Hypercontractivity and logarithmic Sobolev Inequality for non-primitive quantum Markov semigroups and estimation of decoherence rates

Understand

We generalize the concepts of weak quantum logarithmic Sobolev inequality (LSI) and weak hypercontractivity (HC), introduced in the quantum setting by Olkiewicz and Zegarlinski, to the case of non-primitive quantum Markov semigroups (QMS).

• The originality of this work resides in that this new notion of hypercontractivity is given in terms of the so-called amalgamated $\mathbb{L}_p$ norms introduced recently by Junge and Parcet in the context of operator spaces theory.
• We make three main contributions.
• The first one is a version of Gross' integration lemma: we prove that (weak) HC implies (weak) LSI.