), \(2p\), and \(6p\). Applying the direct-sum argument to the resulting injective tensors gives pair trace separation at original length \(6p\). The bijection between blocked words of length \(6\) and original words of length \(6p\) gives pair trace separation for the \(p\)-blocked representatives. The preceding selector argument then applies to \(P^{[p]}\).

The same pairwise equations at original length \(6p\) give selectors for the unblocked representatives directly. For every \(L\geq 6p(g-1)+p\), the remaining prefix has length at least \(p\) and hence spans each full representative matrix algebra. Concatenating the prefix with the selectors proves eventual representative word-tuple separation before blocking.

For the final assertion, first consider one representative \(A_j\). Irreducibility and left-canonicality give a positive peripheral period \(m_j\). The periodic self-overlap formula and the normalized self-overlap hypothesis give, along the same subsequence,

Hence \(m_j=1\). Primitivity then implies normality. Taking a common multiple of the finitely many normality lengths gives a positive length \(L\) at which every representative is block injective. Block \(L+1\) sites. Injectivity propagates from length \(L\) to length \(L+1\), so the blocked representatives are injective. Act on the first letter of each physical block and apply the blocked representative separation followed by the representative-grouped Lemma L. This gives equality of the two induced insertions on every blocked representative. The remaining \(L\) letters of the first block form a spanning family, so Lemma 16.4.19 recovers \(YA_j=ZA_j\) before blocking.

The BNT canonical-form hypotheses give eventual representative word-tuple separation. Apply Theorem 16.4.14.

Pass through the block-diagonal similarity in the normalized BNT-refined horizontal form. Linear combinations of physical letters are covariant under this similarity, and cyclicity of the trace removes it from each closed marked chain. On the representative-indexed sector decomposition one has

Corollary 16.4.21 identifies the two marks on every minimal representative. Taking their weighted direct sums and restoring the common block-diagonal similarity gives \(C^u=E^u\).

For the normalized BNT-refined assertion, positive-length MPV equality gives the first-site identity for the BNT sector decomposition. The representative-grouped Lemma L gives \(YA_j=ZA_j\) for every representative. First-site insertion commutes with the weighted direct sum, and conjugating each copy by its gauge gives \(YM=ZM\) on the original bond space.

For the specialized assertion, positivity and Hermiticity give \(Q_1H^{(N)}=Q_1H^{(N)}Q_1=H^{(N)}Q_1\). These two identities give the corresponding first-site equalities for the BNT sector decomposition. Apply the representative-grouped Lemma L to identify first \(Y_L(Q)A_j\) with \(Y_C(Q)A_j\) and then with \(Y_R(Q)A_j\). Transitivity gives the stated equality for every representative, and the same direct-sum and copy-gauge argument gives the literal equality on \(\widetilde M\).

For literal CPSV canonical form, first-site agreement is a marked-trace identity on the original tensor. Theorem 16.4.17 needs only the identities with a nonempty unmarked tail; it identifies the two linear marks and gives (33). In the invariant-projection case, positivity first yields \(P_1H^{(N)}=P_1H^{(N)}P_1=H^{(N)}P_1\); its matrix coefficients are exactly the required first-site agreement.

This is the per-block-separated form of Lemma L of [ 13 , Appendix C.3, lines 1835–1846 ] , using the finite criterion from Theorem 16.4.8 applied directly to the full block family rather than to a minimal basis of representatives. Expanding the hypothesis over a word of length \(L+1\) beginning with the fixed site and folding each block contribution into a trace pairing against the corresponding first-site insertion tensor yields, upon taking the \(Y\)-\(Z\) difference, a tuple \((\Delta _k)_k\) of block matrices. For every word \(w\) of length \(L\),

The finite block-separation hypothesis applied to the tuple \((\mu _k^{L+1}\Delta _k)_k\) forces each weighted block difference to be zero, and the nonzero weights give the blockwise equality of the two first-site insertions.

Taking adjoints gives \(HP=PHP\), and the conclusion follows by subtraction.

Since \(H\) and \(P\) are Hermitian, taking adjoints gives \(HP=(PH)^\dagger =(PHP)^\dagger =PHP\). Therefore \((\mathbb {1}-P)HP=HP-PHP=0\).

Apply the representative-grouped Lemma L first to the actions \(Y_L,Y_R\) and then to \(Y_L,Y_C\). The BNT canonical-form hypotheses supply the required common block-injectivity length. Transitivity gives the stated equality on every minimal representative; the repeated copies have already been combined through \(\sum _q\mu _{j,q}^{N}\).

Decode the doubled index at every site as a ket–bra pair. The defining trace of the resulting matrix word is exactly the corresponding entry of \(H^{(N)}(M)\). Summing the first doubled index against \(Y_L\), \(Y_R\), or \(Y_C\) leaves the single or double sums in (34).

Write the output and summed doubled indices as \((a,b)\) and \((c,e)\). The Kronecker deltas in \(Y_L\) and \(Y_R\) remove, respectively, the sums over \(e\) and \(c\); \(Y_C\) leaves both multiplications. The resulting coefficients are \(\sum _c P_{a,c}M^{cb}\), \(\sum _e P_{e,b}M^{ae}\), and \(\sum _{c,e}P_{a,c}P_{e,b}M^{ce}\).

Lemma 16.4.29 turns the two operator identities into equality of the corresponding first-site contractions. Equality of matrix product vectors gives these equalities for the chosen canonical-form decomposition. Apply Theorem 16.4.27 with \((Y_L,Y_C,Y_R)\).

Put \(L=N+1\). Splitting off the first site of a density-operator entry gives

Summing the first ket index against \(P\) therefore pairs the operator \((PM)^{ij}\) against the remaining word; the hypothesis \(PM=PMP\) replaces it by \((PMP)^{ij}\), which unfolds to the entry of \(P_1H^{(N+1)}P_1\). This proves \(P_1H^{(N+1)}=P_1H^{(N+1)}P_1\). Since \(H^{(N+1)}\) is positive semidefinite and \(P\) is self-adjoint, Theorem 16.4.26 gives \((\mathbb {1}-P_1)H^{(N+1)}P_1=0\), hence \(H^{(N+1)}P_1=P_1H^{(N+1)}P_1=P_1H^{(N+1)}\).

For every tail length \(N\), Theorem 16.4.33 turns \(PM=PMP\) into the entrywise identities \(P_1H^{(N+1)}=P_1H^{(N+1)}P_1\) and \(P_1H^{(N+1)}=H^{(N+1)}P_1\). These are exactly the hypotheses of Theorem 16.4.31, whose conclusion is the equality of the insertions of \(MP\) and \(PMP\) on every minimal representative.

Diagonalize \(H\). The function \(x\mapsto x^p\) is injective on the non-negative reals, so a matrix commutes with the diagonal matrix of the \(p\)-th powers exactly when it commutes with the diagonal matrix itself.

Diagonalize \(H=U\Lambda U^\dagger \) with \(U\) unitary and \(\Lambda \) the diagonal matrix of eigenvalues \(\lambda _i\geq 0\). Conjugating the hypothesis by \(U\) gives, entrywise for \(Q'=U^\dagger QU\),

Since \(x\mapsto x^p\) is injective on the non-negative reals, every entry \(Q'_{ij}\) with \(\lambda _i\neq \lambda _j\) vanishes, so \(Q'\Lambda =\Lambda Q'\). Conjugating back gives \(QH=HQ\).

Suppose \(Q\) were such a projector. Every \(H^{(N+1)}\) is positive semidefinite, so commutation of \(Q_1\) with \([H^{(N+1)}]^p\) implies commutation with \(H^{(N+1)}\) (Theorem 16.4.36), contradicting \(Q_1H^{(N+1)}\ne H^{(N+1)}Q_1\). This is the contradiction concluding the periodic-sector step in the proof of [ 13 , Proposition 4.13 ] , lines 1892–1893.

A nontrivial \(p\)-periodic vector of \(\widetilde M\) supplies a periodic-sector projector for \(M\), and Theorem 16.4.38 shows that no such projector exists.

The word product of layer-product letters along a ket–bra configuration expands, by induction on the length and the mixed-product property of the Kronecker product, as a sum over configurations \(\kappa \) of the contracted physical index of Kronecker products of the layers’ word products. Taking traces, each summand factors into the product of the two traces, so the entry of \(H^{(N)}(MM')\) at \((\sigma ,\tau )\) equals \(\sum _\kappa H^{(N)}(M)_{\sigma \kappa }H^{(N)}(M')_{\kappa \tau }\), the entry of the matrix product. The power identity follows by induction on \(p\), and positive semidefiniteness of \([H^{(N)}]^p\) follows from that of \(H^{(N)}\).

The letter identity is a direct computation from the layer product: contracting the shared physical index composes the physical-space operators. Induction on \(p\) writes each vertical letter of \(M^{[p]}\) as a length-\(p\) word. Thus, for every bond index \(v\), there is a word \(w\) of length \(p\) such that \(\widetilde{M^{[p]}}_v=A_w\), and \(QA_w=QA_wQ\) implies \(Q\widetilde{M^{[p]}}_v=Q\widetilde{M^{[p]}}_vQ\). Applying this identity to the two one-site actions gives \(QM^{[p]}=QM^{[p]}Q\).

By Theorem 16.4.43 the word invariance gives \(QM^{[p]}=QM^{[p]}Q\) on the stacked tensor, which generates matrix product density operators by Theorem 16.4.42. Theorem 16.4.33 applied to \(M^{[p]}\) yields \(Q_1H^{(N)}(M^{[p]})=H^{(N)}(M^{[p]})Q_1\) for every length, and \(H^{(N)}(M^{[p]})=[H^{(N)}]^p\) by the stacked-layers identity. Together with the assumed non-commutation family this is exactly the periodic-sector projector of Definition 16.4.37.

A nontrivial \(p\)-periodic vector supplies an invariant non-commuting projector, which is a periodic-sector projector by Definition 16.4.44.

The rescaled unital gauge \(K\) of the corner satisfies the hypotheses of the cyclic decomposition of the peripheral spectrum (Theorem D.3.5), and the eigenvalue \(\mu /r\ne 1\) of modulus one (Lemma D.3.6) forces the peripheral cyclic group to have order \(m\ge 2\). Theorem  [ 216 , “Cyclic decomposition for irreducible finite Kraus maps” ] produces orthogonal projections \(P_0,\ldots ,P_{m-1}\) summing to the identity with \(\mathcal{E}_K(P_{k+1})=P_k\), and the letter-level shift \(K_vP_{k+1}=P_kK_v\) holds (Lemma 5.8.1.1). Set \(q=m\) and let \(Q\) be the complement of the support projection \(\pi \) of \(YY^\dagger \) for \(Y=V\rho ^{1/2}P_0\). A word \(A_w\) of \(m\) letters of \(\widetilde M\) satisfies \(A_wY=YG_w\), where \(G_w\) is a scalar multiple of the corresponding word in the letters of \(K\), which commutes with \(P_0\) by the full-period shift; so the range of \(Y\) is invariant and Lemma D.3.3 gives \(QA_w=QA_wQ\). If a single letter also satisfied the invariance, then \(\pi \) would fix \(A_vY\), whose columns lie in the sector \(V\rho ^{1/2}P_{m-1}\) by the cyclic shift; cancelling the injective factor \(V\rho ^{1/2}\) and using orthogonality of \(P_{m-1}\) and \(P_0\) would force \(P_{m-1}K_v=0\) for every letter, hence \(P_{m-1}=\mathcal{E}_K(P_0)=0\), contradicting the nonvanishing of the cyclic projections.

Suppose instead that \(Q_1H^{(N+1)}=H^{(N+1)}Q_1\) for every \(N\geq 0\). Idempotence gives

Taking matrix coefficients yields equality of the one-sided first-site contractions. The literal form of Lemma L in Theorem 16.4.23 identifies the two inserted tensors on the original bond space. By Lemma 16.4.30, this equality is \(Q\widetilde M=Q\widetilde M Q\), contradicting the displacement.

A nontrivial periodic vector supplies an integer \(p\geq 1\) and an orthogonal projector \(Q\) such that every word \(A_w\) of length \(p\) in the letters of \(\widetilde M\) satisfies \(QA_w=QA_wQ\), while \(Q\widetilde M\ne Q\widetilde M Q\) (Theorem 16.4.47). By Theorem 16.4.48, there is a length \(N+1\) for which

The word invariance and the vertical-stack identity give

Since \(H^{(N+1)}\) is positive semidefinite, Theorem 16.4.36 implies \(Q_1H^{(N+1)}=H^{(N+1)}Q_1\), a contradiction. This is the periodic exclusion in [ 13 , Proposition 4.13, lines 1888–1893 ] , with the single noncommuting length supplied by Lemma L.

Suppose first that \(Q_1\) commutes with \(H^{(N+1)}\) for every \(N\). Since \(Q^2=Q\), this gives \(Q_1H^{(N+1)}=Q_1H^{(N+1)}Q_1\) at every positive length. Theorem 16.4.23 and Lemma 16.4.30 then imply \(Q\widetilde M=Q\widetilde MQ\), a contradiction. Thus a displaced idempotent has a noncommuting length \(N+1\).

A nontrivial periodic vector supplies the orthogonal projector \(Q\) of Theorem 16.4.47, invariant under every word of some full period \(p\ge 1\). At the noncommuting length just obtained, the stacked-layer identity and the invariant-projection calculation give \(Q_1[H^{(N+1)}]^p=[H^{(N+1)}]^pQ_1\). Positive semidefiniteness and Theorem 16.4.36 imply that \(Q_1\) commutes with \(H^{(N+1)}\), which yields the contradiction in [ 13 , Proposition 4.13, lines 1888–1893 ] using a single noncommuting length.

Idempotence of \(Q\) upgrades the commutation to the one-sided reduction \(Q_1H^{(2)}=Q_1H^{(2)}Q_1\). Unfolding this identity against every trailing letter of \(M\) gives, for every pair of physical indices \(a,b\), a trace identity pairing \((Q\widetilde M)_{ab}\) against \((Q\widetilde MQ)_{ab}\) over every letter of the doubled-index tensor. Injectivity’s spanning property and nondegeneracy of the trace pairing force the two matrices to agree.

A nontrivial periodic vector supplies the invariant, displaced projector \(Q\) of Theorem 16.4.47. Its word invariance transfers to the stacked tensor, giving commutation with \([H^{(2)}]^p\) unconditionally through the invariant-projection step applied to the stack; positive semidefiniteness of the density operators removes the power, and Theorem 16.4.51 turns the resulting commutation with \(H^{(2)}\) into the letter-level invariance that the displacement forbids.

Theorem 16.4.50 gives the corresponding result for normalized BNT-refined horizontal form.

The trace of a positive semidefinite matrix is a non-negative real, and it vanishes only if the matrix itself is zero.

Compression by a Hermitian matrix preserves positive semidefiniteness, since \(PH^{(N)}P=PH^{(N)}P^\dagger \). By Theorem 16.4.53, a nonzero compression therefore has positive trace. The first-site case follows because \(P_1=P\otimes \mathbb {1}^{\otimes N}\) is Hermitian whenever \(P\) is.

By induction on \(N\): splitting off the first site factors the sum as \((\sum _a M^{aa})\) times the sum over the remaining \(N\) sites. Taking horizontal traces and summing the diagonal entries of \(H^{(N)}\) gives the trace identity.

For a length-\(N\) configuration \(\rho \), write \(E_\rho ^{(N)}=M^{\rho _0\rho _0}\cdots M^{\rho _{N-1}\rho _{N-1}}\). Expanding at the first site gives

Summing over \(a\) and \(\rho \) and collecting closes the remaining loops by Lemma 16.4.56. For the two-sided compression, first-site actions compose site by site, \(P_1Q_1=(PQ)_1\), so cyclicity of the trace and idempotency reduce \(\operatorname{tr}(P_1H^{(N+1)}P_1)\) to \(\operatorname{tr}(P_1H^{(N+1)})\).

Suppose that \(P_1H^{(N+1)}P_1=0\) for every \(N\geq 0\). By the coefficient formula for the doubled-index first-site contraction, the two-sided action \(Y_C(P)\) and the zero action agree on every positive-length matrix product vector of the doubled-index tensor. Literal Lemma L therefore identifies their inserted tensors. For every vertical letter \(v\), the insertion identity then gives

This contradicts the hypothesis in (39).

Normality supplies a letter \(v\) for which \(A_v\ne 0\). Since \(\omega \) is nonzero and \(X\) is invertible, the left-hand side of (41) is nonzero. If (42) vanished, multiplying it by \(V^\dagger \) and \(V\) and using \(V^\dagger V=1\) would contradict (41).

If every compression vanished, its matrix entries would say that the first-site insertion by \(P\) has the same matrix product vectors as zero. Lemma L on the normalized BNT-refined horizontal form would then force the inserted tensor itself to vanish, contrary to the assumed nonzero corner.

By Theorem 16.4.57 the trace of the compression is \(\operatorname{tr}(E_P E^N)\), and the sector-projector property replaces \(E_P\) by \(\mu E_\alpha \).

By Theorem 16.4.54 the nonzero compression has positive trace, which Theorem 16.4.62 identifies with \(\mu _{\alpha ,k}d_\alpha ^{(N+1)}\). In particular \(d_\alpha ^{(N+1)}\ne 0\), so the trace of the compression by the sector projector of \((\alpha ,0)\) at the same length equals \(\mu _{\alpha ,0}d_\alpha ^{(N+1)}\ne 0\); that compression is therefore also nonzero, and its trace is a positive real. Dividing by \(\mu _{\alpha ,0}{\gt}0\) gives \(d_\alpha ^{(N+1)}{\gt}0\), and dividing the first inequality by \(d_\alpha ^{(N+1)}\) gives \(\mu _{\alpha ,k}{\gt}0\).

Divide the Gram identity by \(\omega \).

Commutation with the matrices of \(M_\alpha \) extends to all word evaluations, and after blocking these span the full matrix algebra, so the commuting matrix is central and hence scalar. The Gram matrix \(X^\dagger X\) is positive semidefinite and nonzero, so its scalar value is a positive real \(\omega \); dividing \(X\) by \(\sqrt{\omega }\) makes the Gram matrix the identity.

Compress the first site of the positive operator by \(V\). The resulting matrix is Hermitian. Expanding one matrix entry gives the marked chain on the left and the complex conjugate of the entry with \(r,s\) and the ket–bra words exchanged. Taking the conjugate transpose reverses the tail product, cyclicity of trace returns the marked factor to the first position, and positivity of \(c\) permits cancellation without a phase.

Apply the reflected marked-chain identity to the similar tensor \(XA^vX^{-1}\). Conjugate its two open virtual indices by \(X^\dagger \) and \((X^{-1})^\dagger \). On the original side this gives the Gram dressing of \(A\); on the reflected side the gauge factors cancel, leaving the reflected adjoint of \(A\).

Conjugate the open indices in the reflected marked-chain identity for the first corner by \(X^\dagger \). Its left side becomes the marked chain of \((X^\dagger X)A^v(X^\dagger X)^{-1}\), while the gauge factors on the reflected side cancel and leave the marked chain of \(A^\sharp \). The same calculation with \(Y\) gives the identical reflected chain. Thus the two Gram-dressed marks have equal closed chains for every tail word. The reflected identity also holds for the empty tail, but marked separation uses only nonempty tails, as in the proof of Lemma L in [ 13 , Appendix C.3, lines 1848–1858 ] .

These marks lie in the physical-letter span. Indeed, with \(L_X=V_XX\) and \(R_X=V_X(X^{-1})^\dagger \), their coefficients are \(f^X_{(r,s),(i,j)} =c_X^{-1}\overline{(L_X)_{ir}}(R_X)_{js}\), and similarly for \(Y\). The corner identities show that these linear combinations are precisely the horizontal slices of the two Gram-dressed tensors. Physical-letter marked separation therefore identifies every slice, hence every vertical letter.

Identify the distinguished bond space with that of copy \(q\) and set \(W_{\alpha ,q}=V_{\alpha ,0}\). Since the distinguished gauge is the identity, its corner equation reduces to

yielding (47) because \(c_{\alpha ,0}{\gt}0\).

Theorem 16.4.70 transports the distinguished identity-gauge corner to the bond space of copy \(q\). The grouped-corner Gram-dressing property gives

The left side is \(A_{\alpha ,q}^v\) because \(X_{\alpha ,0}=\mathbb {1}\). This gives (48).

Multiply the dressed-target identity by \(X^\dagger \) on the left and by \((X^{-1})^\dagger \) on the right. The left side becomes \((X^\dagger X)B(X^\dagger X)^{-1}\) and the right side collapses to \(C\). Eliminating the common target for two gauges and multiplying by the inverse Gram matrix gives the relative commutation relation.

By Theorem 16.4.72, the ratio \((Y^\dagger Y)^{-1}(X^\dagger X)\) commutes with every letter of \(M_\alpha \). Theorem 16.4.65 makes this ratio scalar. When the matrix dimension is positive, positive definiteness of the two Gram matrices makes the scalar a positive real number \(\omega \). In dimension zero the equality is vacuous and one may choose \(\omega =1\). Conjugating the relative Gram identity by \(Y^{-1}\) shows that \(XY^{-1}\) has Gram matrix \(\omega \mathbb {1}\); division by \(\sqrt{\omega }\) makes it unitary.

Theorem 16.4.71 shows that conjugation by \(X_{\alpha ,q}^\dagger X_{\alpha ,q}\) fixes every letter of the normal representative. Apply the identity-gauge specialization of Theorem 16.4.73; it gives a real number \(\omega _{\alpha ,q}{\gt}0\) with \(X_{\alpha ,q}^\dagger X_{\alpha ,q}=\omega _{\alpha ,q}\mathbb {1}\). Dividing each gauge by the positive square root of its Gram scalar gives the unitary normalization.

After identifying the copy bond space with the representative bond space, define

Since \(\omega _{\alpha ,q}{\gt}0\), the Gram identity implies

Hence \(W_{\alpha ,q}^\dagger W_{\alpha ,q}=\mathbb {1}\), and inserting the two normalized gauges on either side of the original orthogonality equation preserves pairwise orthogonality. The original intertwining identity gives

The same Gram identity and the positive square-root normalization give the corner equality

Multiplying (56) on the left by \(V_{\alpha ,q}\) and on the right by \(V_{\alpha ,q}^\dagger \) gives

Summing these physical-space identities over all copies and classes, then applying the original grouped reconstruction, proves (55).

Theorem 16.4.74 supplies the positive Gram scalars. Apply Theorem 16.4.76 to the phase-class grouping. Its corner identity preserves the grouped reconstruction and gives (58).

At each site value \(i\), the diagonal tensor agrees with the doubled-index tensor on the diagonal pair, \(M^{ii}=\widehat{M}^{\, (i,i)}\). The matrix product vector is multiplicative over sites, so replacing each factor \(M^{\sigma _k\sigma _k}\) by \(\widehat{M}^{\, (\sigma _k,\sigma _k)}\) reindexes the configuration from \(\sigma \) to \(k\mapsto (\sigma _k,\sigma _k)\), which is the stated equality.

By Lemma 16.4.78, the diagonal matrix product vector equals that of \(\widehat{M}\) on the diagonal-paired configuration, which by hypothesis equals that of the block-diagonal direct sum there. Expanding the direct sum as the weighted sum \(\sum _k\mu _k^NV^{(N)}(A_k)\) and restricting each block to the diagonal pair gives \(\sum _k\mu _k^NV^{(N)}(A_k^{\mathrm{diag}})\), the matrix product vector of the single-spin block-diagonal direct sum.

The four nonzero scalar multiples of the matrix units span the full matrix algebra. The identity \((3/5)^2+(4/5)^2=1\) gives left-canonicality. After diagonal restriction, \(E_{ii}E_{jj}=\delta _{ij}E_{ii}\). Hence every word is either zero or a scalar multiple of some \(E_{ii}\), and therefore has vanishing \((0,1)\) entry. Since \(E_{01}\) does not, it lies outside the span at every blocking length.

By induction on the word.

Word evaluation of the diagonal tensor multiplies the matrices \(M^{\sigma _k\sigma _k}\), which is exactly the operator word evaluation with equal ket and bra indices; taking the trace gives the diagonal entry.

By Lemma 16.4.82, the value is a diagonal entry of \(\rho ^{(N)}(M)\), and the diagonal entries of a positive semidefinite matrix are non-negative.

The vertical direct-sum tensor is the block-diagonal direct sum over the single flattened index \(q\), with flattened weights \(\widetilde\omega _q\) and blocks \(\widetilde A_q\). By the block matrix-product-vector decomposition (Theorem 2.6.9),

The flattening identifies \(q\) with the pair \((\alpha ,j)\) via the bijection

under which \(\widetilde\omega _q=\omega _{\alpha j}\) and \(\widetilde A_q=A_\alpha \). Reindexing the sum along this bijection yields the double sum over \((\alpha ,j)\).

For \(N{\gt}0\) and every configuration \(\sigma \), the two sides expand as

The summand indexed by \(k\) in the first sum equals the summand indexed by \(e(k)\) in the second, so reindexing by \(e\) proves the equality.

The left-invariance hypothesis gives \(P_1H^{(N)}=P_1H^{(N)}P_1\) at every positive length. Since \(H^{(N)}\) is positive and \(P=P^\dagger \),

The matrix coefficients of (62) give agreement of the corresponding first-site actions. In particular,

The original-coordinate marked Lemma L gives equality of their inserted tensors on the original bond space. Reading this equality as a vertical one-site action yields

Translate the left-invariance identity into the one-site ket and bra actions of the original tensor. The normalized BNT-refined version of Lemma L gives the opposite one-sided identity. Translating back to the vertically viewed tensor proves the claim.

Apply the irreducible-corner decomposition under invariant-projector closure to \(\widetilde M\).

For the isometries of Theorem 16.4.88, set \(P_k=V_kV_k^\dagger \). The isometry, orthogonality, completeness, and intertwining identities give the four assertions directly.

Retain exactly the irreducible corners containing a nonzero letter. The Perron–Frobenius eigenvalue of each retained corner is positive, and the absence of nontrivial periodic vectors makes the inverse-square-root rescaling normal. Put the removed square root into \(\nu _k\) and reindex the retained corners. The original complete reducing decomposition expresses each vertical letter as the sum of all compressed corners; every omitted summand vanishes, which gives (65). Thus the retained range projections need not sum to the identity, but their orthogonal complement is a zero sector.

Theorem 16.4.86 gives invariant-projector closure for \(\widetilde M\), and Theorem 16.4.49 excludes its nontrivial periodic vectors. Applying Theorem 7.7.5 then gives normal nonzero corners with positive spectral weights and orthogonal isometries, together with the stated intertwining and compression formulas. Their reconstruction is

Take one representative from each phase class. Theorem 8.2.8 gives equality of bond dimensions and (69) for every copy. Distinct representatives are gauge-phase separated by construction, so Theorem 8.2.13 gives eventual linear independence. The phase-class regrouping of the finite sum gives the BNT expansion. The representative copy is assigned the identity gauge and phase one, while the self-overlap normalization of normal tensors shows that every phase has unit modulus. Substitution into the two intertwining identities, the compression identity, and the reconstruction (65) proves all remaining algebraic assertions without altering any physical isometry. Cyclicity of trace gives the sector-loop identities. The nonzero-corner separation theorem supplies a nonzero compression for every sector, and the sector trace identity and MPDO positivity then make every grouped coefficient positive.

Take one representative from each phase class. Theorem 8.2.8 gives equality of bond dimensions and (71) for every copy. Distinct representatives are gauge-phase separated by construction, so Theorem 8.2.13 gives eventual linear independence. The phase-class regrouping of the finite sum gives the BNT expansion. The representative copy is assigned the identity gauge and phase one, while the self-overlap normalization of normal tensors shows that every phase has unit modulus. Substitution into the two intertwining identities, the compression identity, and the reconstruction (66) proves all remaining algebraic assertions without altering any physical isometry. Cyclicity of trace gives the sector-loop identities. The nonzero-corner separation theorem supplies a nonzero compression for every sector, and the sector trace identity and MPDO positivity then make every grouped coefficient positive.

The \((k,l)\) block of \(UU^\dagger \) is \(W_k^\dagger W_l\), so orthonormality yields (73). The corresponding block of \(UT_vU^\dagger \) is \(W_k^\dagger T_vW_l=W_k^\dagger W_lB_l^v\), which equals \(B_k^v\) for \(k=l\) and zero otherwise, yielding (74). Finally, multiplication of the block diagonal matrix by the block row gives the sum \(\sum _kW_kB_k^vW_k^\dagger \), which is \(T_v\) by hypothesis. The row indices of \(U\) form the dependent disjoint union \(\coprod _k\{ 0,\ldots ,D_k-1\} \). Its canonical order-preserving bijection with \(\{ 0,\ldots ,\sum _kD_k-1\} \) gives the coordinate reindexing required by the vertical canonical-form definition.

Form the block-row map \(U\) from the adjoints \(W_{\alpha ,q}^\dagger \). Theorem 16.4.96 gives

Identifying the dependent pairs \((\alpha ,q)\) and their bond indices with the standard coordinates of the two finite direct sums gives the required matrix dimensions. Positivity of the multiplicities and weights, together with the BNT hypothesis, supplies the remaining conditions of vertical canonical form.

The unit-modulus weight witness in the normalized BNT-refined horizontal form, together with left canonicity of its representative, makes the doubled-index tensor nonzero. Hence its vertical reshaping \(\widetilde M\) is nonzero. If the retained decomposition had no sector, (70) would make every letter of \(\widetilde M\) zero, a contradiction. Passing from the nonempty finite set of retained sectors to its phase partition gives a retained representative.

Write the doubled-index tensor in normalized BNT-refined horizontal form as \(\widehat M=GSG^{-1}\), where \(S\) is the weighted direct sum of its basis tensors. Choose a copy whose weight has unit modulus. Its basis tensor cannot have every letter equal to zero, since left canonicity gives \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\). The corresponding nonzero weight and nonzero letter give \(S\ne 0\), hence \(\widehat M\ne 0\). If \(\widetilde M=0\), the bijection between its vertical index and the two horizontal bond indices would give \(\widehat M=0\), a contradiction.

Put \(C_k^v=\mu _kB_k^v\). Induction on a word \(w\) gives \(V_k^\dagger T^w=C_k^wV_k^\dagger \). For a nonempty word \(w=(v,w')\), reconstruction of its first letter, cyclicity of trace, and \(V_k^\dagger V_k=\mathbb {1}\) therefore give

Thus \(T\) and \(A_{\mathrm{ret}}\) have the same matrix product vectors at every positive length. The spectral BNT hypothesis gives an algebraic BNT for \(A_{\mathrm{ret}}\) by Theorem 8.2.5; in particular, it supplies the required positive bond dimensions. Lemma E.1.5 then gives the same algebraic BNT for \(T\).

For \(Z=X,Y\), set

The corner identity identifies the mark with coefficients \(f^Z\) with the horizontal slice of \((Z^\dagger Z)A^v(Z^\dagger Z)^{-1}\). Self-adjointness of the two corner compressions identifies both closed marked chains with the same reflected chain. The original-coordinate marked Lemma L for literal canonical form therefore gives equality of the two marks, hence (76).

Equation (76) makes the ratio of the two Gram matrices commute with every letter of \(A\). Normality makes this ratio scalar, and positive definiteness makes the scalar a positive real number.

Theorem 16.4.74 gives \(X_{\alpha ,q}^\dagger X_{\alpha ,q}=\omega _{\alpha ,q}\mathbb {1}\) with \(\omega _{\alpha ,q}{\gt}0\). Absorb \(\omega _{\alpha ,q}^{-1/2}X_{\alpha ,q}\) into the physical corner maps. The normalized maps are isometries with orthogonal ranges, intertwine \(\widetilde M\) with the undressed representatives, and preserve the exact reconstruction. The grouped representatives form a BNT, so Theorem 16.4.97 gives the vertical canonical form.

Theorem 16.4.94 supplies the grouped vertical BNT decomposition with physical isometries. The literal pairwise Figure 8 theorem supplies the grouped-corner Gram-dressing property. Apply Theorem 16.4.103.

Theorem 16.4.93 supplies the grouped vertical decomposition. The pairwise Figure 8 theorem supplies its grouped-corner Gram-dressing property. Apply Theorem 16.4.103.

Commutation with each factor implies commutation with their product.

Each word evaluates to the scalar \(\prod _l u_{\sigma _l\tau _l}\) times the one-by-one identity, whose trace is that scalar.

From \(O^\dagger \rho =\mu \rho \) one gets \(\rho O=\bar\mu \rho \). Then \(\rho O\rho \) equals both \(\lambda \rho ^2\) and \(\bar\mu \rho ^2\). If \(\rho \neq 0\) then \(\rho ^2\neq 0\), so \(\lambda =\bar\mu \) and \(O\rho =\rho O\). For unitary \(O\), the relation \(\rho =O^\dagger O\rho =\lambda O^\dagger \rho \) shows that \(O^\dagger \) is a strong symmetry.

For unitary \(O\), \(\rho ^\dagger \rho =(O\rho )^\dagger O\rho =|\lambda |^2\rho ^\dagger \rho \) with \(\rho ^\dagger \rho \neq 0\). The fusion relation follows by applying \(O_aO_b\) to \(\rho ^{(L)}\) in both forms.

If \(O\rho =\lambda \rho \), the matrix \((O-\lambda )\Phi \) has Gram matrix \((O-\lambda )\rho (O-\lambda )^\dagger =0\), so it vanishes. The converse is immediate.

The twisted purifying tensor is a gauge transform of \(A\) up to the scalar \(\zeta _g\), so the purifying state is multiplied by \(\zeta _g^L\). For the weak condition, \(U_g^{\otimes L}\rho U_g^{\otimes L\dagger } =|\zeta _g|^{2L}\rho \), and comparing traces gives \(|\zeta _g|^{2L}=1\) whenever \(\rho \neq 0\).

Let \(B_g\) be the twisted tensor with entries \(B_g^{(i,k)}=\sum _j(U_g)_{ij}A^{(j,k)}\). Acting with a unitary on the spin leg leaves the transfer map unchanged and preserves irreducibility, so \(B_g\) is again normal. By Theorem 16.5.8 there are scalars \(c_g(N)\) with

for every \(N{\gt}0\). Normality of \(B_g\) gives \(O_{B_gB_g}(N)\to 1\), so \(O_{B_gB_g}(N)\neq 0\) for all large \(N\), and then \(c_g(N)\neq 0\). Theorems 8.2.12 and 8.2.8 turn this proportionality into the gauge relation. The converse is Theorem 16.5.9.

The boundary-weighted vector of the doubled tensor at the configuration \(((\sigma _1,\tau _1),\dots ,(\sigma _L,\tau _L))\) is the entry \(\rho ^{(L)}_{\sigma \tau }\). Expanding the trace of a product of linear combinations gives the twisted operator. For unitary \(u\), \([u,\rho ]=0\) is equivalent to \(u\rho u^\dagger =\rho \), and \((\bar u^{\otimes L})^T=(u^{\otimes L})^\dagger \).

The operators are diagonal with entries \(1+s(\boldsymbol \sigma )\), where \(s(\boldsymbol \sigma )=\pm 1\) is the eigenvalue of \(\sigma _z^{\otimes N}\). Conjugation by \(\sigma _x^{\otimes N}\) flips every spin and multiplies \(s\) by \((-1)^N\).

On the support of \(\rho ^{(N)}(R)\) the second qubit of each site equals the first qubit of the next site, so the product of the \(\sigma _z\) signs is a square. Flipping every qubit preserves the support and the weight matrix. At \(N=1\) the entries of \(\rho ^{(1)}\) on the two supported configurations are in the ratio \(16:9\), and those of the flipped operator in the ratio \(9:16\).

On the support of \(\rho ^{(N)}(T)\) the right qubit of each site equals the left qubit of the next, so the product of the \(Z_\ell Z_r\) signs is a square. Flipping the left and right qubits preserves the support and exchanges the two bond matrices, and the flag sign exchanges the two flag signs, which gives the second strong symmetry. The one-site coefficients are diagonal in the flag qubit, which gives the weak \(Z_f\) symmetry. At \(N=1\) the two flag sectors carry entries of opposite sign under both weak symmetries, so neither is strong.