Min- and Max- Relative Entropies and a New Entanglement Monotone

TL;DR

This paper introduces min- and max-relative entropies, explores their properties, and defines a new entanglement monotone, providing a framework that connects these quantities with smooth entropies and the Information Spectrum approach.

Contribution

It presents new relative entropy measures, a novel entanglement monotone, and generalizations that link to smooth entropies and spectral divergence rates.

Findings

Min- and max-relative entropies are formally defined and their properties analyzed.

A new entanglement monotone, the max-relative entropy of entanglement, is introduced.

Smooth versions of these entropies are developed, connecting to the smooth Renyi entropies and spectral divergence rates.

Abstract

Two new relative entropy quantities, called the min- and max-relative entropies, are introduced and their properties are investigated. The well-known min- and max- entropies, introduced by Renner, are obtained from these. We define a new entanglement monotone, which we refer to as the max-relative entropy of entanglement, and which is an upper bound to the relative entropy of entanglement. We also generalize the min- and max-relative entropies to obtain smooth min- and max- relative entropies. These act as parent quantities for the smooth Renyi entropies, and allow us to define the analogues of the mutual information, in the Smooth Renyi Entropy framework. Further, the spectral divergence rates of the Information Spectrum approach are shown to be obtained from the smooth min- and max-relative entropies in the asymptotic limit.