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.
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.
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
[UV], 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
Un and
Vn denote the submatrices of
U and
V, respectively, corresponding to the
n-th equal-temperature block. We prove that
kerUn=kerVn 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 prove a noncommutative
p-Poincaré inequality for GNS-detailed-balance quantum Markov semigroups (QMSs) on non-tracial
σ-finite von Neumann algebras, assuming only the existence of a spectral gap. Extending the semi-commutative results of Huang and Tropp, we first establish the inequality for tracial von Neumann algebras. We use Markov dilations to obtain chain-rule estimates for Dirichlet forms and amalgamated free products to define an appropriate noncommutative derivation. We then extend the result to QMSs satisfying GNS detailed balance on non-tracial
σ-finite von Neumann algebras, using Haagerup's reduction and Kosaki's interpolation theorem. As applications, we recover sub-exponential concentration inequalities and estimate the Lipschitz and completely bounded Lipschitz diameters of the QMSs.
We introduce the notion of the BKM coercivity constant and show that the positivity of this constant governs the exponential relative entropy decay of the semigroup. Indeed, we prove that the modified log-Sobolev constant is equivalent to the BKM coercivity constant up to a constant depending on the asymptotic conditional expectation. We also discover that the BKM coercivity constants for matrix amplifications are already attained with a qubit auxiliary system, which also coincides with the GNS spectral gap of the generating Lindbladian. As a corollary, we establish a criterion that the complete modified log-Sobolev inequality holds if and only if the GNS gap of the semigroup is strictly positive. All the discussion above applies to quantum Markov semigroups that admit a faithful asymptotic conditional expectation as long-time equilibration, with no detailed balance condition assumed. As an application, we show the finite-dimensional Chen-Kastoryano-Gilyén Gibbs samplers always satisfy the complete modified log-Sobolev inequality.
Let
(Tt)t≥0 be a Gaussian quantum Markov semigroup with a faithful normal invariant state
ρ. For every
s∈[0,1], the
s-embedding associated with
ρ induces a contraction semigroup
(Tt(s))t≥0 on the Hilbert--Schmidt space
B2(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)∗: 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)∗+L(s) and obtain an explicit formula for the spectral gap. Using quantum characteristic functions, we represent
Tt(s), up to unitary equivalence, as a complex Gaussian integral operator and prove that its kernel is square-integrable for every
t>0. Hence
Tt(s) is a Hilbert--Schmidt operator on
B2(h) for every
t>0, and every induced semigroup is immediately compact. Consequently,
L(s) and
L(s)∗ have compact resolvent for all
s∈[0,1], and their point spectra exhaust their full spectra. These results provide a full quantum counterpart of classical Ornstein--Uhlenbeck spectral theory.