Abstract
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 offtheshelf 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.
Original language  English 

Pages (fromto)  259296 
Number of pages  38 
Journal  Foundations of Computational Mathematics 
Volume  19 
Issue number  2 
DOIs  
Publication status  Published  1 Apr 2019 
Keywords
 Convex optimization
 Matrix concavity
 Quantum relative entropy
Prizes

SIAM Activity Group on Optimization Best Paper Prize 2020
Saunderson, James (Recipient), 2020
Prize: Prize (including medals and awards)