Dependencies

The connected two-point function is

For every \(\lambda _2\in \mathbb {C}\), define the associated correlation-length quantity by

For a nonzero subleading eigenvalue in the physical range \(0{\lt}|\lambda _2|{\lt}1\), this quantity is positive. If all subleading spectral values vanish and correlations therefore vanish after finitely many transfer steps, the total definition gives the limiting value \(\xi =0\).

For a matrix \(\rho \in M_{D}(\mathbb {C})\) and an auxiliary insertion \(X\in M_{D}(\mathbb {C})\), define

When \(\operatorname{tr}(\rho )=1\), this removes the trace component of \(X\rho \).

Two tensors \(A\) and \(B\) of the same bond dimension \(D\) are gauge equivalent if there exists an invertible matrix \(X \in \mathrm{GL}_D(\mathbb {C})\) such that, for every \(i \in \{ 0,\ldots ,d{-}1\} \),

In tensor-network notation, (8) is represented by

where the black node denotes the tensor \(A\), the upper leg is the physical index \(i\), and the two red side nodes denote the gauge matrices acting on the virtual legs.

A tensor \(A\) is injective if its matrices span the full matrix algebra: \(\operatorname{span}_{\mathbb {C}}\{ A^i : i = 0,\ldots ,d{-}1\} = M_{D}(\mathbb {C})\).

An MPS tensor \(A=(A^i)_i\) is left-canonical when

Equivalently, its Kraus map is trace-preserving.

A tensor \(A\) of physical dimension \(d\) and bond dimension \(D\) is a normal tensor if, after the spectral-radius normalization of  [ 13 ] , (i) \(A\) admits no nontrivial invariant orthogonal projection and (ii) the associated CPM \(\mathcal{E}_A(X)=\sum _i A^iX(A^i)^\dagger \) has spectral radius \(r(\mathcal{E}_A)=1\) and peripheral spectrum \(\sigma _\partial (\mathcal{E}_A)=\{ 1\} \).

Fix an MPS tensor and a right fixed point of its transfer map. Write the tensor as \(A\) and the fixed point as \(\rho ^R\). Under the hypotheses of Theorem 4.3.2, that fixed point exists and is unique up to scaling; we normalize it so that \(\operatorname{tr}(\rho ^R)=1\). For an auxiliary-space insertion \(X\in M_{D}(\mathbb {C})\), define

Let \(\rho \in M_{D}(\mathbb {C})\) have nonzero trace, and set \(P(Z)=\operatorname{tr}(Z)\rho /\operatorname{tr}\rho \). For physical operators \(X,Y\) on independently chosen blocks of lengths \(L_1,L_2\), and \(n\geq 0\) unobserved intermediate sites, define

The centering of the inner insertion includes adjacent blocks, \(n=0\).

For matrices \(\rho ^L,\rho ^R\in M_{D}(\mathbb {C})\) and physical operators \(X,Y\) on independently chosen blocks of lengths \(L_1,L_2\), define

When \(\rho ^L,\rho ^R\) are positive fixed matrices of the dual transfer map and the transfer map, respectively, normalized by \(\operatorname{tr}(\rho ^L\rho ^R)=1\), this is the connected infinite-chain physical expectation with \(n\geq 0\) unobserved intermediate sites. This is the fixed-point contraction of [ 17 , Section II.B.3 ] .

For an observable \(O\) on a block of \(L\) physical spins, define the linear map on the virtual matrix space by

This transfer map also depends linearly on the observable \(O\). Diagrammatically, \(O\) is inserted between the two layers of the doubled chain on its \(L\) sites:

This is the observable transfer map of [ 13 , lines 490–496 ] .

The transfer map associated to a tensor \(A\) is the linear map \(\mathcal{E}_A : M_{D}(\mathbb {C}) \to M_{D}(\mathbb {C})\) defined by

Diagrammatically, the transfer map is the double-layer contraction

in which the upper node denotes \(A\), the lower node denotes \(A^\dagger \), and the physical index is summed over between them.

For auxiliary-space insertions \(X,Y\in M_{D}(\mathbb {C})\) and \(n\ge 0\), define

Equivalently, insert \(X\) at site \(0\), propagate by \(\mathcal{E}_A^n\), then insert \(Y\) and take the trace.

If \(|\lambda _2| {\lt} 1\) and \(\lambda _2 \ne 0\), then \(\xi {\gt} 0\).

If \(B^i = X A^i X^{-1}\) for all \(i\), then \(B^w = X A^w X^{-1}\) for every word \(w\).

Let the products of some fixed number of matrices of \(A\) span \(M_{D}(\mathbb {C})\), and let \(\lambda _1\) be an eigenvalue of \(\mathcal{E}_A\) of largest modulus. Then \(\lambda _1\ne 0\), and there are \(\zeta \ne 0\), an invertible \(X\) and \(L\ge 1\) such that \(B^i=X\zeta A^iX^{-1}\) satisfies the gauge (3) for some \(\rho \), the products of \(L\) matrices of \(B\) span \(M_{D}(\mathbb {C})\), the eigenvalues of \(\mathcal{E}_B\) are those of \(\mathcal{E}_A\) divided by \(\lambda _1\), and \(|\phi _N(B)\rangle =\zeta ^N|\phi _N(A)\rangle \) for every \(N\). In particular, whenever \(|\phi _N(A)\rangle \neq 0\), \(\lvert \langle \phi _N(B) | \psi \rangle \rvert /\lVert \phi _N(B)\rVert =\lvert \langle \phi _N(A) | \psi \rangle \rvert /\lVert \phi _N(A)\rVert \) for every vector \(|\psi \rangle \); when \(|\phi _N(A)\rangle =0\), also \(|\phi _N(B)\rangle =0\). If instead \(A\) is irreducible and \(\mathcal{E}_A\) has the unique eigenvalue \(1\) of largest modulus, the same holds with \(\zeta =1\).

For operators \(X,X'\) on \((\mathbb {C}^d)^{\otimes L_1}\) and \(Y,Y'\) on \((\mathbb {C}^d)^{\otimes L_2}\), with \(X\otimes Y\) acting on \(L_1+L_2\) consecutive sites,

For an operator \(O\) on \((\mathbb {C}^d)^{\otimes N}\) and the unnormalized vector \(|\phi _N(A)\rangle \) of (1), \(\langle \phi _N(A)|O|\phi _N(A)\rangle =\operatorname {tr}\mathcal{E}_O\), the trace of a linear map on \(M_{D}(\mathbb {C})\).

Let \(A\) be normal in trace-preserving gauge, and let \(\rho \succeq 0\) satisfy \(\mathcal{E}_A(\rho )=\rho \) and \(\operatorname{tr}\rho =1\). Then \(\rho _{\operatorname{spec}}(\mathcal{E}_A-P){\lt}1\), where \(P(Z)=\operatorname{tr}(Z)\rho \).

Suppose \(\mathcal{E}_A\) is trace preserving, \(\mathcal{E}_A(\rho )=\rho \), and \(\operatorname{tr}\rho =1\). Then, for all \(n\geq 0\),

For \(n\geq 1\) this also equals \(\operatorname{tr}(\mathcal{E}_X(\mathcal{E}_A-P)^n(\mathcal{E}_Y(\rho )))\), including the equal-support specialization used in preparation estimates.

A normal tensor \(A\) with leading transfer eigenvalue one admits a trace-preserving gauge representative \(B\) with positive trace-one fixed state \(\rho _B\), and positive normalized left/right fixed matrices \(\rho ^L,\rho ^R\) for \(A\). At every separation and for every pair of finite-support physical operators, the left-right connected contraction for \(A\) equals that for \(B,\mathbb {1},\rho _B\).

If \(\operatorname{tr}(\rho )=1\), then

Suppose that \(\rho \) is a right fixed point of the transfer map, \(\mathcal{E}_A(\rho )=\rho \). Then, for every \(n\geq 0\),

When moreover \(\operatorname{tr}(\rho )=1\), the preceding lemma shows that \(Z_\rho (X)\) is traceless. The identity isolates the fixed-point contribution before the remaining transfer iterates are analysed.

For an injective trace-preserving MPS tensor, there exist \(C{\gt}0\) and \(\xi {\gt}0\) such that, for all \(n\ge 0\) and all traceless \(X\in M_{D}(\mathbb {C})\), i.e. \(\operatorname{tr}(X)=0\),

Traceless matrices lie in \(\ker P\), where \(P\) is the fixed-point projection (7). Since \(\mathcal{E}_A-P\) has spectral radius strictly less than \(1\), choose any \(r\) with \(\rho _{\operatorname{spec}}(\mathcal{E}_A-P){\lt}r{\lt}1\). The Gelfand formula then gives exponential decay with \(\xi =-1/\log r\). Choosing a rate strictly above the spectral radius is necessary in general because a Jordan block at the spectral radius can prevent a uniform bound proportional to \(\rho _{\operatorname{spec}}(\mathcal{E}_A-P)^n\).

Let \(A\) be a normal tensor, with leading transfer eigenvalue normalized to one. There are positive definite matrices \(\rho ^L,\rho ^R\) satisfying

and a rate \(0{\lt}r{\lt}1\) such that every pair \(X,Y\) of physical operators on independently chosen finite supports admits a constant \(C_{XY}{\gt}0\) with \(|C_{\rho ^L,\rho ^R}^{A}(X,Y;n)|\leq C_{XY}r^n\) for every \(n\geq 0\). No trace-preserving gauge or diagonalizability hypothesis is imposed. This proves the decay conclusion of [ 17 , Section II.B.3 ] for normal tensors. The more general class whose unique eigenvalue of largest modulus is algebraically simple, with possibly singular fixed matrices, is outside this statement.

Let \(A\) be normal with leading transfer eigenvalue one. There are positive normalized left/right fixed matrices for which every finite-support physical connected correlator has the form

at every \(n\ge 0\). Here \(0\notin S\), each \(\mu \in S\) is an actual transfer eigenvalue of \(A\) with \(|\mu |{\lt}1\), and \(t(n)=0\) for \(n\ge D^2\). A single canonical representative \(B\) supplies all Jordan indices: \(\deg p_\mu \) is strictly below the stabilization index of the generalized \(\mu \)-eigenspace of its complementary transfer map. No diagonalizability, assumed expansion, or trace-preserving condition on the original \(A\) is required.

Let \(X\) be invertible and suppose

for every physical index \(i\). If \(B\) is a CPSV normal tensor, then \(A\) is a CPSV normal tensor.

Fix an injective, trace-preserving MPS tensor \(A\) with positive definite fixed point \(\rho \). Let

be the corresponding fixed-point projection. Then there exist \(C {\gt} 0\) and \(0 {\lt} \delta \le 1\) such that, for all \(n\ge 0\) and \(X\in M_{D}(\mathbb {C})\),

The transfer-map gap \(\delta \) exists by primitivity (the corresponding theorem in the companion quantum-channel volume  [ 216 , “Complementary transfer-map gap for primitive maps” ] , the complementary transfer-map gap).

Let \(T=\mathcal{E}_A-P\) be the complementary transfer map used in the centered physical correlator. There is a finite set \(S\) of eigenvalues of \(T\) and coefficients \(c_{\mu j}\) such that, at every separation \(n\ge 0\),

where \(k_\mu \) is the least exponent at which the generalized \(\mu \)-eigenspace stabilizes. Terms with \(j{\gt}n\) have zero binomial coefficient; the displayed exponent uses truncated natural subtraction. This formula includes \(n=0\) and \(\mu =0\), with \(0^0=1\).

Suppose \(B^i=UA^iU^{-1}\) with \(U\) invertible, and put \(\widetilde\rho ^L=U^{-\dagger }\rho ^LU^{-1}\) and \(\widetilde\rho ^R=U\rho ^RU^\dagger \). Then, for every pair of independently supported physical operators and every \(n\geq 0\),

The identity does not require stationarity or positivity of the boundary matrices.

The same correlator has the all-separation form

with \(\deg p_\mu {\lt}k_\mu \) and \(t(n)=0\) for \(n\ge k_0\).

For any tensor \(A\) and matrix \(\rho \) of nonzero trace, every rate \(r{\gt}\rho _{\operatorname{spec}}(\mathcal{E}_A-P)\) and every pair of physical operators on independent finite supports have a constant \(C_{XY}{\gt}0\) such that, for all \(n\geq 0\),

If \(A\) is normal in trace-preserving gauge and \(\rho \succeq 0\) is a normalized fixed state, one can choose a single rate \(\rho _{\operatorname{spec}}(\mathcal{E}_A-P){\lt}r{\lt}1\) for every finite support and every pair of observables. Thus the fixed-point infinite-chain connected correlations decay exponentially, without a diagonalizability assumption. The source’s pure-exponential expansion needs the Jordan-block correction recorded in [ 94 ] .

Suppose \(B^i=UA^iU^{-1}\) with \(U\) invertible. For every physical operator \(O\) on \(L\) sites and every \(Z\in M_{D}(\mathbb {C})\),

This includes \(L=0\); no unitarity or normalization is assumed.

Suppose \(\mathcal{E}_A\) is trace preserving, \(\mathcal{E}_A(\rho )=\rho \), and \(\operatorname{tr}\rho =1\). For every pair of independently supported physical operators and every \(n\geq 0\), \(C_{\mathbb {1},\rho }^{A}(X,Y;n)=C_{\mathrm{phys}}(X,Y;n)\).

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 \(N\to \infty \).

Assume \(D \ge 1\). Let \(A\) be an injective MPS tensor with \(\sum _i (A^i)^\dagger A^i = \mathbb {1}\). Then the transfer map \(\mathcal{E}_A\) has a unique positive semidefinite fixed point \(\rho \) up to scaling, and \(\rho \) is positive definite.

For an injective trace-preserving MPS tensor, there exists \(\delta {\gt}0\) such that every eigenvalue \(\mu \ne 1\) of the transfer map satisfies \(|\mu |\le 1-\delta \).

Let \(A\) and \(B\) be injective normalized MPS tensors that are not gauge-phase equivalent. Then \(\rho _{\operatorname{spec}}(F_{AB}){\lt}1\).