7 Peripheral Channel Structure and Transfer-Operator Gaps
This chapter studies two spectral phenomena for finite-dimensional quantum channels and MPS transfer maps, following [ PGVWC07 ] and [ Wol12 ] .
The first is peripheral structure: for an irreducible unital Schwarz map with a faithful adjoint fixed point, the eigenvalues on the unit circle form a finite cyclic group (Theorem 7.1.1), and under trace preservation the order of this group divides the bond dimension \(D\) (Theorem 7.9.1). Removing this period by unitary conjugation and cyclic corner projections produces the periodicity-free cyclic-sector decomposition (Theorem 7.9.1.1) used by the canonical-form reduction of Chapter 9.
The second is the mixed transfer operator \(F_{AB}\) of two MPS tensors \(A\) and \(B\), and its spectral radius \(\rho _{\operatorname{spec}}(F_{AB})\). We show that \(\rho _{\operatorname{spec}}(F_{AB})\leq 1\) under trace-preserving normalization (Theorem 7.4.1), that equality forces \(A\) and \(B\) to be gauge-phase equivalent (Theorem 7.4.4), and that mismatched bond dimensions always force a strict gap (Theorem 7.5.2). Since the MPV overlap \(O_{AB}(N)\) equals the operator trace \(\operatorname{Tr}(F_{AB}^N)\) (Theorem 7.3.1), these gaps translate directly into the overlap-decay theorems (Theorem 7.6.3 and its rectangular and irreducible-trace-preserving variants) used throughout the fundamental theorem. The complementary case — a single tensor whose transfer map has a trivial peripheral spectrum away from the fixed-point projection — gives the rank-one Perron limit \(\operatorname{Tr}(\mathcal{E}_A^n)\to 1\) (Theorem 7.11.1) and hence primitive self-overlap convergence \(O_{AA}(N)\to 1\) (Theorem 7.11.2).
Throughout, \(\rho _{\operatorname{spec}}(\cdot )\) denotes the spectral radius of a linear endomorphism of a finite-dimensional matrix space: \(\rho _{\operatorname{spec}}(F_{AB}){\lt}1\) for a mixed transfer operator and \(\rho _{\operatorname{spec}}(\mathcal{E}_A-P){\lt}1\) for a transfer map after removing its fixed-point projection \(P\). The unqualified phrase spectral gap is reserved for the energy gap of a many-body Hamiltonian, discussed later in the full blueprint.
The algebraic mixed-transfer identities in the first two sections do not use normalization. Starting with the spectral-radius estimates, we assume the trace-preserving normalization
Right-canonical (unital) and left-canonical (trace-preserving) gauges are generally different similarity transforms of the same tensor, recorded in Remark 7.4.5; both are used below, depending on which fixed point — of the transfer map or of its adjoint — drives the gauge.
The proofs of the technical mixed-transfer identities, Frobenius-norm estimates, gauge-rigidity intertwiners, spectral-radius decay estimates, the rank-one Perron projection, and the cyclic-decomposition machinery removing peripheral periodicity are collected in Appendix C.
7.1 Peripheral eigenvalue group structure
This section proves the peripheral spectrum structure of [ Wol12 , Theorem 6.6 ] for irreducible Schwarz maps: the peripheral eigenvalues form a finite cyclic group. The trace-preserving refinement that the order divides the bond dimension is proved in Theorem 7.9.1. The invertibility of peripheral eigenvectors and the product closure of the peripheral eigenvalues used below are proved in Section C.6.
Let \(E\) be an irreducible unital Schwarz map on \(M_{D}(\mathbb {C})\) with a positive-definite adjoint fixed point (trace-preservation is not required). Then there exist \(m \ge 1\) and a primitive \(m\)-th root of unity \(\gamma \) such that the peripheral eigenvalues of \(E\) are exactly \(\{ 1,\gamma ,\gamma ^2,\ldots ,\gamma ^{m-1}\} \).
The peripheral eigenvalues form a finite subset of \(\mathbb {C}\), contain \(1\), and are closed under multiplication. Finiteness and product closure imply closure under inversion, so they form a finite subgroup of \(\mathbb {C}^\times \). Every finite subgroup of \(\mathbb {C}^\times \) is cyclic. A generator \(\gamma \) has order equal to the group cardinality \(m\), and the group is \(\{ 1,\gamma ,\gamma ^2,\ldots ,\gamma ^{m-1}\} \).
7.2 Mixed transfer operator
The mixed transfer operator (or cross transfer operator) for two MPS tensors \(A\) and \(B\) of the same bond dimension \(D\) is the linear map \(F_{AB}:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) defined by
For MPS tensors \(A\) of bond dimension \(D_1\) and \(B\) of bond dimension \(D_2\), possibly with \(D_1\neq D_2\), the rectangular mixed transfer operator is the map \(F_{AB}^{\mathrm{rect}}: M_{D_1\times D_2}(\mathbb {C})\to M_{D_1\times D_2}(\mathbb {C})\) defined by
The self-transfer identity \(F_{AA}=\mathcal{E}_A\), the iterated word expansion of \(F_{AB}\), and the trace-expansion lemma over matrix units are proved in Section C.1.
7.3 MPV overlap as transfer trace
Recall from (??) that \(O_{AB}(N)=\sum _{\sigma \in \{ 0,\ldots ,d{-}1\} ^N} V^{(N)}(A)_\sigma \overline{V^{(N)}(B)_\sigma }\). This section rewrites that scalar as the operator trace of the mixed transfer power.
For tensors \(A\) and \(B\) of bond dimension \(D\geq 1\),
Section C.1 expands the operator trace over matrix units (Lemma C.1.3) and the transfer power by words (Lemma C.1.2); reassembling the sums over matrix-unit indices gives
The rectangular case, Lemma 7.3.2, is the same argument over the rectangular matrix-unit basis.
For tensors \(A\) of bond dimension \(D_1\geq 1\) and \(B\) of bond dimension \(D_2\geq 1\),
7.4 Eigenvalue bound and transfer-operator gap
The eigenvalue bound uses the Frobenius (Hilbert–Schmidt) norm \(\| {\cdot }\| _F\) on the finite-dimensional matrix algebra and the trace positivity of two positive semidefinite matrices; both are proved in Section C.2. For the rigidity theorem, one gauges \(A\) and \(B\) separately using positive fixed points of their transfer maps to obtain unital Kraus families, and then applies the Kadison–Schwarz equality argument ( [ Wol12 , Theorem 5.3 ] ) to a block embedding of a mixed-transfer eigenvector.
Let \(A\) and \(B\) be normalized MPS tensors. Every eigenvalue \(\mu \) of \(F_{AB}\) satisfies \(|\mu |\leq 1\).
Let \(A\) and \(B\) be normalized MPS tensors of bond dimensions \(D_1\geq 1\) and \(D_2\geq 1\). Every eigenvalue \(\mu \) of \(F_{AB}^{\mathrm{rect}}\) satisfies \(|\mu |\leq 1\).
A uniform Frobenius bound \(\| (F_{AB}^{\mathrm{rect}})^n(X)\| _F^2\leq D_1^2\| X\| _F^2\) follows from the word expansion, the Frobenius submultiplicativity \(\| MN\| _F\leq \| M\| _F\| N\| _F\), and Cauchy–Schwarz applied to the trace-preserving normalization of \(A\) and \(B\) (Section C.2). If \(F_{AB}^{\mathrm{rect}}(X)=\mu X\) with \(X\neq 0\), iterating gives \(|\mu |^{2n}\| X\| _F^2\leq D_1^2\| X\| _F^2\) for every \(n\), forcing \(|\mu |\leq 1\).
For normalized tensors, \(\rho _{\operatorname{spec}}(F_{AB})\leq 1\).
The spectral radius is the maximum of \(|\mu |\) over the eigenvalues, and each satisfies \(|\mu |\leq 1\) by Theorem 7.4.1.
Let \(A\) and \(B\) be injective normalized MPS tensors. If \(\rho _{\operatorname{spec}}(F_{AB})\geq 1\), then \(A\) and \(B\) are gauge-phase equivalent.
Since \(\rho _{\operatorname{spec}}(F_{AB})\leq 1\) by Lemma 7.4.3, the hypothesis gives an eigenvalue \(\mu \) with \(|\mu |=1\) and a nonzero eigenvector \(X\) satisfying \(F_{AB}(X)=\mu X\). The proof proceeds in five steps, elaborated in Section C.3.
Step 1 (Separate gauging). Choose positive definite fixed points \(\rho _A\) and \(\rho _B\) of \(\mathcal{E}_A\) and \(\mathcal{E}_B\) by Theorem 6.4.2. Gauge the tensors separately to unital Kraus families by Theorem 6.5.1: \(A'^i=\rho _A^{-1/2}A^i\rho _A^{1/2}\) and similarly for \(B\). Set \(X'=\rho _A^{-1/2}X\rho _B^{-1/2}\).
Step 2 (Block Kraus family and off-diagonal embedding). Form
The matrices \(\{ C^i\} \) form a unital Kraus family on \(M_{2D}(\mathbb {C})\), and \(E_C(Y)=\mu Y\).
Step 3 (Kadison–Schwarz equality and Kraus relations). The positive definite fixed points of the adjoint maps give a faithful weighted trace for the block map. Applying the Kadison–Schwarz inequality to \(Y\) and pairing its positive semidefinite gap with this fixed point shows that the gap vanishes. The Kraus relation of Theorem 5.2.2 then gives \(Y(C^i)^\dagger =\mu (C^i)^\dagger Y\). Applying the same argument to the complementary product gives \(C^iY=\mu YC^i\).
Step 4 (Intertwining identity and invertibility). Expanding the block relations from Step 3 gives
The corresponding multiplicative identities make \(X'^\dagger X'\) and \(X'X'^\dagger \) nonzero positive semidefinite fixed points of the appropriate transfer maps. Injectivity upgrades them to positive definite matrices, so \(X'\) is invertible.
Step 5 (Skolem–Noether conclusion). The intertwining identity and invertibility of \(X'\) give an algebra isomorphism between the full matrix algebras generated by the two families. By Theorem 3.2.4, this isomorphism is inner. Undoing the separate gauges gives \(B^i=\overline{\mu }XA^iX^{-1}\) for every \(i\), which is the gauge-phase relation (??).
The gauging is applied separately to \(\mathcal{E}_A\) and \(\mathcal{E}_B\); the mixed transfer map \(F_{AB}\) need not be simultaneously unital and trace-preserving.
The right-canonical gauge used here, \(A'^i=\rho ^{-1/2}A^i\rho ^{1/2}\), makes \(\{ A'^i\} \) a unital Kraus family: \(\sum _iA'^i(A'^i)^\dagger =\mathbb {1}\). The convention \(\widetilde A^i=\rho ^{1/2}A^i\rho ^{-1/2}\) makes the family trace-preserving: \(\sum _i(\widetilde A^i)^\dagger \widetilde A^i=\mathbb {1}\). Both conventions appear in practice: the unital form is used in the transfer-operator gap proof above, while the trace-preserving form is used in the canonical-form reduction of Chapter 9.
Let \(A\) and \(B\) be injective normalized MPS tensors that are not gauge-phase equivalent. Then \(\rho _{\operatorname{spec}}(F_{AB}){\lt}1\).
7.5 Transfer-operator gap — rectangular case
Let \(A\) and \(B\) be injective tensors of bond dimensions \(D_1\) and \(D_2\). If there are \(X\in M_{D_1\times D_2}(\mathbb {C})\setminus \{ 0\} \) and \(\mu \in \mathbb {C}\) such that, for every physical index \(i\),
then \(D_1=D_2\).
The first relation in (??) shows that \(\ker (X)\) is invariant under every \((B^i)^\dagger \): if \(Xv=0\), then \(X(B^i)^\dagger v=\mu (A^i)^\dagger Xv=0\). Since the \(\{ B^i\} \) span the full matrix algebra, so do the \(\{ (B^i)^\dagger \} \) (conjugate transposition is a conjugate-linear bijection and therefore preserves complex linear spans), so \(\ker (X)\) is invariant under every matrix; injectivity of the resulting map forces \(\ker (X)=0\), giving \(D_2\leq D_1\). Taking adjoints of the second relation gives \(X^\dagger (A^i)^\dagger =\overline\mu (B^i)^\dagger X^\dagger \), so the same argument applied to \(X^\dagger \) shows \(\ker (X^\dagger )\) is invariant under every \((A^i)^\dagger \) and hence trivial, giving \(D_1\leq D_2\).
Let \(A\) be an injective normalized tensor of bond dimension \(D_1\) and \(B\) an injective normalized tensor of bond dimension \(D_2\), with \(D_1\neq D_2\). Then \(\rho _{\operatorname{spec}}(F_{AB}^{\mathrm{rect}}){\lt}1\).
Theorem 7.4.2 gives \(|\mu |\leq 1\) for every eigenvalue of \(F_{AB}^{\mathrm{rect}}\). If its spectral radius were \(1\), there would be a modulus-one eigenvector \(X\in M_{D_1\times D_2}(\mathbb {C})\setminus \{ 0\} \).
Gauge \(A\) and \(B\) to unital families \(A'\) and \(B'\) and transport \(X\) to a nonzero rectangular intertwiner \(X'\) satisfying (??). Lemma 7.5.1 would then give \(D_1=D_2\), contradicting the hypothesis.
If \(D_1\neq D_2\) and both tensors are injective and normalized, then \(O_{AB}(N)\to 0\) as \(N\to \infty \).
The gap \(\rho _{\operatorname{spec}}(F_{AB}^{\mathrm{rect}}){\lt}1\) from Theorem 7.5.2 implies \((F_{AB}^{\mathrm{rect}})^N\to 0\), so its operator trace tends to zero. The identity (??) therefore gives \(O_{AB}(N)\to 0\).
7.6 MPV overlap decay
The Banach-algebra fact that powers below unit spectral radius vanish is proved in Section C.4.
Let \(a\) be an element of a complex Banach algebra with \(a^2=a\) and \(\rho _{\operatorname{spec}}(a){\lt}1\). Then \(a=0\).
Lemma C.4.1 gives \(a^n\to 0\). Idempotence gives \(a^{n+1}=a\) for every \(n\), so uniqueness of limits yields \(a=0\).
Let \(A\) and \(B\) be injective normalized tensors that are not gauge-phase equivalent. Then \(F_{AB}^n(X)\to 0\) for every \(X\in M_{D}(\mathbb {C})\).
Let \(A\) and \(B\) be injective normalized tensors of the same bond dimension that are not gauge-phase equivalent. Then \(O_{AB}(N)\to 0\) as \(N\to \infty \).
By (??) and (??),
Theorem 7.6.2 makes every summand tend to zero. Since the sum is finite, \(O_{AB}(N)\to 0\).
The block-separation consequences of this gap — cross-correlation decay between distinct blocks and the persistence of self-correlation at a fixed point — used in the canonical-form separation of Chapter 9, are recorded in Section C.4.
7.7 Transfer-operator gap under irreducible-TP hypotheses
The preceding transfer-operator gap results require injectivity of both tensors. The following variants weaken this to irreducibility together with the trace-preserving normalization (??), following Cirac et al. [ CPGSV17 ] and the irreducibility theory of [ Wol12 , Section 6.2 ] . The argument replaces the Kraus-commutation step with a Cauchy–Schwarz rigidity argument for the Perron–Frobenius fixed point.
Let \(A\) and \(B\) be irreducible trace-preserving normalized MPS tensors of bond dimension \(D\). If \(\rho _{\operatorname{spec}}(F_{AB})\geq 1\), then \(A\) and \(B\) are gauge-phase equivalent.
Choose a modulus-one eigenvector as in Theorem 7.4.4. The proof follows its five steps, with one change. Separate gauging, the block Kraus family, and the Kadison–Schwarz equality proceed as before. The equality makes the relevant positive products of the transported intertwiner nonzero positive semidefinite fixed points. Under irreducibility, Theorem 6.2.4 makes these fixed points positive definite, so the transported intertwiner is invertible. Lemma C.3.1 supplies (??), and Lemma C.3.2 then gives gauge-phase equivalence.
Let \(A\) and \(B\) be irreducible trace-preserving normalized MPS tensors of the same bond dimension that are not gauge-phase equivalent. Then \(\rho _{\operatorname{spec}}(F_{AB}){\lt}1\).
Let \(A\) and \(B\) be irreducible trace-preserving normalized tensors of the same bond dimension that are not gauge-phase equivalent. Then \(O_{AB}(N)\to 0\) as \(N\to \infty \).
Theorem 7.7.2 gives \(\rho _{\operatorname{spec}}(F_{AB}){\lt}1\), so \(F_{AB}^N\to 0\). The overlap–trace identity (??) gives \(O_{AB}(N)\to 0\).
Let \(A\) and \(B\) be irreducible trace-preserving normalized tensors of bond dimensions \(D_1\) and \(D_2\), respectively, with \(D_1\neq D_2\). Then \(\rho _{\operatorname{spec}}(F_{AB}^{\mathrm{rect}}){\lt}1\).
Suppose instead that \(\rho _{\operatorname{spec}}(F_{AB}^{\mathrm{rect}})=1\), and choose a modulus-one eigenvector \(X\in M_{D_1\times D_2}(\mathbb {C})\setminus \{ 0\} \). Gauge \(A\) and \(B\) separately to unital families \(A'\) and \(B'\) and transport \(X\) to \(X'\neq 0\). The matrix \(X'X'^\dagger \) is a nonzero positive semidefinite fixed point of the appropriate transfer map on \(M_{D_1}(\mathbb {C})\), so Theorem 6.2.4 makes it positive definite. Thus \(X'\) has full row rank and \(D_1\leq D_2\). Similarly, \(X'^\dagger X'\) is positive definite, so \(X'\) has full column rank and \(D_2\leq D_1\). Hence \(D_1=D_2\), a contradiction.
If \(D_1\neq D_2\) and both tensors are irreducible and trace-preserving normalized, then \(O_{AB}(N)\to 0\) as \(N\to \infty \).
Theorem 7.7.4 gives \(\rho _{\operatorname{spec}}(F_{AB}^{\mathrm{rect}}){\lt}1\), so \((F_{AB}^{\mathrm{rect}})^N\to 0\). The rectangular overlap–trace identity (??) gives \(O_{AB}(N)\to 0\).
7.8 Overlap rigidity for normal blocks
The BNT matching argument ultimately turns a non-decaying mixed overlap into a gauge-phase match. The following two results isolate that spectral step.
Let \(A\) and \(B\) have the same bond dimension. If \(\| \langle V^{(N)}(A) | V^{(N)}(B) \rangle \| \to 1\), then the mixed transfer map has spectral radius at least \(1\).
If the mixed-transfer spectral radius were less than \(1\), the iterated mixed transfer map would tend to zero in operator norm, hence so would its trace. By (??), the overlap would tend to zero, contradicting the hypothesis.
Let \(A\) and \(B\) be irreducible MPS tensors of the same bond dimension with \(\sum _i(A^i)^\dagger A^i=\mathbb {1}\) and \(\sum _i(B^i)^\dagger B^i=\mathbb {1}\). If \(\| \langle V^{(N)}(A) | V^{(N)}(B) \rangle \| \to 1\), then \(A\) and \(B\) are gauge-phase equivalent.
7.9 Peripheral spectral refinements
Let \(E\) be an irreducible unital trace-preserving Schwarz map on \(M_{D}(\mathbb {C})\) with a positive-definite adjoint fixed point. Then there exist \(m\geq 1\) and a primitive \(m\)-th root of unity \(\gamma \) such that \(m\mid D\) and the peripheral eigenvalues of \(E\) are exactly \(\{ 1,\gamma ,\gamma ^2,\ldots ,\gamma ^{m-1}\} \).
Apply Theorem 7.1.1 for the cyclic structure. Divisibility \(m\mid D\) follows from the cyclic decomposition: \(m\) mutually orthogonal projections summing to \(\mathbb {1}\) each have equal trace by trace preservation, so \(m\, \operatorname{tr}(P_0)=D\) with \(\operatorname{tr}(P_0)\in \mathbb {N}\).
Let \(E\) be an irreducible unital trace-preserving Schwarz map on \(M_{D}(\mathbb {C})\) with a positive-definite adjoint fixed point. Let \(\gamma \) be a primitive \(m\)-th root of unity such that the peripheral eigenvalues are exactly \(\{ \gamma ^k:k\in \{ 0,\ldots ,m{-}1\} \} \). Then \(m\mid D\).
The cyclic decomposition gives orthogonal projections \(P_0,\ldots ,P_{m-1}\) with \(\sum _kP_k=\mathbb {1}\) and \(E(P_{k+1})=P_k\). Trace preservation gives \(\operatorname{tr}(P_{k+1})=\operatorname{tr}(P_k)\) for every \(k\), hence
Since the trace of an orthogonal projection is an integer, \(m\mid D\).
7.9.1 Cyclic decomposition of irreducible Schwarz maps
The construction — normalizing a peripheral eigenvalue to a unitary eigenvector, forming its discrete Fourier (cyclic) spectral projections, and the supporting corner-algebra apparatus (restriction to an invariant corner, corner rank, and the compression isometry \(PM_{D}(\mathbb {C})P\cong M_{n}(\mathbb {C})\)) — is given in Section C.6.
Under irreducibility and faithful fixed point hypotheses, there is a full cyclic projection decomposition realizing the peripheral period.
Let \(P_0,\ldots ,P_{m-1}\) be orthogonal projections summing to the identity and cyclically permuted by a Kraus map, \(\sum _vK_vP_{k+1}K_v^\dagger =P_k\). Then the projections are mutually orthogonal, none of them vanishes, and each Kraus operator intertwines consecutive projections:
Consequently, a product of \(\ell \) Kraus operators moves each projection back by \(\ell \) steps; in particular, a product of \(m\) Kraus operators commutes with every projection.
Compressing \(\sum _vK_vP_{k+1}K_v^\dagger =P_k\) by \(\mathbb {1}-P_k\) exhibits a vanishing sum of positive semidefinite matrices, so each summand vanishes: \((\mathbb {1}-P_k)K_vP_{k+1}=0\). Mutual orthogonality follows by compressing the resolution of the identity by a fixed \(P_k\), another vanishing sum of positive semidefinite matrices. Expanding \(K_vP_{k+1}\) and \(P_kK_v\) through the resolution of the identity, the off-diagonal compressions vanish by orthogonality, leaving (??). If some projection vanished, the cyclic action would propagate the vanishing around the cycle, contradicting the resolution of the identity. The word statement follows by induction on the length.
The primitivity and irreducibility of the restricted sector dynamics, the invariance of each cyclic corner under the period, and the closure of the peripheral eigenvalues under products and inverses are proved in Section C.6.
Each peripheral eigenspace is one-dimensional.
Given a peripheral unitary eigenvector \(U\) for \(\gamma \), the multiplicative-domain identity gives \(E(XU^\dagger )=XU^\dagger \) for any \(\gamma \)-eigenvector \(X\). By the scalar fixed-point lemma for irreducible unital maps, \(XU^\dagger =c\mathbb {1}\), hence \(X=cU\) and the eigenspace is one-dimensional.
7.10 Self-overlap and the period-one criterion
Let \(D{\gt}0\) and \(m{\gt}0\). Let \(A\) be a left-canonical irreducible MPS tensor with bond dimension \(D\), and suppose that the peripheral eigenvalues of its transfer map are exactly \(\{ z\in \mathbb {C}:z^m=1\} \). Then \(O_{AA}(mn)\longrightarrow m\).
Apply Theorem 9.11.1 to the period-\(m\) blocking \(A^{[m]}\). It gives \(m\) nonzero cyclic-sector tensors \(C_0,\ldots ,C_{m-1}\), each left-canonical, whose unit-weight direct sum has the same MPV family as \(A^{[m]}\). The cyclic-sector relations make every \(C_u\) primitive and irreducible. Distinct sectors are not gauge-phase equivalent. Hence the primitive self-overlap limit and the equal- and unequal-dimension overlap-decay theorems give
Expanding the overlap of the unit-weight direct sum and taking the finite sum of these limits yields
Finally, blocking identifies \(O_{A^{[m]}A^{[m]}}(n)=O_{AA}(mn)\), which proves the claim.
Let \(A\) be an irreducible left-canonical tensor of positive bond dimension. If \(O_{AA}(N)\longrightarrow 1\), then the transfer map of \(A\) is primitive and \(A\) is normal.
Let \(m\) be the peripheral period. Along its multiples, \(O_{AA}(mn)\longrightarrow m\). The assumed full limit gives \(O_{AA}(mn)\to 1\), hence \(m=1\). The transfer map is therefore primitive, and irreducibility and left-canonicality give normality.
Let \(A\) be an irreducible left-canonical tensor of positive bond dimension. If \(O_{AA}(N)\to 1\), then \(A\) is normal.
This is the normality conclusion of Theorem 7.10.2.
7.11 Primitive overlap convergence (complementary transfer-map gap form)
The theorems in this section are stated directly in terms of the complementary map \(E-P\). This is the analytic form needed for the later overlap argument, corresponding to the primitive-case specialization of [ Wol12 , Theorem 6.7 ] where \(T_\phi \) is a rank-one projection. For primitive normalized transfer maps, Chapter 4 supplies this complementary transfer-map input under explicit hypotheses; the rank-one Perron projection \(P=P_\rho \), its trace, and the concrete irreducible instantiation of the complementary spectral-radius gap are given in Section C.5.
Let \(E\) be trace-preserving with fixed point \(\rho \), where \(\operatorname{tr}(\rho )\neq 0\), and let \(P\) be the fixed-point projection of Definition 4.9.1. If \(\rho _{\operatorname{spec}}(E-P){\lt}1\), then \(\operatorname{Tr}(E^n)\to 1\) as \(n\to \infty \).
Let \(A\) be a normalized MPS tensor with a nonzero PSD fixed point \(\rho \) of the transfer map. If \(\rho _{\operatorname{spec}}(\mathcal{E}_A-P){\lt}1\), where \(P\) is the fixed-point projection, then \(O_{AA}(N)\to 1\) as \(N\to \infty \).
By the overlap–trace identity (??) with \(A=B\), one has \(O_{AA}(N)=\operatorname{Tr}(\mathcal{E}_A^N)\). By Theorem 7.11.1, \(\operatorname{Tr}(\mathcal{E}_A^N)\to 1\).