Relative Entropy Convergence for Depolarizing Channels

TL;DR

This paper analyzes the convergence behavior of depolarizing quantum channels using relative entropy, computes the key log-Sobolev-1 constant, and applies it to improve entropy inequalities and bounds for tensor powers.

Contribution

It provides the explicit computation of the log-Sobolev-1 constant for depolarizing channels and applies this to enhance entropy inequalities and analyze tensor powers.

Findings

Computed the log-Sobolev-1 constant for depolarizing channels.

Improved the concavity inequality of von-Neumann entropy.

Established a uniform lower bound for tensor powers using Shearer's inequality.

Abstract

We study the convergence of states under continuous-time depolarizing channels with full rank fixed points in terms of the relative entropy. The optimal exponent of an upper bound on the relative entropy in this case is given by the log-Sobolev-1 constant. Our main result is the computation of this constant. As an application we use the log-Sobolev-1 constant of the depolarizing channels to improve the concavity inequality of the von-Neumann entropy. This result is compared to similar bounds obtained recently by Kim et al. and we show a version of Pinsker's inequality, which is optimal and tight if we fix the second argument of the relative entropy. Finally, we consider the log-Sobolev-1 constant of tensor-powers of the completely depolarizing channel and use a quantum version of Shearer's inequality to prove a uniform lower bound.