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 23.1.4 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 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 23.2.2, \(\mathcal{E}_M\) 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 10.5.3 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 23.4.13 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 10.5.3 to the nonzero weights above each fixed representative, exactly as in Theorem 23.4.11. This selects a sufficiently large \(L\) with \(c_j^{(L+1)}\neq 0\) and proves \(C_j=E_j\).

Split the finite sum over listed coordinates into active and inactive parts. The inactive marked blocks vanish. Reindex the active part by the class-copy pairs \((j,q)\) and use the bond-dimension identifications to replace each active 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 (??).

Theorem 23.4.15 rewrites (??) into marked-trace agreement for \(P_{\mathrm{act}}\). Its representatives have eventual word-tuple separation, and every copy weight is nonzero. Theorem 23.4.14 therefore gives \(C_j=E_j\).

First-site agreement for \(T_{\mathrm{grp}}\) also holds for \(P_{\mathrm{act}}\) because their matrix product vectors agree at every positive length. Applying Theorem 23.4.11 gives (??).

Induction on \(w\) in (??) gives \(CA^w=V^\dagger EB^wV\). Cyclicity of trace and \(VV^\dagger =\mathbb {1}\) give (??). Linearity of the physical mark gives the grouped marked tensor, with inactive blocks killed by their zero weights.

Apply (??) to both sides of (??), remove the block gauge by cyclicity of trace, and use Theorem 23.4.16 on the active representatives. The resulting equality of the full grouped marked tensors, followed by (??), 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 (??) 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 (??) 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 verbatim 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 23.4.19 recovers \(YA_j=ZA_j\) before blocking.

The BNT canonical-form hypotheses give eventual representative word-tuple separation. Apply Theorem 23.4.14.

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 23.4.21 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 23.4.17 identifies the two linear marks and gives (??). 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 [ CPGSV16 , Appendix C.3, lines 1835–1846 ] , using the finite criterion from Theorem 23.4.8 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.

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 (??).

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 23.4.28 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 23.4.26 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 23.4.25 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 23.4.32 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 23.4.30, whose conclusion is the equality of the insertions of \(MP\) and \(PMP\) on every minimal representative.

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 23.4.34), contradicting \(Q_1H^{(N+1)}\ne H^{(N+1)}Q_1\). This is the contradiction concluding the periodic-sector step in the proof of [ CPGSV16 , Proposition 4.13 ] , lines 1892–1893.

A nontrivial \(p\)-periodic vector of \(\widetilde M\) supplies a periodic-sector projector for \(M\), and Theorem 23.4.36 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 23.4.41 the word invariance gives \(QM^{[p]}=QM^{[p]}Q\) on the stacked tensor, which generates matrix product density operators by Theorem 23.4.40. Theorem 23.4.32 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 23.4.35.

A nontrivial \(p\)-periodic vector supplies an invariant non-commuting projector, which is a periodic-sector projector by Definition 23.4.42. The exclusion of periodic vectors is then Theorem 23.4.38.

The rescaled unital gauge \(K\) of the corner satisfies the hypotheses of the cyclic decomposition of the peripheral spectrum (Theorem E.3.5), and the eigenvalue \(\mu /r\ne 1\) of modulus one (Lemma E.3.6) forces the peripheral cyclic group to have order \(m\ge 2\). Theorem 7.9.1.1 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 7.9.1.2). 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 E.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 23.4.23 identifies the two inserted tensors on the original bond space. By Lemma 23.4.29, 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 23.4.45). By Theorem 23.4.46, 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 23.4.34 implies \(Q_1H^{(N+1)}=H^{(N+1)}Q_1\), a contradiction. This is the periodic exclusion in [ CPGSV16 , 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 23.4.23 and Lemma 23.4.29 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 23.4.45, 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 23.4.34 imply that \(Q_1\) commutes with \(H^{(N+1)}\), which is the required contradiction. This uses the pointwise contradiction in [ CPGSV16 , Proposition 4.13, lines 1888–1893 ] ; the stronger all-length inequality printed there is unnecessary.

A nontrivial \(p\)-periodic vector supplies the invariant, displaced projector of Theorem 23.4.45; the non-commutation hypothesis upgrades the displacement to the all-length non-commutation family, so periodic vectors supply cyclic projectors and Theorem 23.4.44 applies.

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 23.4.45. 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 23.4.51 turns the resulting commutation with \(H^{(2)}\) into the letter-level invariance that the displacement forbids.

Theorem 23.4.48 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 23.4.53, 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 23.4.56. 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 (??).

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

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 23.4.57 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 23.4.54 the nonzero compression has positive trace, which Theorem 23.4.62 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 \((\alpha ,1)\) at the same length equals \(\mu _{\alpha ,1}d_\alpha ^{(N+1)}\ne 0\); that compression is therefore also nonzero, and its trace is a positive real. Dividing by \(\mu _{\alpha ,1}{\gt}0\) gives \(d_\alpha ^{(N+1)}{\gt}0\), and dividing the first inequality by \(d_\alpha ^{(N+1)}\) gives \(\mu _{\alpha ,k}{\gt}0\).

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.

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.

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.

Transport the distinguished corner of the phase class along the equality of its bond dimension with that of the copy \(q\). Its distinguished gauge is the identity. Apply Theorem 23.4.66 to this corner and the corner indexed by \(q\). The Gram conjugation belonging to the distinguished corner is the identity, which gives the stated equality.

The distinguished copy has identity gauge. Use the equality of bond dimensions to regard its positive corner equation on the bond space of the copy \(q\). The literal pairwise Figure 8 theorem compares the two corners. Its distinguished Gram conjugation is the identity, which gives (??).

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 23.4.69, the ratio \((Y^\dagger Y)^{-1}(X^\dagger X)\) commutes with every letter of \(M_\alpha \). Theorem 23.4.64 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.

Transport the normal representative to the bond space of the chosen copy. Theorem 23.4.67 states that conjugation by \(X_{\alpha ,k}^\dagger X_{\alpha ,k}\) fixes every letter of the transported tensor. Apply Theorem 23.4.70 with the second gauge equal to the identity. The resulting Gram identity has a positive real scalar, and division by its positive square root gives the stated unitary.

The representative on this bond space is normal. Apply normal Gram rigidity to (??) with the distinguished gauge equal to the identity. This gives (??) with \(\omega _{\alpha ,q}{\gt}0\). Dividing by its positive square root gives a unitary matrix.

Each \(Q_{\alpha ,q}\) is unitary, so right multiplication by it preserves the isometry equation and the pairwise orthogonality of the physical sector maps. In the intertwining relation, the scalar normalization commutes with every letter and the adjacent factors \(X_{\alpha ,q}^{-1}X_{\alpha ,q}\) cancel. The Gram identity gives \(X_{\alpha ,q}^\dagger =\omega _{\alpha ,q}X_{\alpha ,q}^{-1}\); hence

Substitution into every summand of the grouped reconstruction proves the final identity. Transport along the equality of bond dimensions changes only the coordinate labels.

For each copy, put \(Q_{\alpha ,q}=\omega _{\alpha ,q}^{-1/2}X_{\alpha ,q}\) and \(W_{\alpha ,q}=V_{\alpha ,q}Q_{\alpha ,q}\). Unitarity of \(Q_{\alpha ,q}\) preserves the isometry equation and pairwise orthogonality. Substituting \(W_{\alpha ,q}=V_{\alpha ,q}Q_{\alpha ,q}\) into the intertwining identity gives

Thus the adjacent factors \(X_{\alpha ,q}^{-1}X_{\alpha ,q}\) cancel and leave the undressed representative \(M_\alpha ^v\). The Gram identity \(X_{\alpha ,q}^\dagger X_{\alpha ,q} =\omega _{\alpha ,q}\mathbb {1}\) also gives

Hence conjugation by \(W_{\alpha ,q}\) reproduces the original grouped corner. Summing these equalities gives (??).

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 23.4.75, 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 23.4.79, 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 2.5.9),

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 (??) give agreement of the corresponding first-site actions. 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 23.4.85, 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 (??). Thus the retained range projections need not sum to the identity, but their orthogonal complement is a zero sector.

Theorem 23.4.83 gives invariant-projector closure for \(\widetilde M\), and Theorem 23.4.47 excludes its nontrivial periodic vectors. Applying Theorem 9.7.4 then gives normal nonzero corners with positive spectral weights and orthogonal isometries, together with the stated intertwining, compression, and reconstruction formulas.

Take one representative from each phase class. Theorem 10.2.6 gives equality of bond dimensions and (??) for every copy. Distinct representatives are gauge-phase separated by construction, so Theorem 10.2.11 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 (??) 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 10.2.6 gives equality of bond dimensions and (??) for every copy. Distinct representatives are gauge-phase separated by construction, so Theorem 10.2.11 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 (??) 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 \(W_k^\dagger W_l\), hence the first identity follows from orthonormality. The corresponding block of \(UT_vU^\dagger \) is \(W_k^\dagger T_vW_l=W_k^\dagger W_lB_l^v\), which is \(B_k^v\) for \(k=l\) and zero otherwise. 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 23.4.92 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, (??) 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 10.2.3; in particular, it supplies the required positive bond dimensions. Lemma F.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 (??).

Equation (??) 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.

Write \(A\) for the doubled-index MPS tensor of \(M\). If \(A=0\), take no retained sectors; the unique map onto the zero-dimensional sector space gives the conclusion. Suppose henceforth that \(A\ne 0\). Apply Theorem 10.6.1.9 to its literal canonical-form data. The active listed indices are identified with pairs \((\alpha ,k)\) consisting of a phase class and a copy, and every active copy retains its original nonzero coefficient \(\nu _{\alpha ,k}\), its bond dimension, a unit phase, and an invertible gauge \(X_{\alpha ,k}\). The chosen tensors \(C_\alpha \) form a basis of normal tensors. Adjoin the inactive zero-weight indices to these pairs and call the resulting grouped index set \(\mathcal G\). There is an explicit equivalence from the dependent coordinates of \(\mathcal G\) to the original flattened listed coordinates. If \(T_{\mathrm{grp}}\) is the grouped direct sum, reindexed along this equivalence, then there are a block-diagonal gauge \(G\) and the original coisometry \(V\) such that

and

The active summands of \(T_{\mathrm{grp}}\) are \(\nu _{\alpha ,k}\zeta _{\alpha ,k}C_\alpha ^i\), with \(|\zeta _{\alpha ,k}|=1\). Every inactive summand is the original listed block with coefficient zero. Thus the grouped coordinates are exact without deleting the inactive complement or introducing a selecting coisometry.

Let \(H^{(N)}\) be the positive operator generated horizontally by \(\widetilde M\). If an orthogonal projector \(P\) on the physical space satisfies \(P\widetilde M=P\widetilde MP\), and \(P_1\) acts on the first site, then self-adjointness gives

Write \(Y_R(P)\) and \(Y_C(P)\) for the right and central first-site contractions determined by \(P\). The active representative sector decomposition has weights \(\lambda _{\alpha ,k}=\nu _{\alpha ,k}\zeta _{\alpha ,k}\), all nonzero, and its representatives have simultaneous word-tuple separation at every sufficiently large length. It has the same matrix product vectors as \(T_{\mathrm{grp}}\) at every positive length. Hence Theorem 23.4.16 gives

on every active representative. Theorem 23.4.17 then uses the listed block gauge and the ambient coisometry, with \(VV^\dagger =\mathbb {1}\), to give the original-space inserted-tensor equality. Theorem 23.4.83 therefore yields

Thus every one-sided invariant subspace is reducing. The inactive listed coordinates remain present with weight zero throughout this argument; no insertion identity is asserted for an individual inactive block.

Theorem 23.4.47 excludes nontrivial periodic vertical sectors. Its proof uses the displaced cyclic projector \(Q\) and one length \(N+1\) for which \(Q_1H^{(N+1)}\ne H^{(N+1)}Q_1\); full-period invariance gives commutation with \([H^{(N+1)}]^p\), and positivity gives the contradictory commutation with \(H^{(N+1)}\) itself.

Combining this exclusion with invariant-projector closure gives the normal corners of Theorem 23.4.88. Thus there are positive weights \(\nu _k\), normal tensors \(B_k\), and isometries \(V_k\) with pairwise orthogonal ranges such that

This completes periodic exclusion and the normal vertical block decomposition under the literal canonical-form hypothesis.

Theorem 23.4.90 groups these normal corners into actual vertical representative families over tensors \(M_\alpha \). It retains the physical isometries, both intertwining identities, exact corner compression, and the literal reconstruction. Every effective coefficient \(\mu _{\alpha ,k}=\nu _{\alpha ,k}\zeta _{\alpha ,k}\) is positive. Thus the phase-class grouping and the positivity argument of source lines 1898–1902 are complete under the literal canonical-form hypothesis.

On these actual grouped corners, Theorem 23.4.68 gives

Theorem 23.4.72 then gives a real number \(\omega _{\alpha ,k}{\gt}0\) such that

and \(\omega _{\alpha ,k}^{-1/2}X_{\alpha ,k}\) is unitary. These are the actual-grouped Figure 8 and gauge-normalization conclusions of Proposition 4.13. They are distinct from the tensor-attached two-site marked problem in Appendix C.4.

Theorem 23.4.74 absorbs these unitary bond gauges into the mutually orthogonal physical sector maps. The resulting maps intertwine \(\widetilde M\) with the undressed representatives and reconstruct every vertical letter exactly. Theorem 23.4.96 identifies the representatives as a basis of normal tensors for \(\widetilde M\). Taking the adjoints of the normalized sector maps as the block-row map \(U\), Theorem 23.4.93 gives (??), and the sector intertwinings give (??). Invariant-projector closure makes the retained support reducing, while the complement is the all-zero sector. Summing the sector reconstructions therefore gives the exact identity (??).

The preceding paragraphs recast the source proof on the active retained coordinates required by the literal canonical-form definition.

Apply Theorem 23.4.89 once and keep its representatives \(M_\alpha \), multiplicities \(r_\alpha \), exact positive weights \(\mu _{\alpha ,q}\), and physical sector maps. For these same witnesses, Theorem 23.4.96 proves that \(\{ M_\alpha \} _\alpha \) is a BNT for \(\widetilde M\).

Theorem 23.4.73 absorbs each normalized bond gauge into its physical map and gives maps \(W_{\alpha ,q}\) such that

The canonical equivalence between the dependent disjoint union \(\coprod _\alpha \{ 0,\ldots ,r_\alpha -1\} \) and its standard finite index set reindexes the nested reconstruction sum without changing any weight or sector. The hypotheses of Theorem 23.4.93 now hold, and that theorem gives the vertical canonical form of \(M\).