B Peripheral Channel Structure and Transfer-Operator Gaps: Supporting Results
The spectral theory of MPS transfer maps in Chapter 5 relies on several structural steps: algebraic mixed-transfer identities, peripheral intertwiners for gauge rigidity, Banach-algebra decay bounds for overlaps, the rank-one Perron projection for primitive channels, and cyclic-sector decompositions.
B.1 Mixed-transfer powers and overlap traces
The mixed transfer operator \(F_{AB}\) of Definition 5.1.1 and its rectangular form \(F_{AB}^{\mathrm{rect}}\) of Definition 5.1.2 satisfy exact word expansions and trace identities.
Setting \(B=A\) recovers the ordinary transfer map: \(F_{AA}=\mathcal{E}_A\).
For every \(X\in M_{D}(\mathbb {C})\) and \(N\geq 0\),
where \(A^\sigma =A^{\sigma _1}\cdots A^{\sigma _N}\) denotes the word evaluation of Definition 2.1.2.
Induct on \(N\). The base case is \(F_{AB}^0(X)=X\). For the inductive step, apply \(F_{AB}\) to each summand in (1) and reindex using \(\{ 0,\ldots ,d{-}1\} ^{N+1}\cong \{ 0,\ldots ,d{-}1\} \times \{ 0,\ldots ,d{-}1\} ^N\).
Theorem 5.2.2 combines Lemma B.1.2 with Lemma [ 216 , “Trace expansion over matrix units” ] over the rectangular matrix-unit basis: expanding \(\operatorname{Tr}(T)\) for \(T=(F_{AB}^{\mathrm{rect}})^N\) by the trace-expansion formula of [ 216 , “Trace expansion over matrix units” ] , applying (1) to each \(T(E_{pq})\), and reassembling the sums over matrix-unit indices gives
Theorem 5.2.1 is the square case \(D_1=D_2=D\) of this statement, so it needs no separate expansion.
B.2 Auxiliary trace positivity
B.3 Rectangular spectral-radius formulation
For normalized tensors of bond dimensions \(D_1\) and \(D_2\), \(\rho _{\operatorname{spec}}(F_{AB}^{\mathrm{rect}})\leq 1\).
This is the MPS formulation of the corresponding theorem in the companion quantum-channel volume [ 216 , “Spectral-radius bound for a trace-preserving mixed Kraus map” ] : the rectangular mixed transfer is the mixed Kraus map, and the two tensor normalization identities are its trace-preserving hypotheses.
B.4 Peripheral intertwiners and gauge rigidity
Transporting a modulus-one mixed-transfer eigenvector through separate unital gauges produces Kraus-level intertwiners that force equal bond dimension once invertible, establishing the gauge rigidity of Theorem 5.6.1 and Theorem 5.6.4.
Let \(A\) be a normalized tensor of bond dimension \(D_1\) and \(B\) a normalized tensor of bond dimension \(D_2\). Suppose \(\rho _A,\rho _B\) are fixed points of the respective transfer maps and \(S_A,S_B\) are invertible matrices with \(S_AS_A^\dagger =\rho _A\) and \(S_BS_B^\dagger =\rho _B\) (for instance the positive definite Perron–Frobenius fixed points of Theorem 4.3.2 and their square roots, as in the right-canonical gauge of Theorem [ 216 , “Right-canonical (unital) gauge” ] ). Gauge to unital families \(A'^i=S_A^{-1}A^iS_A\) and \(B'^i=S_B^{-1}B^iS_B\). If \(X\in M_{D_1\times D_2}(\mathbb {C})\setminus \{ 0\} \) satisfies \(F_{AB}^{\mathrm{rect}}(X)=\mu X\) with \(|\mu |=1\), then \(X'=S_A^{-1}X(S_B^\dagger )^{-1}\) is nonzero, the families \(A'\) and \(B'\) are unital, and, for every \(i\),
Embed the gauged families and transported eigenvector into the block matrices
on \(M_{D_1+D_2}(\mathbb {C})\); the matrices \(\{ C^i\} \) form a unital Kraus family, and \(E_C(Y)=\mu Y\). 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 \(C^iY=\mu YC^i\). Expanding these block relations gives (2).
Let \(A\) and \(B\) be tensors of the same bond dimension. Suppose invertible gauges take them to \(A'\) and \(B'\), and suppose there are an invertible matrix \(X'\) and a scalar \(\mu \) with \(|\mu |=1\) such that \(A'^iX'=\mu X'B'^i\) for every physical index \(i\). Then \(A\) and \(B\) are gauge-phase equivalent.
If the bond dimension is zero, the bond spaces are empty, and the unit scalar together with the unique identity gauge satisfies the relation. Now suppose the bond dimension is positive, and write \(S_A,S_B\) for the two gauges. The corresponding gauged-intertwining theorem in the companion quantum-channel volume [ 216 , the square gauge construction ] forms the inverse pair
Transporting the assumed relation through \(S_A\) and \(S_B\) gives \(A^iX=\mu XB^i\). Multiplication by \(X^{-1}\) therefore yields
Since \(|\mu |=1\), the scalar \(\mu ^{-1}\) is nonzero, so this is the required gauge-phase equivalence.
Combining these two lemmas proves Theorem 5.6.1: Lemma B.4.1 supplies the intertwining relations (2) for the separately gauged, unital families. The Kadison–Schwarz equality used inside its own proof makes \(X'X'^\dagger \) and \(X'^\dagger X'\) nonzero positive semidefinite fixed points of the gauged transfer maps; under irreducibility, Theorem 4.1.1 upgrades these to positive definite matrices, so the transported intertwiner \(X'\) is invertible. Lemma B.4.2 then converts the invertible intertwiner, together with the two gauge matrices, into a single invertible matrix realizing the gauge-phase relation \(B^i=\overline\mu XA^iX^{-1}\) via the Skolem–Noether theorem (Theorem [ 216 , “Skolem–Noether” ] ).
Lemma 5.4.1 gives the companion kernel-invariance argument, independent of irreducibility: if a modulus-one rectangular intertwiner satisfying (6) and (7) exists between injective tensors of bond dimensions \(D_1\) and \(D_2\), then (6) shows that \(\ker (X)\) is invariant under every \((B^i)^\dagger \); since the \(\{ B^i\} \) span the full matrix algebra, so do the \(\{ (B^i)^\dagger \} \), so \(\ker (X)\) is trivial and \(D_2\leq D_1\). Taking adjoints of (7) and applying the same argument to \(X^\dagger \) shows that \(\ker (X^\dagger )\) is invariant under every \((A^i)^\dagger \) and hence trivial, giving \(D_1\leq D_2\). Combined with Lemma B.4.1, this proves \(\rho _{\operatorname{spec}}(F_{AB}^{\mathrm{rect}}){\lt}1\) whenever \(A\) and \(B\) are injective tensors of unequal bond dimension, which is exactly Theorem 5.4.2.
B.5 Spectral-radius decay and overlap limits
The spectral-radius gap \(\rho _{\operatorname{spec}}(F_{AB}) {\lt} 1\) translates into operator-norm decay and asymptotic orthogonality of matrix product vectors through standard Banach-algebra convergence.
Theorem 5.4.3 applies Lemma [ 216 , “Powers vanish below unit spectral radius” ] to the rectangular mixed transfer operator and then evaluates its powers at a fixed matrix. Theorem 5.5.4 combines the gap \(\rho _{\operatorname{spec}}(F_{AB}){\lt}1\) of Theorem 5.3.6 with the equal-bond-dimension specialization of this rectangular decay theorem. Since \(\operatorname{Tr}\) is continuous on the finite-dimensional matrix-endomorphism algebra, this gives \(\operatorname{Tr}(F_{AB}^N)\to 0\) term by term over the finitely many matrix-unit entries in the trace-expansion formula of [ 216 , “Trace expansion over matrix units” ] ; combined with the overlap–trace identity (4), this proves Theorem 5.5.5.
If \(A\) and \(B\) are injective normalized tensors that are not gauge-phase equivalent, then \(\operatorname{tr}(F_{AB}^N(X))\to 0\) as \(N\to \infty \) for every \(X\in M_{D}(\mathbb {C})\).
Theorem 5.5.4 gives \(F_{AB}^N(X)\to 0\) in operator norm, and continuity of the trace gives the conclusion.
If \(\rho \) is a fixed point of \(\mathcal{E}_A\), then \(\operatorname{tr}(\mathcal{E}_A^N(\rho ))=\operatorname{tr}(\rho )\) for every \(N\geq 0\). In particular, this applies to the distinguished Perron–Frobenius fixed point from Theorem 4.3.2.
Since \(\mathcal{E}_A(\rho )=\rho \), one has \(\mathcal{E}_A^N(\rho )=\rho \) for every \(N\). Taking the trace proves the claim.
B.6 Rank-one Perron projection
Let \(E:M_{D}(\mathbb {C})\to M_{D}(\mathbb {C})\) be a trace-preserving linear map with a fixed point \(\rho \neq 0\), \(\operatorname{tr}(\rho )\neq 0\), underlying the primitive overlap convergence in Theorem 5.10.2. Write \(P:=P_\rho \) for the fixed-point projection of the corresponding definition in the companion quantum-channel volume [ 216 , “Fixed-point projection” ] ,
and \(N:=E-P\) for the complementary part. The power decomposition (the corresponding theorem in the companion quantum-channel volume [ 216 , “Power decomposition” ] ) gives \(E^n=P+N^n\) for \(n\geq 1\).
Under the hypothesis \(\rho _{\operatorname{spec}}(N){\lt}1\) of Theorem 5.10.1, Lemma [ 216 , “Powers vanish below unit spectral radius” ] gives \(N^n\to 0\), so \(\operatorname{Tr}(E^n)=\operatorname{Tr}(P)+\operatorname{Tr}(N^n)\to \operatorname{Tr}(P)=1\) by Lemma [ 216 , “Trace of the fixed-point projection” ] . This is exactly the mechanism recorded in Theorem 5.10.1.
When \(E=\mathcal{E}_A\) is the transfer map of a normalized MPS tensor and \(\rho \) is its Perron–Frobenius fixed point (Theorem 4.3.2, Chapter 4), the overlap–trace identity (4) with \(A=B\) gives \(O_{AA}(N)=\operatorname{Tr}(\mathcal{E}_A^N)\), so Theorem 5.10.1 yields the primitive self-overlap limit of Theorem 5.10.2. The hypothesis \(\rho _{\operatorname{spec}}(N){\lt}1\) is exactly the complementary transfer-map gap that the complementary spectral-radius theorem for primitive maps (the corresponding theorem in the companion quantum-channel volume [ 216 , “Complementary transfer-map gap for primitive maps” ] ) supplies: for a primitive \(E\) all eigenvalues bounded by \(1\) in modulus and no trace-zero fixed point, every eigenvalue of \(N=E-P\) has modulus strictly less than \(1\), and since \(M_{D}(\mathbb {C})\) is finite-dimensional this bounds the spectral radius \(\rho _{\operatorname{spec}}(N)\) itself.
Let \(A\) be an irreducible normalized MPS tensor whose transfer map \(\mathcal{E}_A\) is primitive. Then there is a nonzero positive semidefinite fixed point \(\rho \) of \(\mathcal{E}_A\) with \(\operatorname{tr}(\rho )\neq 0\) such that, for \(P:=P_\rho \) and \(N:=\mathcal{E}_A-P\), \(\rho _{\operatorname{spec}}(N){\lt}1\).
This concretely instantiates the hypothesis of Theorem 5.10.1 and, via Theorem 5.10.2, gives the primitive self-overlap limit for every irreducible peripherally primitive normalized MPS tensor.
B.7 Periodicity removal
The cyclic decomposition for irreducible finite Kraus maps proceeds by normalizing a peripheral eigenvector to a unitary, forming its discrete Fourier spectral projections, and tracking the cyclic shift of the resulting corners under the channel.
Combining these lemmas proves Theorem [ 216 , “Cyclic decomposition for irreducible finite Kraus maps” ] : Lemma [ 216 , “Normalized peripheral unitary” ] supplies a unitary generator of exact order \(m\) for a primitive \(m\)-th peripheral root, and Lemma [ 216 , “Cyclic projections from a peripheral unitary” ] builds the \(m\) Fourier projections out of its powers, using Lemma [ 216 , “Powers on a peripheral-unitary orbit” ] to identify how the channel shifts them.
B.7.1 Cyclic corners and restricted primitivity
Corner preservation, the associated corner subspace, irreducibility of the restricted map, and corner rank are defined in the companion quantum-channel volume [ 216 , “Corner preservation” and the following corner definitions ] . We use those definitions here in the MPS periodicity-removal argument.
B.7.2 Group structure and divisibility
Theorem [ 216 , “Peripheral eigenspaces are one-dimensional” ] follows the same pattern: given a peripheral unitary eigenvector \(U\) for \(\gamma \) (Lemma [ 216 , “Peripheral unitary eigenvector” ] ), the multiplicative-domain identity gives \(E(XU^\dagger )=XU^\dagger \) for any \(\gamma \)-eigenvector \(X\), so the scalar fixed-point lemma for irreducible unital maps (Lemma [ 216 , “Scalar fixed points for irreducible unital maps” ] ) gives \(XU^\dagger =c\mathbb {1}\), hence \(X=cU\) and the eigenspace is one-dimensional.
The divisibility statements of Theorem [ 216 , “Transfer-map peripheral eigenvalues form a cyclic group” ] and Theorem [ 216 , “Transfer-map period divides bond dimension” ] follow from Lemma [ 216 , “Cyclic projections from a peripheral unitary” ] and Theorem [ 216 , “Rank equals trace for an orthogonal projection” ] : the \(m\) cyclic Fourier projections \(P_0,\ldots ,P_{m-1}\) sum to \(\mathbb {1}\) and are cyclically permuted by \(E\), so trace preservation gives \(\operatorname{tr}(P_{k+1})=\operatorname{tr}(P_k)\) for every \(k\), hence \(D=\operatorname{tr}(\mathbb {1})=\sum _{k=0}^{m-1}\operatorname{tr}(P_k)=m\, \operatorname{tr}(P_0)\). Since the trace of an orthogonal projection is a natural number (Theorem [ 216 , “Rank equals trace for an orthogonal projection” ] ), \(m\mid D\).
This periodicity-removal chain — unitary normalization, Fourier projections, cyclic Kraus shift (Lemma 5.8.1.1), the blocked cyclic corners of this section, and restricted primitivity — is exactly the machinery behind the period-removing blocking theorems of Chapter 7: [ 244 , Theorem 6.6 ] supplies the cyclic peripheral spectrum of the adjoint transfer map, and Theorem 7.3.1.1 (Section 7.3.1) and Theorem 7.11.1 apply it to realize the cyclic-sector decomposition of a blocked irreducible tensor with rectangular support isometries.