On the modified logarithmic Sobolev inequality for the heat-bath dynamics for 1D systems

Abstract

The mixing time of Markovian dissipative evolutions of open quantum manybody systems can be bounded using optimal constants of certain quantum functional inequalities, such as the modified logarithmic Sobolev constant. For classical spin systems, the positivity of such constants follows from a mixing condition for the Gibbs measure, via quasi-factorization results for the entropy. Inspired by the classical case, we present a strategy to derive the positivity of the modified logarithmic Sobolev constant associated to the dynamics of certain quantum systems from some clustering conditions on the Gibbs state of a local, commuting Hamiltonian. In particular we show that for the heat-bath dynamics for 1D systems, the modified logarithmic Sobolev constant is positive under the assumptions of a mixing condition on the Gibbs state and a strong quasi-factorization of the relative entropy.

Approximate tensorization of the relative entropy for noncommuting conditional expectations

“…In this section, we prove our main results concerning approximate tensorization of the relative entropy (also known in the literature as quasi-factorization [10], [6], [2]). In particular, we relate the weak and strong constants to properties of the subalgebras.…”

“…when all algebras are commutative), and when M ≡ C1 H [10]. These inequalities, termed as approximate tensorization of the relative entropy (also known in the literature as quasi-factorization of the relative entropy, [10], [6], [2]), take the following form…”

“…Typically, c = 0 at infinite temperature, and remains small for conditional expectations onto far apart regions and at high enough temperature. Such an inequality was recently generalized to the quantum setting in [7,6,2]. In [17], a different extension of SSA in the case of noncommuting squares was proposed, with an additive error term that measures the distance from being a commuting square.…”

Ricci curvature of quantum channels on non-commutative transportation metric spaces

“…The efficiency of the Gibbs sampler depends on the time it takes to reach the approximating state. In the recent years, various Gibbs samplers were proposed in the literature [44,14,8,19,7]. In [53,65,29], the authors prove curvature lower bounds for different Gibbs samplers and Lipschitz constants.…”

Deviation bounds and concentration inequalities for quantum noises

“…Proofs of LSI/MLSI for quantum interacting spin systems (a.k.a. Gibbs samplers) have recently attracted the attention of the community [6,7,2]. More recently, it was shown that Gibbs states over arbitrary graphs satisfy a transportation cost inequality at large enough temperature [21].…”