Entropy Production of Doubly Stochastic Quantum Channels

TL;DR

This paper establishes universal bounds on entropy production in doubly stochastic quantum channels using logarithmic-Sobolev inequalities, with implications for quantum capacity limits and extending to discrete-time channels.

Contribution

It introduces a new comparison method for logarithmic-Sobolev constants and provides bounds that are invariant under tensor powers, advancing understanding of quantum entropy dynamics.

Findings

Universal lower bound on logarithmic-Sobolev constant for quantum channels

Improved bounds on quantum capacities using entropy production estimates

Extension of discrete-time inequalities to quantum channels

Abstract

We study the entropy increase of quantum systems evolving under primitive, doubly stochastic Markovian noise and thus converging to the maximally mixed state. This entropy increase can be quantified by a logarithmic-Sobolev constant of the Liouvillian generating the noise. We prove a universal lower bound on this constant that stays invariant under taking tensor-powers. Our methods involve a new comparison method to relate logarithmic-Sobolev constants of different Liouvillians and a technique to compute logarithmic-Sobolev inequalities of Liouvillians with eigenvectors forming a projective representation of a finite abelian group. Our bounds improve upon similar results established before and as an application we prove an upper bound on continuous-time quantum capacities. In the last part of this work we study entropy production estimates of discrete-time doubly-stochastic quantum channels by extending the framework of discrete-time logarithmic-Sobolev inequalities to the quantum case.