Twisting commutes with rescaling, \(\widetilde{(cA)}_g=c\, \widetilde A_g\), and the \(N\)-site coefficients of \(cB\) are \(c^N\) times those of \(B\). If \(B^i=XA^iX^{-1}\), then \((\widetilde B_g)^i=\sum _j U(g)_{ij}XA^jX^{-1}=X(\widetilde A_g)^iX^{-1}\), so \(\widetilde B_g\) is gauge equivalent to \(\widetilde A_g\), and Theorem 2.4.19 gives \(\mathcal{V}(B)=\mathcal{V}(A)=\mathcal{V}(\widetilde A_g)=\mathcal{V}(\widetilde B_g)\).

Expand both sides of (2) using (1) and \(U(gh)=U(g)\, U(h)\), then swap the order of summation.

Since \(U(1)=\mathbb {1}_d\), each component in (1) reduces to \(\sum _j\delta _{ij}\, A^j=A^i\).

The twisted tensor \(\widetilde A_g\) is the physical rotation of \(A\) by \(U(g)\), and \(U^{[L]}(g)=U(g)^{\otimes L}\). Hence (3) is the case \(W=U(g)\), \(C=A\) of Theorem G.1.1.3.

Equation (3) identifies the twisted blocked tensor with the blocking of the twisted tensor. Equality of MPV families is preserved by blocking, so the defining equality for \(A\) yields the defining equality for \(A^{[L]}\).

By (3) the left side is a product of \(L\) matrices \((\widetilde A_g)^{i_k}=R\, A^{i_k}R^{-1}\), and the inner factors \(R^{-1}R\) cancel.

On-site symmetry gives \(\mathcal{V}(A)=\mathcal{V}(\widetilde{A}_g)\), and the single-block Fundamental Theorem turns this equality of MPV families into gauge equivalence.

The physical action is transferred to the virtual level by the same local move that replaces the twisted tensor by a gauge-conjugated one:

\(\qquad \leadsto \qquad \) .

Apply Lemma 10.1.10 to obtain \(X(g)\), then set \(\varphi (g)=1\) in (4).

The matrix \(Z=Y^{-1}X\) commutes with every \(A^i\). Since \(A\) is injective, \(\{ A^i\} \) spans \(M_{D}(\mathbb {C})\), so \(Z\) lies in the centre of \(M_{D}(\mathbb {C})\). By Lemma 2.1.5, \(Z\) is a scalar matrix. Since \(Z\) is invertible, that scalar is nonzero, and \(Y=u\cdot X\).

The matrix \(Z=YX^{-1}\) satisfies \(ZA^i=c\, A^iZ\) for every \(i\), hence \(ZM=c\, MZ\) for every matrix \(M\) by linearity. Taking \(M=\mathbb {1}\) gives \((1-c)Z=0\), so \(c=1\) because \(Z\) is invertible. Then \(Z\) commutes with \(M_{D}(\mathbb {C})\), so it is a nonzero scalar by Lemma 2.1.5.

By induction on \(w\): inserting \(XX^{-1}\) and \(YY^{-1}\) between the letters gives \(X^{-1}A^{iw}X=(X^{-1}A^iX)(X^{-1}A^wX)\), and likewise for \(Y\).

By Lemma 10.1.15 the relation propagates to words of length \(N\) with the scalar \(c^N\). For an injective blocking length \(N\), Lemma 10.1.14 applied to the blocked tensor gives \(X=u\, Y\). Substituting back, \((1-c)\, Y^{-1}A^iY=0\) for every \(i\), and some \(A^i\) is nonzero because the words of length \(N\) span the full matrix algebra; hence \(c=1\).

Expand \(\rho (g)\, (\rho (h)\rho (k))\) and \((\rho (g)\rho (h))\rho (k)\) using the projective multiplication law (6) and equate the results via matrix associativity. The hypothesis \(D {\gt} 0\) allows cancellation of the invertible matrix factor, giving (7).

The twist composition law makes \(X_hX_g\) and \(X_{gh}\) implement the same twist. Gauge uniqueness gives a nonzero scalar relating them. Reindex by inversion to obtain the projective multiplication law.

Since \(A\) is injective and on-site symmetric, each twisted tensor \(\widetilde{A}_g\) is gauge equivalent to \(A\) (Lemma 10.1.10). Choose \(X(g) \in \mathrm{GL}_{D}(\mathbb {C})\) satisfying (8) for all \(i\).

By functoriality (Lemma 10.1.4), the product \(X(h)\, X(g)\) also intertwines \(A\) with \(\widetilde{A}_{gh}\). Since \(X(gh)\) does the same, gauge uniqueness (Lemma 10.1.13) gives a nonzero scalar \(\omega '(h,g)\) such that

Defining \(\rho (g):=X(g^{-1})\) converts (9) to the standard law (6), where \(\omega (g,h):=\omega '(g^{-1},h^{-1})\).

Theorem 10.1.21 gives \(\rho \) and \(\omega \). By Lemma 10.1.19, the cocycle identity (7) follows from associativity of matrix multiplication and invertibility of the representation matrices.

Gauge uniqueness gives nonzero scalars \(f(k)\) with \(X_2(k)=f(k)\, X_1(k)\) for every physical twist \(k\). Since the projective representation is \(\rho _a(g)=X_a(g^{-1})\), setting \(k=g^{-1}\) gives \(\rho _2(g)=f(g^{-1})\rho _1(g)\). Equivalently, for each fixed \(g\) the two local gauges

  and   

describe the same twisted tensor, so they differ by a scalar. Set \(\alpha (g):=f(g^{-1})\). Substituting \(\rho _2(g)=\alpha (g)\rho _1(g)\) into the projective law (6) and cancelling the invertible matrix \(X_1((gh)^{-1})=\rho _1(gh)\) yields

Thus \(\omega _2\) is the coboundary multiple of \(\omega _1\) with cochain \(\alpha \). This is precisely the orientation required by (11) for the stated relation \(\omega _2\sim \omega _1\). Equivalently,

so the inverse cochain also gives the symmetric conclusion \(\omega _1\sim \omega _2\).

Associativity gives the scalar identity of Lemma 10.1.19. Because the inclusion \(\mathbb {C}^\times \hookrightarrow \mathbb {C}\) is injective, the same identity holds in \(\mathbb {C}^\times \), which is precisely Definition 10.2.4.

This follows from the definition of projective equivalence as cohomology-class equality for factor systems.

Write \(\omega _1(g,h)=\varphi (g)\varphi (h)\varphi (gh)^{-1}\omega _2(g,h)\) for a coboundary witness \(\varphi \). Substituting into the ratio and using \(gh=hg\) and the commutativity of \(\mathbb {C}^{\times }\) gives

where the \(\varphi \)-factors cancel since \(\varphi (gh)=\varphi (hg)\).

The trivial cocycle has commutator phase \(1\) at every pair, so by Theorem 10.2.1.2 and (15), any cocycle cohomologous to it has \(\beta =1\) at every commuting pair. Contrapositively, \(\beta _\omega (g,h)\neq 1\) for one commuting pair forces \(\omega \) to be non-cohomologous to the trivial cocycle.

The one-site insertion is \(\mathcal{E}_u\). Splitting the last physical factor and inducting on \(L\) gives \(\mathcal{E}_{u^{\otimes L}}=\mathcal{E}_u^L\), including \(L=0\). Expanding the physical numerator and squared norm as endomorphism traces gives (20).

The transfer powers converge pointwise to \(P(X)=\operatorname{tr}(\Lambda X)\mathbb {1}\). Finite dimensionality gives convergence of their traces, and the trace of the rank-one map \(P\) is \(\operatorname{tr}\Lambda =1\). Thus \(Z_L=\operatorname {Tr}(\mathcal{E}_A^L)\to 1\).

Put \(F=\mathcal{E}_O\) and \(P(X)=\operatorname{tr}(\Lambda X)\mathbb {1}\). The exact numerator is \(\operatorname {Tr}(F\mathcal{E}_A^n)\). Pointwise convergence and finite dimensionality give \(\operatorname {Tr}(F\mathcal{E}_A^n)\to \operatorname {Tr}(FP) =\operatorname{tr}[\Lambda F(\mathbb {1})]\). The denominator tends to one by Theorem 10.3.1.2. Tensor-product insertions give the stated physical string specializations.

The exact iterate identity is \(\mathcal{E}_u^L=\mu ^L L_V\mathcal{E}_A^L L_{V^\dagger }\), where \(L_V(X)=VX\). Since \(L_{V^\dagger }L_V=\mathbb {1}\), cyclicity of the endomorphism trace gives the numerator identity. A nonzero periodic vector has \(Z_L\ne 0\), so (20) gives (22).

Below unit spectral radius, the twisted transfer-power traces tend to zero, while \(Z_L\to 1\). At unit radius the intertwiner gives (22) on the eventual nonzero tail. If \(\mu ^L\to z\), then \(|z|=1\) and comparison with the one-step shift gives \(z\mu =z\), forcing \(\mu =1\).

Let \(\mathcal{E}_u(V)=\lambda V\). Multiplication by \(\Lambda V^\dagger \), the trace, and Cauchy–Schwarz give

Faithfulness of \(\Lambda \) yields \(|\lambda |\le 1\). Equality in Cauchy–Schwarz is equivalent to (23); conversely that relation makes \(e^{i\theta }\) an eigenvalue. It also gives \(\mathcal{E}_A(V^\dagger V)=V^\dagger V\). The fixed space of the pure canonical transfer map is one-dimensional, so \(V^\dagger V=c\mathbb {1}\) for some \(c{\gt}0\). Replacing \(V\) by \(c^{-1/2}V\) makes it unitary without changing the intertwining relation.

If \(V,V'\) are peripheral eigenmatrices with eigenvalues \(e^{i\theta },e^{i\theta '}\), respectively, the equality relations give the exact identity

Purity leaves only the peripheral eigenvalue \(1\), hence \(\theta '=\theta \) modulo \(2\pi \). The one-dimensional fixed space then gives \(V^\dagger V'\in \mathbb {C}\mathbb {1}\), proving uniqueness of the peripheral eigenvalue and its eigendirection.

At a fixed unitary \(u\), physical endpoints have positive limiting string magnitude exactly when there are \(V\), \(|\mu |=1\), and \(n,m\) with \(\mathcal{E}_u(V)=\mu V\) and a nonzero displayed trace coefficient. Positive limiting magnitude excludes spectral radius below one; peripheral rigidity and the physical endpoint selection rule give this equivalence. The trace coefficient also implies \(V\ne 0\). Set \(u_0=\mu ^{-1}u\). This is unitary and remains nonscalar, since \(u_0=c\mathbb {1}\) would imply \(u=(\mu c)\mathbb {1}\). Its eigenmatrix is fixed. Conversely a fixed eigenmatrix is a peripheral eigenmatrix with \(\mu =1\), so the same endpoint criterion supplies the endpoints.

For matrices \(B,C\) indexed by the same letters, the entry \((s,t)\) of \(P_BP_C^\dagger \) is \(\sum _{a,b}(B_s)_{ab}\overline{(C_t)_{ab}} =\operatorname{tr}(B_sC_t^\dagger )\). Multiplying by \(K\) on the right replaces each \(B_s\) by \(\Lambda A_s\), so the entry \((s,t)\) of \(P_NKP_N^\dagger \) is \(\operatorname{tr}(\Lambda A_sA_t^\dagger )\), which is (27).

Expand \(\operatorname{tr}(\varrho _NO)=\sum _{s,t}(\varrho _N)_{s,t}O_{t,s}\) and substitute (27). Linearity of the trace gives (28).

Positivity and the rank bound follow from \(\varrho _N=P_NKP_N^\dagger \) with \(K\geq 0\) on the \(D^2\)-dimensional virtual pair space. Its trace is \(\operatorname{tr}(\Lambda \mathcal{E}_A^N(\mathbb {1}))=1\). Summation over the rightmost letter in (27) inserts \(\sum _i A_iA_i^\dagger =\mathbb {1}\). Cyclicity of the trace makes summation over the leftmost letter insert \(\sum _iA_i^\dagger \Lambda A_i=\Lambda \).

Put \(F_N=P_N\sqrt K\), so \(\varrho _N=F_NF_N^\dagger \). The matrix \(G_N=F_N^\dagger F_N\) has dimension \(D^2\) and satisfies \(\operatorname{tr}G_N=1\) and \(\operatorname{tr}(G_N^2)=\operatorname{tr}(\varrho _N^2)\). Cauchy–Schwarz gives \((\operatorname{tr}G_N)^2\leq D^2\operatorname{tr}(G_N^2)\).

Cyclicity of the trace moves the physical factors in \(\operatorname{tr}(P_N^AK(P_N^A)^\dagger P_N^BK(P_N^B)^\dagger )\) onto the virtual pair space. Expanding the mixed transfer iterate gives (32). Since all factors in (31) have dimension \(D^2\), entrywise decay of \(G_N\) forces the scalar overlap to vanish.

Physical rotation multiplies the blocked physical matrix by \(u^{\otimes N}\). In the virtual-gauge case, \(B_s=\mu ^NVA_sV^\dagger \), so the phases cancel in \(B_sB_t^\dagger \). Cyclicity of the trace then replaces \(V^\dagger \Lambda V\) by \(\Lambda \).

By Lemma 10.3.3.1, the source purity hypothesis makes \(\mathcal{E}_A\) irreducible. If \(\rho (\mathcal{E}_u)=1\), Theorem 10.3.3.3 supplies a unitary \(V\) and a phase \(\mu \) satisfying \(B_i=\sum _j u_{ij}A_j=\mu VA_iV^\dagger \). Lemma 10.3.3.2 then gives \(V^\dagger \Lambda V=\Lambda \) for the faithful normalized stationary dual density. Theorem 10.3.2.7 now proves physical invariance.

Conversely, invariance gives \(\varrho _N(B,\Lambda )=\varrho _N(A,\Lambda )\). The mixed map is the twisted transfer map:

Hence (30) and (31) give \(D^{-2}\leq \operatorname{tr}(\varrho _N^2)=\operatorname{tr}(KG_NKG_N^\dagger )\). If \(\rho (\mathcal{E}_u){\lt}1\), its powers tend to zero, hence \(G_N\to 0\) by (32). This contradicts the positive uniform bound. The twisted spectral-radius upper bound of Theorem 10.3.3.3 therefore forces \(\rho (\mathcal{E}_u)=1\).

Both sides express the same local picture with opposite orientation:

= .

The two formulas are the same identity read from opposite sides.

Starting from

= ,

one inserts \(V^\dagger V = \mathbb {1}\) so that each local transfer cell is written in terms of conjugated Kraus operators, then replaces those operators using the intertwining relation. After that, the physical \(u\)-boxes disappear by the unitary Kraus-mixing identity.

Set \(B^i := V\, A^i\, V^\dagger \). The transfer-map covariance relation

=

implies that the two \(d\)-element Kraus families \(B\) and \(A\) define the same transfer map. Rectangular Kraus freedom (the corresponding theorem in the companion quantum-channel volume  [ 216 , “Rectangular Kraus freedom” ] ) converts \(\mathcal{E}_B = \mathcal{E}_A\) into a unitary Kraus mixing matrix \(u\) with \(B^i = \sum _j u_{ij} A^j\).

By Lemma G.4.2, the twisted transfer map is the mixed transfer operator \(F_{A,B_u}\), as in (6). Because \(B_u\) is obtained from \(A\) by unitary mixing of the Kraus operators, the transfer maps of \(A\) and \(B_u\) coincide, so the two adjoint channels share a common positive definite fixed point. Gauging both families by that fixed point puts them in a common trace-preserving gauge. Theorem 5.3.1 then applies to the mixed transfer operator in that gauge, and similarity preserves the eigenvalue \(\lambda \).

By Lemma G.4.2, the twisted transfer map is the mixed transfer operator \(F_{A,B_u}\). Because \(B_u\) is obtained from \(A\) by unitary mixing of the Kraus operators, the transfer maps of \(A\) and \(B_u\) coincide, so the two adjoint channels share a common positive definite fixed point. Gauging both families by that fixed point puts them in a common trace-preserving gauge. In that gauge, (37) becomes a modulus-one peripheral eigenvalue for an irreducible mixed transfer operator, so Theorem 5.6.1 yields gauge-phase equivalence. Undoing the common trace-preserving gauge gives the claim for \(A\) and \(B_u\).

If \(\rho (\mathcal{E}_u) {\lt} 1\), Lemma G.6.1 forces every virtual-boundary sequence to decay to zero, contradicting Definition 10.3.2.10. Hence \(\rho (\mathcal{E}_u) \ge 1\). Theorem 10.3.2.17 shows that every eigenvalue has modulus at most \(1\), so some peripheral eigenvalue has modulus exactly \(1\). Applying Theorem 10.3.2.18 then gives the gauge-phase equivalence.

Start from an invertible intertwiner \(X\) and a nonzero gauge scalar \(\zeta \) between \(A\) and \(B_u\). The positive matrix \(Q:=XX^\dagger \) satisfies \(\mathcal{E}_A(Q)=|\zeta |^2Q\). Since \(\mathcal{E}_A(\mathbb {1})=\mathbb {1}\), irreducibility forces the two positive eigenvalues to agree, so \(|\zeta |=1\) and \(Q\) is a positive fixed point of \(\mathcal{E}_A\). Uniqueness of positive fixed points for an injective normalized tensor then makes \(Q\) a positive scalar multiple of the identity. Rescaling \(X\) by the square root of that scalar produces a unitary \(V\), and the original intertwining relation becomes (38), with \(|\mu |=1\).

Expand \(\mathcal{E}_u(V)\) using (16), insert the intertwining relation into each local term, and use \(V^\dagger V = \mathbb {1}\) together with \(\mathcal{E}_A(\mathbb {1})=\mathbb {1}\).

Relation (40) first shows that \(V\Lambda V^\dagger \) is another positive fixed point of the adjoint transfer map. It has the same trace as \(\Lambda \), so uniqueness of the normalized positive fixed point gives \(V\Lambda V^\dagger =\Lambda \). Multiplying by \(V^\dagger \) on the left and by \(V\) on the right yields (41).

Choose the boundary matrices \(X = V^\dagger \) and \(Y = V\), where \(V\) is the unitary from the virtual local-covariance datum. By Theorem 10.3.2.21, one has \(\mathcal{E}_u^L(V)=\mu ^L V\). Therefore

whose norm is constantly \(1\).

The forward direction is Theorem 10.3.2.21. For the reverse direction, start from a unitary peripheral eigenmatrix. Theorem 10.3.2.18 gives gauge-phase equivalence with the companion tensor; then Theorem 10.3.2.20 yields a virtual unitary and phase satisfying the local covariance relation, and Theorem 10.3.2.22 gives the boundary-state invariance needed for virtual local covariance.

If virtual-boundary nondecay holds, Theorem 10.3.2.19 gives a gauge-phase equivalence with the companion tensor. Theorem 10.3.2.20 extracts a virtual unitary and phase from that equivalence, and Theorem 10.3.2.22 shows that the same unitary preserves the boundary state, so one obtains virtual local covariance. Conversely, Lemma 10.3.2.23 turns virtual local covariance into a nondecaying virtual-boundary sequence.

Virtual-boundary nondecay first gives gauge-phase equivalence with the companion tensor by Theorem 10.3.2.19. Theorem 10.3.2.20 then normalizes the resulting intertwiner to a unitary and produces the phase \(\mu \).

Apply the channel library’s spectral characterization of irreducibility with right Perron matrix \(\mathbb {1}\), left Perron matrix \(\Lambda \), and Perron value \(1\). Positivity and unitality give the spectral radius \(1\); fixed-space simplicity gives the required uniqueness.

For \(\mathcal{E}_u(X)=\lambda X\), phase-retaining conjugation gives \(\mathcal{E}_A(V^\dagger X)=\mu ^{-1}\lambda V^\dagger X\). This matrix is nonzero. Peripheral primitivity forces \(\mu ^{-1}\lambda =1\), and irreducibility makes the resulting fixed matrix scalar. Multiplication by \(V\) recovers the claimed direction.

Write \(S_N=\ell (\mathcal{E}_u^N(Y))\), where \(Y=\mathcal{E}_y(\mathbb {1})\) and \(\ell (X)=\operatorname{tr}(\Lambda \mathcal{E}_x(X))\). Gelfand’s formula gives \(\| \mathcal{E}_u^N\| \leq Mr^N\) for a constant \(M{\gt}0\). Thus \(|S_N|\leq \| \ell \| M\| Y\| r^N\), including \(N=0\).

Apply the complementary-remainder argument to the adjoint Kraus family. It gives \(\rho (\mathcal{E}_A-P){\lt}1\). Unitality and stationarity give \((\mathcal{E}_A-P)^N=\mathcal{E}_A^N-P\) at positive lengths, so Gelfand’s formula proves the operator estimate. Apply the bounded linear functional \(G\mapsto \operatorname {Tr}(FG)\) and use \(\operatorname {Tr}(FP)=\operatorname{tr}(\Lambda F(\mathbb {1}))\) to obtain (42). No diagonalizability is needed.

The exact identity \(\mu ^{-N}\mathcal{E}_u^N(Y)=V\mathcal{E}_A^N(V^\dagger Y)\) expresses the error as \(\ell ((\mathcal{E}_A^N-P)(V^\dagger \mathcal{E}_y(\mathbb {1})))\), with \(\ell (X)=\operatorname{tr}(\Lambda \mathcal{E}_x(VX))\). Apply Theorem 10.3.5.2.

Apply (42) to the identity insertion and to \(F=\mathcal{E}_O\). Choose a tail on which \(|Z_L|\geq 1/2\). For \(s=\operatorname{tr}(\Lambda \mathcal{E}_O(\mathbb {1}))\) and \(a_n=\operatorname {Tr}(\mathcal{E}_O\mathcal{E}_A^n)\), division gives \(|a_n/Z_{k+n}-s|\leq 2(|a_n-s|+|s|\, |Z_{k+n}-1|)\). Since \(r^{k+n}\leq r^n\), this proves (44). For the full ring, apply Gelfand’s bound to \(\mathcal{E}_u^L\), take its trace, and divide by the same denominator bounded away from zero.

Consider the increasing spaces

Stationarity gives \(S_\ell \subseteq S_{\ell +1}\). Equality at one step propagates through the recurrence \(S_{\ell +1}=\sum _{i,j}A_iS_\ell A_j^\dagger \). By Theorem 17.2.16, normality and invertibility of \(\Omega \) make one of these spaces all of \(M_{D}(\mathbb {C})\). A dimension argument therefore gives \(S_{D^2}=M_{D}(\mathbb {C})\). The coefficients of a spanning expansion are the entries of \(O\) in (45).

Conjugate transposition reverses the order of the letters and maps one word span bijectively onto the other.

Apply stationary endpoint realization to \(A\) at \(\mathbb {1}\) to obtain \(\mathcal{E}_y^{[q]}(\mathbb {1})=Y\). Apply it to the adjoint letters at the stationary matrix \(\Lambda \) to represent \(\Lambda X\). Reversing both physical words and using trace cyclicity gives \(\operatorname{tr}[\Lambda \mathcal{E}_x^{[q]}(Z)]=\operatorname{tr}(\Lambda XZ)\) for every \(Z\). These identities prove (46) for every \(N\) without blocking the middle string.

Exact realization first supplies arbitrary block observables for which each coefficient is \(\operatorname{tr}\Lambda =1\). Matrix units span the block observable space, so each nonzero coefficient is nonzero on some tensor product of one-site matrix units. Decompose each factor as \(H+iK\) with \(H,K\) Hermitian. Multilinearity lets one choose either \(H\) or \(K\) at each site while retaining a nonzero coefficient. Exact realization of arbitrary virtual boundaries does not require those representatives themselves to be Hermitian products.

Canonical purity gives irreducibility and peripheral primitivity. Apply primitive-channel word-span fullness to the adjoint family and reverse words to obtain normality. Peripheral rigidity supplies a unitary \(V\) and a phase \(\mu \). The preceding theorem gives nonzero endpoint coefficients \(\alpha _y,\beta _x\) for Hermitian product endpoints. The demodulated virtual transfer limit yields \(|S_N^{[D^2]}|\to |\alpha _y|\, |\beta _x|{\gt}0\).

Expanding the Gram product gives \(G_{(a,b),(c,e)}=\sum _i(A^i)_{ab}\overline{(A^i)_{ce}}\), which is (48). For any doubled matrix \(X\) and pair indices \(p,q\), the two actions of \(L_H\) are

Substitute (48) and \(\overline{H_{br}}=H_{rb}\) into the difference of these sums to obtain (49). Realignment permutes the four indices, so the two commutators vanish simultaneously.

The first identity in (50) is matrix multiplication with \((K_0^\top )_{ij}=(K_0)_{ji}\). The second expands as \(\sum _{c,e}V_{ac}(A^i)_{ce}\overline{V_{be}}\). For the infinitesimal identity use \(\overline{H_{be}}=H_{eb}\) in the second tensor-factor action. Taking adjoints of the tensor factors gives \(L_H^\dagger =L_H\).

By Theorem 10.3.7.2, \(L_H\) commutes with \(G=FF^\dagger \). The Hermitian lift criterion of the companion quantum-channel volume  [ 216 , “Hermitian intertwiners of rectangular matrices” ] gives

Set \(K=K_0^\top \). Theorem 10.3.7.3 gives the infinitesimal relation, and \(K\) is Hermitian because \(K_0\) is. Intertwining passes to powers and then to the exponential series:

The tensor-exponential identity is proved in [ 216 , “Hermitian exponential paths” ] . Applying (50) yields (51). The same exponential identities give continuity, unitarity, and the group laws for all real parameters.

The support convention in (52) is \(K_0|_{\ker F}=0\). For the physical generator this becomes \(K|_{\overline{\ker F}}=0\): if \(Fv=0\), then \(K\overline v=\overline{K_0v}=0\). Thus \(u(t)\) is the identity on \(\overline{\ker F}\).

Multiplying each commutator by \((A^i)^\dagger \) and summing gives

Stationarity implies \(\operatorname{tr}(\Lambda \mathcal{E}_A(H))=\operatorname{tr}(\Lambda H)\). Taking the \(\Lambda \)-weighted trace therefore gives \(0=c\operatorname{tr}\Lambda =c\). Hence \(H\) is fixed by \(\mathcal{E}_A\) and is scalar by hypothesis.

Apply Theorem 10.3.4.4 with \(x=y=\mathbb {1}\) and phase \(\mu =1\). Since \(\mathcal{E}_1=\mathcal{E}_A\), unitality and stationarity give

Hermiticity of \(\Lambda \) and trace cyclicity imply \(\operatorname{tr}(\Lambda V^\dagger )=\overline{\operatorname{tr}(\Lambda V)}\). The factor \(\mu ^N\) is identically one, so no phase adjustment occurs in (53).

The function \(t\mapsto \operatorname{tr}(\Lambda V(t))\) is continuous at zero and has value \(1\) there. The complement of \(\{ 0\} \) is open in \(\mathbb {C}\).

Obtain \(K\) from Theorem 10.3.7.4. Purity makes every fixed matrix scalar. If \(K=c\mathbb {1}\), the infinitesimal relation in (51) becomes \(HA^i-A^iH=cA^i\) for every \(i\). Theorem 10.3.7.5 would force \(H\in \mathbb {C}\mathbb {1}\), a contradiction. Thus \(K\) is nonscalar.

The nonscalar-exponential theorem in [ 216 , “Hermitian exponential paths” ] gives \(u(t)\notin \mathbb {C}\mathbb {1}\) throughout a punctured neighborhood of zero. Theorem 10.3.7.7 gives \(\operatorname{tr}(\Lambda V(t))\ne 0\) in a neighborhood of zero. Canonical purity implies irreducibility and peripheral primitivity, as in Theorem 10.3.6.5. Intersect the two neighborhoods and apply Theorem 10.3.7.6 to obtain (54). Taking absolute values gives the positive limiting magnitude required for physical string order, with the same fixed operators \(x=y=\mathbb {1}\).

Diagonal insertions are invariant under the physical rephasing. Direct calculation gives

Iteration and the normalized trace give \(-4/9\). The positive limit is constant, so no asymptotic approximation is needed. The twist has distinct diagonal entries and is therefore nonscalar.

Every \(\mathcal E_y(\mathbb {1})\) is diagonal. The twist sends \(\mathbb {1}\) to zero, \(\sigma _z\) to \(-\sigma _x\), and \(\sigma _x\) to zero. Thus two iterations kill the right-endpoint image, and every later correlator vanishes.

The explicit endpoint maps satisfy \(\mathcal E_y^{(2)}(\mathbb {1})=-\sigma _y\) and \(\mathcal E_x^{(2)}(\sigma _y)=-\mathbb {1}\), while \(\mathcal E_{-\sigma _x}(\sigma _y)=\sigma _y\). Both minus signs cancel, and the normalized trace is one. The same fixed-point calculation applies to the blocked twist, whose nonzero off-diagonal entry makes it nonscalar.

The transfer map fixes the scalar matrices and multiplies traceless matrices by \(-1/3\). Induction proves (57), whose endomorphism trace is (58). For \(L\geq 2\), the absolute value of the correction to one is at most \(1/3\), so the norm is nonzero. Direct substitution gives \(Z_0=4\) and \(Z_1=0\). The virtual symmetry \(V=\sigma _z\) has phase one, so Theorem 10.3.1.4 applies at all the nonzero lengths. At length one the normalized expectation is zero.

Every positive-length coefficient of the controlled-\(Z\) state has squared modulus \(2^{-L}\), and there are \(2^L\) such coefficients. The empty product has trace two. The virtual symmetry \(V=\sigma _y\) implements \(u=-\sigma _x\) with phase one, so Theorem 10.3.1.4 gives the overlap.

Expand the two virtual indices in each letter product and collect them in the tensor product. Physical rotation is multiplication of \(M\) on its left. Finally use \(X^{-1}=X^\dagger \) for unitary \(X\).

Injectivity gives \(V^*V=I\) and \(P{\gt}0\). Every \(P_\gamma \) is positive definite, so \(P_\gamma ^{-1}V^*\) is a left inverse of \(M_\gamma \). Its rows express every matrix unit as a linear combination of the tensor letters, proving injectivity. Continuity and the endpoint identities follow from the affine formula. Unitary covariance gives \([M^*M,K]=0\); functional calculus gives \([P,K]=0\). Cancellation of the invertible factor \(P\) gives \(UV=VK\), and hence the covariance of the entire path. For bond conjugation, \(K=X^T\otimes X^*\). Since \(X^{-1}=X^*\), this is also covariance by a virtual gauge change.

The preceding theorem gives a continuous injective family on the compact interval \([0,1]\). The compact-family parent-gap theorem applies to this family. Covariance by bond conjugation gives gauge equivalence under physical rotation, so the canonical parent interactions commute with the physical tensor powers.

The injective local ground-space projections vary continuously, so their complementary projections do as well. Their norms are at most one. Compactness gives a uniform norm gap above the kernel. Positivity of the parent Hamiltonian makes all its eigenvalues nonnegative; applying the norm bound to a unit eigenvector orthogonal to the kernel shows that the same positive constant is a lower bound for every nonzero eigenvalue. The nonzero periodic MPS vector is a zero eigenvector. Finally, covariance makes every local parent projection, and hence its periodic sum, commute with the physical symmetry.

Apply the preceding construction to the injective polar path. The endpoint identities give the canonical parents of \(W\) and \(A\); unitary bond-conjugation covariance is preserved by the entire interpolation.

Physical rotation is a finite sum of fixed scalar multiples of the tensor letters, hence is continuous. Composition of physical rotations and \(VE=EU\) give \(V(EA)=E(UA)\). Summing the covariance of \(A\) gives \(E(UA)^i=\sum _jE_{ij}XA^jX^{-1}=X(EA)^iX^{-1}\).

Continuity is preserved by physical rotation. Because \(E^\dagger E=I\), every original letter is a linear combination of the embedded letters, so the embedded family is injective. The intertwining identity transports virtual covariance. The canonical injective-path theorem then supplies continuity of the parent projections, norm at most one, and a positive gap uniform in the parameter and every ring length \(N\ge 2\). Apply this construction to the polar family and use its endpoint identities to obtain the second path.

Put \(P=FF^\dagger \). Then \(P^2=P=P^\dagger \) and \(E^\dagger E=FJ^\dagger JF^\dagger +(I-P)^2=P+I-P=I\). Also \(EF=(J,0)\). Taking adjoints of \(U_gF=FS_g\) and multiplying by the unitary actions gives \(F^\dagger U_g=S_gF^\dagger \). Thus \(U_gP=PU_g\) and \(V_gJF^\dagger =JF^\dagger U_g\), which give the two components of the intertwining identity. Apply these identities to the polar factor; reading its entries as tensor letters gives the claimed endpoint.

The covariance of \(A\) reads \(U_gM=M(X_g^T\otimes X_g^\dagger )\) for its physical matrix \(M\). Both actions are unitary, so its polar isometric factor \(F\) has the same intertwining identity. The frame-embedding theorem gives \(E^\dagger E=I\) and \((V_g\oplus U_g)E=EU_g\). Apply the embedded polar-path construction and use \(EF=(J,0)\) to identify the initial tensor.

Tensoring with identities, taking direct sums, and changing finite coordinates preserve self-adjointness and idempotence. The corresponding Euclidean linear operators are symmetric projections. For each cyclic sector configuration, neighboring terms act on the independent edge factors \((r_n,l_{n+1})\) and therefore commute. Products of commuting orthogonal projections are positive, so expanding the square of their sum gives \(H_N^2\ge H_N\). A nonnegative eigenvalue then satisfies \(\lambda ^2\ge \lambda \), hence is zero or at least one.

The normalized bond vector is fixed by the direct-sum bond action. Its penalty therefore commutes with the action. Conjugation by the incoming-bond permutation gives the physical action and interaction; the full-window embedding at the origin is the identity.

Regroup the two physical sites by their sector pair. The included complement \(I-h_B\) occupies only the active pair. Every other sector has zero complement and hence identity interaction. Changing back to the original physical coordinates gives the stated equality.

The isometric local conjugation agrees with the original term on the included chain. Its complement penalty vanishes there. Summing over all translates gives the Hamiltonian identity. Finally, \(E_N^\dagger E_N=I\) preserves nonvanishing.

The canonical-parent decomposition under a physical isometry reads \(h_{EA}=Th_AT^\dagger +I-TT^\dagger \), which is the extension formula. The difference \(k'-h'=T(k-h)T^\dagger \) is positive whenever \(k-h\) is positive.

For commuting orthogonal projections, the kernel of their sum is the range of the product of their complementary projections. Since \(I-h'=T(I-h)T^\dagger \), the complete enlarged product is \(E_N\) times the original product times \(E_N^\dagger \). An isometric conjugation maps a matrix range onto its isometric image, so the two kernel identities give the claim.

Each letter is \(c\sum _{a,b}J_{i,(a,b)}|a\rangle \langle b|\), so its physical matrix is \(cJ\). Physical covariance therefore gives \(U(cJ)=(cJ)(X^T\otimes Y)\). Cancel \(c\ne 0\).

The identity \(EF=(J,0)\) identifies the physical matrices. Multiplication by a nonzero scalar preserves every boundary space and therefore its orthogonal complement projection.

The embedded normalized fixed-point tensors have virtual covariance on their respective summands. Their physical matrices are nonzero scalar multiples of \(J_0\) and \(J_1\), so the frame-covariance identity gives both intertwining identities.

First-summand inclusion is isometric and intertwines a unitary action with its direct sum. Apply the frame-embedding theorem to each \(F_j,J_j\) and retain the additional zero summand. The coordinate-frame identities give the required target covariance. A unitary virtual matrix has its adjoint as its general linear inverse, so both descriptions of the pair action agree.

The complement term vanishes on \(F_j\). The remaining component is \(J_j\), followed by the zero physical summands. Use the normalized fixed-point frame identity and invariance of boundary spaces under nonzero scalar multiplication.

Embed each polar deformation by its fixed physical isometry. Its intertwining identity transports symmetry, and the exact polar endpoint identifies the normalized fixed-point canonical parent.

Perron–Frobenius theory gives \(\Lambda {\gt}0\) and \(r{\gt}0\) with \(\mathcal{E}_A^\dagger (\Lambda )=r\Lambda \). Trace duality and unitality give

Since \(\operatorname{tr}\Lambda {\gt}0\), \(r=1\). The boundary-state theorem makes any virtual gauge \(X\) a nonzero scalar multiple \(cW\) of a unitary \(W\). The scalar cancels in conjugation. On-site symmetry supplies a gauge for each group element by the injective fundamental theorem, and the preceding argument supplies the unitary choices.

Perron normalization gives \(B\) by nonzero rescaling and virtual gauge. Both operations preserve the local MPS spaces, hence all canonical parent interactions; they also preserve on-site symmetry and injectivity. The preceding theorem supplies unitary virtual covariance for \(B\). Apply the symmetric polar-path construction and replace its final interaction by the equal canonical interaction of \(A\).

Conjugation preserves the projective multiplication law and commutes with physical twisting. Scalar rescaling also commutes with twisting. Apply gauge independence on \(B\) to compare the transported representation with any other choice.

Choose unitary virtual covariance for \(B\) and assemble these same matrices into a projective representation. Polar covariance preserves the matrices throughout the deformation, hence also their factor system. Gauge preparation identifies this class with \([\omega _A]\); gauge independence compares it with any endpoint choices.

First deform each blocked injective MPS to its isometric form; the deformation preserves the symmetry relation

On an interval where the minimal injective bond dimension is fixed, continuous tensors with exact symmetry admit continuous virtual gauges by Theorem 10.4.1.52. Their factor systems have constant cohomology class by the determinant argument. Symmetry of a unique ground-state line initially gives a scalar action; its character must first be removed by a fixed physical phase gauge. Changes of minimal bond dimension require an additional step: derive continuous balanced canonical data from the physical gap and restrict the virtual action to the invariant bond subspace surviving the limit. The algebraic restriction is Theorem 10.4.1.54, but the required continuity implication remains unformalized [ 101 ] .

Conversely, let the endpoint isometric forms have virtual actions \(V_g^0\) and \(V_g^1\) in the same cohomology class. Embed their bond spaces into the direct sum and interpolate between the maximally entangled bonds by

The commuting parent Hamiltonians along this interpolation remain gapped, with the uniform spectral estimate of Lemma 10.4.9.18. Their symmetry is

Equality of the cohomology classes is exactly what makes the cross-term action \(\widehat U^{\mathrm{path}}\) a linear representation. Conjugating back by the endpoint isometries gives the common direct-sum physical representation and the required symmetric gapped path.

The virtual representation theorem produces \(\rho (g^{-1}) \in \mathrm{GL}_{D}(\mathbb {C})\) satisfying \(\sum _j U(g)_{ij}\, A^j = \rho (g^{-1})\, A^i\, \rho (g^{-1})^{-1}\). A direct computation using \(\mathcal{E}_A(\mathbb {1})=\mathbb {1}\) shows that \(\mathcal{E}_{U(g)}(\rho (g^{-1}))=\rho (g^{-1})\), so \(\rho (g^{-1})\) is a nonzero eigenvector with eigenvalue \(1\). Theorem 10.3.2.18 gives gauge-phase equivalence with the companion tensor, and Theorem 10.3.2.20 extracts a virtual unitary and phase. By Theorem 10.3.2.22, this unitary preserves the boundary state, so Lemma 10.3.2.23 gives virtual-boundary nondecay.

Let \(\lambda \) witness \(\omega _1\sim \omega _0\), so that \(\omega _1(g,h)=\lambda _g\lambda _h\lambda _{gh}^{-1}\omega _0(g,h)\), and put \(\psi _g=\lambda _g/|\lambda _g|\). The unit-phase map \(z\mapsto z/|z|\) is multiplicative, so

Hence \(\psi _gV_g^0\) has factor system \(\omega _1/|\omega _1|\), and multiplication by a unit-modulus scalar preserves unitarity. For a unitary virtual action, \(1=\| V_gV_h\| =|\omega (g,h)|\, \| V_{gh}\| =|\omega (g,h)|\). Hence its factor system equals its pointwise unit phase.

The product with its adjoint is \((V_g^0(V_g^0)^*)\oplus (V_g^1(V_g^1)^*)=I_{D_0}\oplus I_{D_1}\).

If \(\alpha _t=\operatorname {Ad}Y\), matrix-unit multiplication gives \(X(t)=(Y^{-1}X_0)_{aa}Y\). This proves the intertwining identity, and gives \(X(t_0)=X_0\). Continuity follows entrywise from the formula. Its determinant is nonzero near \(t_0\).

Injectivity gives full column rank of \(P\), so \(P^\dagger P\) is positive definite. Transpose the left-inverse identity to obtain \(P^TC_A=I\). Inversion is continuous on invertible matrices. If \(B^i=YA^iY^{-1}\), the coefficient expansion of \(M\) becomes \(YMY^{-1}\) after replacing the letters.

Construct the continuous inner action from inverse tensor coefficients, then apply the matrix-unit conjugator formula. Exact on-site symmetry gives pointwise gauge equivalence by the injective fundamental theorem, so the argument applies to each twist.

The neighborhoods where local gauges are invertible cover the interval. Choose a finite subdivision subordinate to this cover. At each junction, injectivity implies that the two invertible gauges differ by a nonzero scalar. Rescale the next gauge by that constant and glue the continuous pieces. The same scalar matching gives the prescribed initial gauge.

A matrix \(X\) commuting with a rank-one idempotent \(P\) satisfies \(XP=\operatorname {tr}(XP)P\). This scalar belongs to the finite spectrum of \(X\). For fixed \(U_g\), its trace against \(P_t\) varies continuously and hence is constant on the connected parameter space. The representation law and \(P_t\ne 0\) give the character law. Unitarity puts each scalar on the unit circle.