On equalities in two entropic inequalities

TL;DR

This paper establishes a criterion for when the constrained Holevo capacity equals quantum mutual information in quantum channels, with applications to Bosonic Gaussian channels and conditions for equality involving input entropy.

Contribution

It provides a simple criterion for local equality between key quantum information measures and characterizes states where this equality holds, especially in Gaussian channels.

Findings

Equality set determined by the kernel of the channel

Non-Gaussian mixed states are the only solutions for certain Gaussian channels

Conditions for equality between Holevo capacity and input von Neumann entropy

Abstract

A simple criterion for local equality between the constrained Holevo capacity and the quantum mutual information of a quantum channel is obtained. It implies that the set of all states for which this equality holds is determined by the kernel of the channel (as a linear map). Applications to Bosonic Gaussian channels are considered. It is shown that for a Gaussian channel having no completely depolarizing components the above characteristics may coincide only at non-Gaussian mixed states and a criterion of existence of such states is given. All the obtained results may be reformulated as conditions for equality between the constrained Holevo capacity of a quantum channel and the input von Neumann entropy.