On the asymmetry of the relative entropy

The quantum relative entropy $S(\rho||\sigma)$ is a widely used dissimilarity measure between quantum states, but it has the peculiarity of being asymmetric in its arguments. We quantify the amount of asymmetry by providing a sharp upper bound in terms of two parameters: the trace norm distance between the two states, and the smallest of the smallest eigenvalues of both states. The bound is essentially the asymmetry between two binary distributions governed by these two parameters.