Estimating the decoherence time using non-commutative Functional Inequalities

TL;DR

This paper extends non-commutative functional inequalities to non-primitive quantum Markov semigroups, providing tools to estimate decoherence times and analyzing their algebraic structures, with implications for quantum error correction.

Contribution

It generalizes key inequalities to broader quantum semigroup classes and links these to decoherence and error correction, including new quantum scenarios.

Findings

Positivity of Poincaré constant linked to spectral gap.

Positivity of modified log-Sobolev constant under $\\\mathbb{L}_1$-regularity.

Strong $\mathbb{L}_p$-regularity holds under detailed balance conditions.

Abstract

We generalize the notions of the non-commutative Poincar\'e and modified log-Sobolev inequalities for primitive quantum Markov semigroups (QMS) to not necessarily primitive ones. These two inequalities provide estimates on the decoherence time of the evolution. More precisely, we focus on an algebraic definition of environment-induced decoherence in open quantum systems which happens to be generic on finite dimensional systems and describes the asymptotic behavior of any QMS. An essential tool in our analysis is the explicit structure of the decoherence-free algebra generated by the QMS, a central object in the study of passive quantum error correction schemes. The Poincar\'e constant corresponds to the spectral gap of the QMS, which implies its positivity, while we prove that the modified log-Sobolev constant is positive under the $\mathbb L_1$-regularity of the Dirichlet form, a condition that also appears in the primitive case. We furthermore prove that strong $\mathbb L_p$-regularity holds for QMS that satisfy a strong form of detailed balance condition for $p\geq1$. The latter condition includes all known cases where this strong regularity was proved. Finally and to emphasize the mathematical interest of this study compared to the classical case, we focus on two truly quantum scenarios, one exhibiting quantum coherence, and the other, quantum correlations.