The matrix logarithm, when applied to symmetric positive definite matrices, is known to satisfy a notable concavity property in the positive semidefinite (Loewner) order. This concavity property is a cornerstone result in the study of operator convex functions and has important applications in matrix concentration inequalities and quantum information theory.

In this talk I will show that certain rational approximations of the matrix logarithm remarkably preserve this concavity property and moreover, are amenable to semidefinite programming. Such approximations allow us to use off-the-shelf semidefinite programming solvers for convex optimization problems involving the matrix logarithm. These approximations are also useful in the scalar case and provide a much faster alternative to existing methods based on successive approximation for problems involving the exponential/relative entropy cone. I will conclude by showing some applications to problems arising in quantum information theory.

This is joint work with James Saunderson (Monash University) and Pablo Parrilo (MIT)