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arXiv:2405.04947 · Journal of Functional Analysis

The spectral gap of a Gaussian quantum Markovian generator

Franco Fagnola, Damiano Poletti, Emanuela Sasso, Veronica Umanità

Published Jul 2, 2025• 4 citations

Abstract

Gaussian quantum Markov semigroups are the natural non-commutative extension of classical Ornstein-Uhlenbeck semigroups. They arise in open quantum systems of bosons where canonical non-commuting random variables of positions and momenta come into play. If there exists a faithful invariant density we explicitly compute the optimal exponential convergence rate, namely the spectral gap of the generator, in non-commutative L 2 spaces determined by the invariant density showing that the exact value is the lowest eigenvalue of a certain matrix determined by the diffusion and drift matrices. The spectral gap turns out to depend on the non-commutative L 2 space considered, whether the one determined by the so-called GNS or KMS multiplication by the square root of the invariant density. In the first case, it is strictly positive if and only if there is the maximum number of linearly independent noises. While, we exhibit explicit examples in which it is strictly positive only with KMS multiplication. We do not assume any symmetry or quantum detailed balance condition with respect to the invariant density.

DOI: 10.1016/j.jfa.2025.111119

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Let $(T_t)_{t\geq 0}$ be a Gaussian quantum Markov semigroup with a faithful normal invariant state $ρ$. For every $s\in[0,1]$, the $s$-embedding associated with $ρ$ induces a contraction semigroup $(T_t^{(s)})_{t\geq 0}$ on the Hilbert--Schmidt space $\mathcal B_2(\mathsf h)$; let $L^{(s)}$ denote its generator. Without assuming symmetry or quantum detailed balance, we determine the full spectra of $L^{(s)}$ and $L^{(s)\ast}$: they consist, respectively, of the non-negative integer combinations of the eigenvalues of the phase-space drift matrix and of their complex conjugates. We also diagonalize the self-adjoint closure of $L^{(s)\ast}+L^{(s)}$ and obtain an explicit formula for the spectral gap. Using quantum characteristic functions, we represent $T_t^{(s)}$, up to unitary equivalence, as a complex Gaussian integral operator and prove that its kernel is square-integrable for every $t>0$. Hence $T_t^{(s)}$ is a Hilbert--Schmidt operator on $\mathcal B_2(\mathsf h)$ for every $t>0$, and every induced semigroup is immediately compact. Consequently, $L^{(s)}$ and $L^{(s)\ast}$ have compact resolvent for all $s\in[0,1]$, and their point spectra exhaust their full spectra. These results provide a full quantum counterpart of classical Ornstein--Uhlenbeck spectral theory.

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Gaussian quantum Markov semigroups are the natural non-commutative extension of classical Ornstein-Uhlenbeck semigroups. They arise in open quantum systems of bosons where canonical non-commuting random variables of positions and momenta come into play. If there exists a faithful invariant density we explicitly compute the optimal exponential convergence rate, namely the spectral gap of the generator, in non-commutative L 2 spaces determined by the invariant density showing that the exact value is the lowest eigenvalue of a certain matrix determined by the diffusion and drift matrices. The spectral gap turns out to depend on the non-commutative L 2 space considered, whether the one determined by the so-called GNS or KMS multiplication by the square root of the invariant density. In the first case, it is strictly positive if and only if there is the maximum number of linearly independent noises. While, we exhibit explicit examples in which it is strictly positive only with KMS multiplication. We do not assume any symmetry or quantum detailed balance condition with respect to the invariant density.

Let

(T_t)_{t\geq 0}

be a Gaussian quantum Markov semigroup with a faithful normal invariant state

ρ

. For every

s\in[0,1]

, the

s

-embedding associated with

ρ

induces a contraction semigroup

(T_t^{(s)})_{t\geq 0}

on the Hilbert--Schmidt space

\mathcal B_2(\mathsf h)

; let

L^{(s)}

denote its generator. Without assuming symmetry or quantum detailed balance, we determine the full spectra of

L^{(s)}

and

L^{(s)\ast}

: they consist, respectively, of the non-negative integer combinations of the eigenvalues of the phase-space drift matrix and of their complex conjugates. We also diagonalize the self-adjoint closure of

L^{(s)\ast}+L^{(s)}

and obtain an explicit formula for the spectral gap. Using quantum characteristic functions, we represent

T_t^{(s)}

, up to unitary equivalence, as a complex Gaussian integral operator and prove that its kernel is square-integrable for every

t>0

. Hence

T_t^{(s)}

is a Hilbert--Schmidt operator on

\mathcal B_2(\mathsf h)

for every

t>0

, and every induced semigroup is immediately compact. Consequently,

L^{(s)}

and

L^{(s)\ast}

have compact resolvent for all

s\in[0,1]

, and their point spectra exhaust their full spectra. These results provide a full quantum counterpart of classical Ornstein--Uhlenbeck spectral theory.

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.

We establish a relation between the exponential decay rates of quantum Markov semigroups with respect to different inner products. More precisely, it was conjectured by Fagnola, Poletti, Sasso and Umanità that for a Gaussian quantum Markov semigroup, the exponential decay rate with respect to the KMS inner product is bounded below by the exponential decay rate for the GNS inner product. We show that this is indeed the case and not limited to Gaussian quantum Markov semigroups, but holds for quantum Markov semigroups with a faithful normal invariant state on arbitrary von Neumann algebras. Additionally, the KMS inner product can be replaced by a whole class of inner products induced by operator monotone functions.

We determine the spectrum of the predual generator of an irreducible Gaussian quantum Markov semigroup on trace-class operators, the natural space for density operators. The drift matrix governs the linear part of the Gaussian evolution and organizes the spectral classification. When the linear flow generated by this matrix is not periodic, the spectrum is the closed left half-plane. If all drift eigenvalues have negative real part, every interior point is an eigenvalue. There are no trace-class eigenvectors if some drift eigenvalue has positive real part or all have zero real part. For periodic flow, the spectrum consists of unions of horizontal half-lines or filled parabolic regions. Examples show that irreducibility cannot in general be omitted. We apply the results to oscillators, free particles, and boson chains.

We investigate symmetric Gaussian Quantum Markov Semigroups (G-QMSs), a class of quantum dynamical semigroups acting on the algebra of bounded operators over a bosonic Fock space, whose generators are given in generalized GKLS form with linear and quadratic terms in canonical operators. Our main goal is to characterize the symmetry of such semigroups with respect to a faithful invariant Gaussian state, in terms of explicit algebraic conditions on the generator’s parameters. We provide necessary and sufficient conditions for GNS-symmetry using the action of the semigroup on Weyl operators, and show that under natural assumptions, every symmetric G-QMS is unitarily equivalent to a direct sum of Ornstein–Uhlenbeck-type semigroups. The results are further linked to modular theory by confirming the well-known fact that GNS-symmetry implies commutation with the modular automorphism group. Technical appendices discuss the role of Bogoliubov and metaplectic transformations in simplifying the structure of symmetric semigroups.