5 Peripheral Channel Structure and Transfer-Operator Gaps
The spectral theory of finite-dimensional quantum channels and MPS transfer maps rests on two principal structures, following Pérez-García et al. [ 228 ] and Wolf [ 244 ] : the peripheral spectrum and the mixed transfer operator.
First, for an irreducible unital Schwarz map with a faithful adjoint fixed point, the peripheral eigenvalues on the unit circle form a finite cyclic group (Theorem [ 216 , “Peripheral eigenvalues: cyclic structure for Kraus maps” ] ). Under trace preservation, the order of this group divides the bond dimension \(D\) (Theorem [ 216 , “Transfer-map peripheral eigenvalues form a cyclic group” ] ). Unitary conjugation and cyclic corner projections yield the periodicity-free cyclic-sector decomposition (Theorem [ 216 , “Cyclic decomposition for irreducible finite Kraus maps” ] ) used in the canonical-form reduction of Chapter 7.
Second, the mixed transfer operator \(F_{AB}\) of two MPS tensors \(A\) and \(B\) governs the asymptotic overlap of matrix product vectors. Under trace-preserving normalization, the spectral radius satisfies \(\rho _{\operatorname{spec}}(F_{AB})\leq 1\) (Theorem 5.3.1); for injective (or irreducible trace-preserving) tensors, equality \(\rho _{\operatorname{spec}}(F_{AB})=1\) forces \(A\) and \(B\) to be gauge-phase equivalent (Theorem 5.3.4), while unequal bond dimensions force a strict gap \(\rho _{\operatorname{spec}}(F_{AB}){\lt}1\) (Theorem 5.4.2). Because the MPV overlap \(O_{AB}(N)\) equals the operator trace \(\operatorname{Tr}(F_{AB}^N)\) (Theorem 5.2.1), these spectral gaps imply the overlap-decay theorems (Theorem 5.5.5 and its rectangular and irreducible-trace-preserving variants). For a single primitive tensor whose transfer map has trivial peripheral spectrum away from the fixed-point projection, the complementary gap yields the rank-one Perron limit \(\operatorname{Tr}(\mathcal{E}_A^n)\to 1\) (Theorem 5.10.1) and primitive self-overlap convergence \(O_{AA}(N)\to 1\) (Theorem 5.10.2).
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 subtracting its fixed-point projection \(P\). The unqualified phrase spectral gap is reserved for the energy gap of a many-body Hamiltonian.
The algebraic mixed-transfer identities hold without normalization. The subsequent spectral-radius estimates assume the trace-preserving normalization
Right-canonical (unital) and left-canonical (trace-preserving) gauges are distinct similarity transformations of a tensor (Remark 5.3.5), used according to whether the fixed point of the transfer map or that of its adjoint governs the transformation.
Supporting proofs for the mixed-transfer identities, operator-norm estimates, gauge intertwiners, and cyclic-decomposition details appear in Appendix B.
5.1 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 B.1.
5.2 MPV overlap as transfer trace
The overlap \(O_{AB}(N)=\sum _{\sigma \in \{ 0,\ldots ,d{-}1\} ^N} V^{(N)}(A)_\sigma \overline{V^{(N)}(B)_\sigma }\) (20) coincides with the operator trace of the \(N\)-th power of the mixed transfer operator.
For tensors \(A\) and \(B\) of bond dimension \(D\geq 1\),
Specialize Lemma 5.2.2 to \(D_1=D_2=D\).
For tensors \(A\) of bond dimension \(D_1\geq 1\) and \(B\) of bond dimension \(D_2\geq 1\),
Section B.1 expands the operator trace over the rectangular matrix-unit basis \(E_{pq}\), \(0\leq p\leq D_1-1\), \(0\leq q\leq D_2-1\) (Lemma [ 216 , “Trace expansion over matrix units” ] ), and the transfer power by words (Lemma B.1.2); reassembling the sums over matrix-unit indices gives
5.3 Eigenvalue bound and transfer-operator gap
For trace-preserving Kraus maps, every eigenvalue of the rectangular mixed transfer operator has modulus at most \(1\). When the spectral radius reaches \(1\), Kadison–Schwarz rigidity ( [ 244 , Theorem 5.3 ] ) applied to a block embedding of the eigenvector forces gauge-phase equivalence between two injective (or irreducible trace-preserving) tensors.
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 \(|\mu |\leq 1\).
This is the MPS formulation of the corresponding theorem in the companion quantum-channel volume [ 216 , “Eigenvalue bound for a trace-preserving mixed Kraus map” ] : the map \(F_{AB}^{\mathrm{rect}}\) is the mixed Kraus map of the two matrix families, and the normalizations of \(A\) and \(B\) are exactly its two trace-preserving hypotheses.
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 5.3.1.
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 5.3.3, 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 B.4.
Step 1 (Separate gauging). Choose positive definite fixed points \(\rho _A\) and \(\rho _B\) of \(\mathcal{E}_A\) and \(\mathcal{E}_B\) by Theorem 4.3.2. Gauge the tensors separately to unital Kraus families by Theorem [ 216 , “Right-canonical (unital) gauge” ] : \(A'^i=\rho _A^{-1/2}A^i\rho _A^{1/2}\) 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 the corresponding theorem in the companion quantum-channel volume [ 216 , “KS equality implies Kraus commutation” ] 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 [ 216 , “Skolem–Noether” ] , 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 (11).
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 7.
Let \(A\) and \(B\) be injective normalized MPS tensors that are not gauge-phase equivalent. Then \(\rho _{\operatorname{spec}}(F_{AB}){\lt}1\).
5.4 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 \(D_1=D_2\).
Relation (6) 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 (7) 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 5.3.2 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 (2). Lemma 5.4.1 would then give \(D_1=D_2\), contradicting the hypothesis.
Let \(A\) and \(B\) have bond dimensions \(D_1\) and \(D_2\), respectively. If \(\rho _{\operatorname{spec}}(F_{AB}^{\mathrm{rect}}){\lt}1\), then, for every \(X\in M_{D_1\times D_2}(\mathbb {C})\),
This is the decay mechanism for the off-diagonal transfer operators in [ 13 , Appendix B, equations (B.3)–(B.4) and the paragraph following equation (B.6) ] .
The Gelfand formula implies convergence of \((F_{AB}^{\mathrm{rect}})^n\) to zero in operator norm. Evaluation at a fixed rectangular matrix is continuous, and therefore preserves this limit.
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 5.4.2 and Lemma [ 216 , “Powers vanish below unit spectral radius” ] imply operator-norm convergence \((F_{AB}^{\mathrm{rect}})^N\to 0\), so its operator trace tends to zero. The identity (5) therefore gives \(O_{AB}(N)\to 0\).
5.5 MPV overlap decay
The spectral radius on complex Banach algebras satisfies two standard identities used in the overlap and fixed-point analysis.
Let \(a\) belong to a complex Banach algebra and let \(c\in \mathbb {C}\setminus \{ 0\} \). Then
Multiplication by \(c\) sends the spectrum of \(a\) onto the spectrum of \(ca\). Taking the supremum of the moduli of the spectral values gives the formula.
Let \(a\) be a nonzero element of a complex normed algebra. If \(a^2=a\), then
The identity \(a^2=a\) gives \(a(a-1)=0\). Since \(a\ne 0\), the element \(a-1\) is not invertible, so \(1\) belongs to the spectrum of \(a\). Every spectral value \(\lambda \) satisfies \(\lambda ^2=\lambda \), hence the spectrum is contained in \(\{ 0,1\} \).
The Banach-algebra fact that powers below unit spectral radius vanish is proved in Section B.5.
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 [ 216 , “Powers vanish below unit spectral radius” ] 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 \(N\to \infty \).
Cross-correlation decay between distinct blocks and the persistence of self-correlation at a fixed point, used in the canonical-form separation of Chapter 7, are established in Section B.5.
5.6 Transfer-operator gap under irreducible-TP hypotheses
Under the trace-preserving normalization (1), the transfer-operator gap and overlap-decay theorems extend from injective tensors to irreducible tensors, following Cirac et al. [ 13 ] and Wolf [ 244 , Section 6.2 ] . In place of the full matrix algebra span, the argument uses the strict positive definiteness of 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 5.3.4. 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 4.1.1 makes these fixed points positive definite, so the transported intertwiner is invertible. Lemma B.4.1 supplies (2), and Lemma B.4.2 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 \(N\to \infty \).
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 4.1.1 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 \(N\to \infty \).
5.7 Overlap rigidity for normal blocks
A non-decaying overlap between irreducible tensors forces unit spectral radius for their mixed transfer operator, establishing gauge-phase equivalence.
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 (4), 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.
5.8 Peripheral spectral refinements
The results in this section concerning peripheral eigenvalues, cyclic projections, and peripheral eigenspaces are stated for completely positive finite Kraus maps, rather than the general Schwarz-map setting of Wolf Theorem 6.6; see [ 18 ] .
5.8.1 Cyclic decomposition of irreducible finite Kraus maps
The cyclic decomposition proceeds by normalizing a peripheral eigenvalue to a unitary eigenvector, constructing the discrete Fourier spectral projections, and restricting the map to the resulting invariant corners \(PM_{D}(\mathbb {C})P\cong M_{n}(\mathbb {C})\); details appear in Section B.7.
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:
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 (8). If some projection vanished, the cyclic action would propagate the vanishing around the cycle, contradicting the resolution of the identity.
Under the hypotheses of Lemma 5.8.1.1, a product of \(\ell \) Kraus operators moves each cyclic projection back by \(\ell \) steps. In particular, every product of \(m\) Kraus operators commutes with every cyclic projection.
Induct on the length of the word using (8).
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 B.7.
5.9 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 7.11.1 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 5.9.2.
5.10 Primitive overlap convergence (complementary transfer-map gap form)
For a primitive quantum channel or transfer map, removing the rank-one Perron projection \(P\) leaves a complementary map \(E-P\) with spectral radius strictly less than \(1\) [ 244 , Theorem 6.7 ] . Trace convergence and self-overlap asymptotics follow from this spectral gap, with the rank-one projection \(P=P_\rho \) and its trace properties developed in Section B.6 following [ 216 , “Complementary transfer-map gap for primitive maps” ] .
Let \(E\) be trace-preserving with fixed point \(\rho \), where \(\operatorname{tr}(\rho )\neq 0\), and let \(P\) be the fixed-point projection of the corresponding definition in the companion quantum-channel volume [ 216 , “Fixed-point projection” ] . If \(\rho _{\operatorname{spec}}(E-P){\lt}1\), then \(\operatorname{Tr}(E^n)\to 1\) as \(n\to \infty \).
By the power decomposition (established in the companion quantum-channel volume [ 216 , “Power decomposition” ] ), \(E^n=P+(E-P)^n\) for \(n\geq 1\). Since \(\rho _{\operatorname{spec}}(E-P){\lt}1\), the Gelfand formula gives \((E-P)^n\to 0\), so \(\operatorname{Tr}(E^n)\to \operatorname{Tr}(P)\). Lemma [ 216 , “Trace of the fixed-point projection” ] (Section B.6) gives \(\operatorname{Tr}(P)=1\).
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 \(N\to \infty \).