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