  The matrix logarithm, when applied to Hermitian positive definite matrices,
is concave with respect to the positive semidefinite order. This operator
concavity property leads to numerous concavity and convexity results for other
matrix functions, many of which are of importance in quantum information
theory. In this paper we show how to approximate the matrix logarithm with
functions that preserve operator concavity and can be described using the
feasible regions of semidefinite optimization problems of fairly small size.
Such approximations allow us to use off-the-shelf semidefinite optimization
solvers for convex optimization problems involving the matrix logarithm and
related functions, such as the quantum relative entropy. The basic ingredients
of our approach apply, beyond the matrix logarithm, to functions that are
operator concave and operator monotone. As such, we introduce strategies for
constructing semidefinite approximations that we expect will be useful, more
generally, for studying the approximation power of functions with small
semidefinite representations.
