arXiv:2505.08991

Spectral Gap Bounds for Quantum Markov Semigroups via Correlation Decay

Lucia, Angelo, David Pérez-Garcı́a, Antonio Pérez

Published May 13, 2025• 0 citations

Abstract

Starting from an arbitrary full-rank state of a lattice quantum spin system, we define a "canonical purified Hamiltonian" and characterize its spectral gap in terms of a spatial mixing condition (or correlation decay) of the state. When the state considered is a Gibbs state of a local, commuting Hamiltonian at positive temperature, we show that the spectral gap of the canonical purified Hamiltonian provides a lower bound to the spectral gap of a large class of reversible generators of quantum Markov semigroup, including local and ergodic Davies generators. As an application of our construction, we show that the mixing condition is always satisfied for any finite-range 1D model, as well as by Kitaev's quantum double models.

DOI: 10.48550/arxiv.2505.08991

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arXiv:2505.08991

Spectral Gap Bounds for Quantum Markov Semigroups via Correlation Decay

Lucia, Angelo, David Pérez-Garcı́a, Antonio Pérez

Published May 13, 2025• 0 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.2505.08991

Related papers

Double Localization for Quantum Gibbs Sampler Gaps: From an Abstract Framework to Finite-Group Models

Ryu Hayakawa, Angus Southwell, Caesnan M. G. Leditto +2 more

We introduce double localization, a framework for proving spectral gaps of quantum Gibbs samplers through two complementary operations: geometric localization, which selects updates supported in small spatial regions, and interface localization, which focuses on a model-defined subspace of observables while remaining geometrically global. Our abstract gap theorem combines a global bound on this subspace, called the interface, with local quantum gap estimates under quantitative coupling and geometric assembly conditions. The resulting explicit lower bound controls the gap of the full dynamics without requiring the interface to be invariant under the generator. This separation allows global estimates, including classical comparison, to be combined with local control of the remaining quantum directions. We apply the framework to Hamiltonians built from the vertex terms of Kitaev's finite-group quantum double construction, without plaquette terms, on finite simple three-regular graphs of girth at least six. For the specified local Davies dynamics, we prove an unconditional spectral-gap lower bound by a positive constant independent of graph size for every fixed nontrivial finite group and each fixed inverse temperature $β$ with $0\leβJ<\log(5/3)$ , where $J>0$ is the coupling strength. For the smallest non-Abelian group $S_3$ , double localization yields a positive lower bound uniform in both graph size and the entire interval $0\leβJ\le\log 3$ , providing a guarantee that does not follow directly from existing results. This demonstrates how model-specific finite calculations can extend spectral-gap guarantees for the full quantum dynamics.We introduce double localization, a framework for proving spectral gaps of quantum Gibbs samplers through two complementary operations: geometric localization, which selects updates supported in small spatial regions, and interface localization, which focuses on a model-defined subspace of observables while remaining geometrically global. Our abstract gap theorem combines a global bound on this subspace, called the interface, with local quantum gap estimates under quantitative coupling and geometric assembly conditions. The resulting explicit lower bound controls the gap of the full dynamics without requiring the interface to be invariant under the generator. This separation allows global estimates, including classical comparison, to be combined with local control of the remaining quantum directions. We apply the framework to Hamiltonians built from the vertex terms of Kitaev's finite-group quantum double construction, without plaquette terms, on finite simple three-regular graphs of girth at least six. For the specified local Davies dynamics, we prove an unconditional spectral-gap lower bound by a positive constant independent of graph size for every fixed nontrivial finite group and each fixed inverse temperature $β$ with $0\leβJ<\log(5/3)$, where $J>0$ is the coupling strength. For the smallest non-Abelian group $S_3$, double localization yields a positive lower bound uniform in both graph size and the entire interval $0\leβJ\le\log 3$, providing a guarantee that does not follow directly from existing results. This demonstrates how model-specific finite calculations can extend spectral-gap guarantees for the full quantum dynamics.

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Despite extensive study, our understanding of quantum Markov chains remains far less complete than that of their classical counterparts. [Temme'13] observed that the Davies Lindbladian, a well-studied model of quantum Markov dynamics, contains an embedded classical Markov generator, raising the natural question of how the convergence properties of the quantum and classical dynamics are related. While [Temme'13] showed that the spectral gap of the Davies Lindbladian can be much smaller than that of the embedded classical generator for certain highly structured Hamiltonians, we show that if the spectrum of the Hamiltonian does not contain long arithmetic progressions, then the two spectral gaps must be comparable. As a consequence, we prove that for a large class of Hamiltonians, including those obtained by perturbing a fixed Hamiltonian with a generic external field, the quantum spectral gap remains within a constant factor of the classical spectral gap. Our result aligns with physical intuition and enables the application of classical Markov chain techniques to the quantum setting. The proof is based on showing that any ``off-diagonal'' eigenvector of the Davies generator can be used to construct an observable which commutes with the Hamiltonian and has a Lindbladian Rayleigh quotient which can be upper bounded in terms of that of the original eigenvector's Lindbladian Rayleigh quotient. Thus, a spectral gap for such observables implies a spectral gap for the full Davies generator.Despite extensive study, our understanding of quantum Markov chains remains far less complete than that of their classical counterparts. [Temme'13] observed that the Davies Lindbladian, a well-studied model of quantum Markov dynamics, contains an embedded classical Markov generator, raising the natural question of how the convergence properties of the quantum and classical dynamics are related. While [Temme'13] showed that the spectral gap of the Davies Lindbladian can be much smaller than that of the embedded classical generator for certain highly structured Hamiltonians, we show that if the spectrum of the Hamiltonian does not contain long arithmetic progressions, then the two spectral gaps must be comparable. As a consequence, we prove that for a large class of Hamiltonians, including those obtained by perturbing a fixed Hamiltonian with a generic external field, the quantum spectral gap remains within a constant factor of the classical spectral gap. Our result aligns with physical intuition and enables the application of classical Markov chain techniques to the quantum setting. The proof is based on showing that any ``off-diagonal'' eigenvector of the Davies generator can be used to construct an observable which commutes with the Hamiltonian and has a Lindbladian Rayleigh quotient which can be upper bounded in terms of that of the original eigenvector's Lindbladian Rayleigh quotient. Thus, a spectral gap for such observables implies a spectral gap for the full Davies generator.

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On the existence of the KMS spectral gap in Gaussian quantum Markov semigroups

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In arXiv:2405.04947, it was shown that a Gaussian quantum Markov semigroup on the $d$ -mode bosonic Fock space with a unique faithful normal invariant state has a positive GNS spectral gap if and only if the matrix $[U \overline{V}]$ , formed from the coefficients of the Kraus operators, has $2d$ linearly independent columns. In this paper, we establish the corresponding criterion for the KMS spectral gap. After standardizing the invariant Gaussian state and grouping the modes according to their inverse temperatures, let $U_n$ and $V_n$ denote the submatrices of $U$ and $V$ , respectively, corresponding to the $n$ -th equal-temperature block. We prove that $\ker U_n=\ker V_n$ for every $n$ , and that the KMS spectral gap is positive if and only if this common kernel is trivial for every equal-temperature block. Thus, whereas the GNS spectral gap requires a global maximal-rank condition on the Kraus coefficients, the KMS spectral gap requires maximal column rank only within each equal-temperature block.In arXiv:2405.04947, it was shown that a Gaussian quantum Markov semigroup on the $d$-mode bosonic Fock space with a unique faithful normal invariant state has a positive GNS spectral gap if and only if the matrix $[U \overline{V}]$, formed from the coefficients of the Kraus operators, has $2d$ linearly independent columns. In this paper, we establish the corresponding criterion for the KMS spectral gap. After standardizing the invariant Gaussian state and grouping the modes according to their inverse temperatures, let $U_n$ and $V_n$ denote the submatrices of $U$ and $V$, respectively, corresponding to the $n$-th equal-temperature block. We prove that $\ker U_n=\ker V_n$ for every $n$, and that the KMS spectral gap is positive if and only if this common kernel is trivial for every equal-temperature block. Thus, whereas the GNS spectral gap requires a global maximal-rank condition on the Kraus coefficients, the KMS spectral gap requires maximal column rank only within each equal-temperature block.

math.FACross-listedarXiv:2512.23414Dec 30, 20250 citations

β

with

0\leβJ<\log(5/3)

, where

J>0

is the coupling strength. For the smallest non-Abelian group

S_3

, double localization yields a positive lower bound uniform in both graph size and the entire interval

0\leβJ\le\log 3

, providing a guarantee that does not follow directly from existing results. This demonstrates how model-specific finite calculations can extend spectral-gap guarantees for the full quantum dynamics.

The mean-field Heisenberg ferromagnet is a quantum spin model on the complete graph with isotropic spin-1/2 interactions. This non-commuting Hamiltonian is permutation and

\mathsf{SU}(2)

invariant, and its Gibbs states undergo an

\mathsf{SU}(2)

symmetry breaking phase transition at inverse temperature

β=2

. We consider the associated Davies generator, a canonical model of open-system thermalization, and prove tight asymptotic estimates for its spectral gap at all noncritical temperatures. For fixed

β<2

, the gap as a function of number of qubits

n

Θ(1)

, while for fixed

β>2

the gap is

Θ(n^{-1})

. The matching upper bound of the spectral gap is witnessed by the total magnetization order parameter, suggesting that the low-temperature (

β>2

) slowdown is associated with broken continuous symmetry. Two key ingredients in our approach are a comparison argument, which introduces auxiliary generators to bound dissipation on nontrivial representations of the symmetry groups

\mathsf{SU}(2)

and

\mathsf{S}_n

, and a decomposition of the space of observables into spherical tensor operators to reveal a form of monotonicity.

In arXiv:2405.04947, it was shown that a Gaussian quantum Markov semigroup on the

d

-mode bosonic Fock space with a unique faithful normal invariant state has a positive GNS spectral gap if and only if the matrix

[U \overline{V}]

, formed from the coefficients of the Kraus operators, has

2d

linearly independent columns. In this paper, we establish the corresponding criterion for the KMS spectral gap. After standardizing the invariant Gaussian state and grouping the modes according to their inverse temperatures, let

U_n

and

V_n

denote the submatrices of

U

and

V

, respectively, corresponding to the

n

-th equal-temperature block. We prove that

\ker U_n=\ker V_n

for every

n

, and that the KMS spectral gap is positive if and only if this common kernel is trivial for every equal-temperature block. Thus, whereas the GNS spectral gap requires a global maximal-rank condition on the Kraus coefficients, the KMS spectral gap requires maximal column rank only within each equal-temperature block.