The conditional entropy power inequality for quantum additive noise channels

TL;DR

This paper establishes a quantum conditional Entropy Power Inequality for additive noise channels, demonstrating its optimality and applying it to derive new information-theoretic inequalities and analyze quantum semigroup convergence.

Contribution

It proves the quantum conditional Entropy Power Inequality for additive noise channels and shows its asymptotic optimality, extending classical inequalities to the quantum conditional setting.

Findings

The inequality is optimal and achievable asymptotically with Gaussian states.

New quantum conditional entropy inequalities are derived.

A simple proof for the convergence rate of the quantum Ornstein-Uhlenbeck semigroup is provided.

Abstract

We prove the quantum conditional Entropy Power Inequality for quantum additive noise channels. This inequality lower bounds the quantum conditional entropy of the output of an additive noise channel in terms of the quantum conditional entropies of the input state and the noise when they are conditionally independent given the memory. We also show that this conditional Entropy Power Inequality is optimal in the sense that we can achieve equality asymptotically by choosing a suitable sequence of Gaussian input states. We apply the conditional Entropy Power Inequality to find an array of information-theoretic inequalities for conditional entropies which are the analogues of inequalities which have already been established in the unconditioned setting. Furthermore, we give a simple proof of the convergence rate of the quantum Ornstein-Uhlenbeck semigroup based on Entropy Power Inequalities.