On a quantum version of Shannon's conditional entropy

Abstract

In this article we propose a quantum version of Shannon's conditional entropy. Given two density matrices $ρ$ and $σ$ on a finite dimensional Hilbert space and with $S(ρ)=-\trρ\lnρ$ being the usual von Neumann entropy, this quantity $S(ρ|σ)$ is concave in $ρ$ and satisfies $0\le S(ρ|σ)\le S(ρ)$, a quantum analogue of Shannon's famous inequality. Thus we view $S(ρ|σ)$ as the entropy of $ρ$ conditioned by $σ$.