Stability of logarithmic Sobolev inequalities under a noncommutative change of measure

Abstract

We generalize Holley-Stroock's perturbation argument from commutative to quantum Markov semigroups. As a consequence, results on (complete) modified logarithmic Sobolev inequalities and logarithmic Sobolev inequalities for self-adjoint quantum Markov process can be used to prove estimates on the exponential convergence in relative entropy of quantum Markov systems which preserve a fixed state. This leads to estimates for the decay to equilibrium for coupled systems and to estimates for mixed state preparation times using Lindblad operators. Our techniques also apply to discrete time settings, where we show that the strong data processing inequality constant of a quantum channel can be controlled by that of a corresponding unital channel.

26 pages

References in corpus (4)

• Hypercontractivity of quasi-free quantum semigroups
• Complete entropic inequalities for quantum Markov chains
• On the modified logarithmic Sobolev inequality for the heat-bath dynamics for 1D systems
• Estimating the decoherence time using non-commutative Functional Inequalities

Cited by in corpus (9)

• Complete entropic inequalities for quantum Markov chains
• Entropy decay for Davies semigroups of a one dimensional quantum lattice
• Quantum concentration inequalities
• The modified logarithmic Sobolev inequality for quantum spin systems: classical and commuting nearest neighbour interactions
• Approximate tensorization of the relative entropy for noncommuting conditional expectations
• Quasi-factorization and Multiplicative Comparison of Subalgebra-Relative Entropy
• Geometric Approach Towards Complete Logarithmic Sobolev Inequalities
• Additivity of quantum capacities in simple non-degradable quantum channels
• Efficient and simple Gibbs state preparation of the 2D toric code via duality to classical Ising chains