# Semidefinite approximations of the matrix logarithm

## TL;DR

This paper develops semidefinite approximation methods for the matrix logarithm that preserve its operator concavity, enabling efficient convex optimization involving this function in quantum information and related fields.

## Contribution

It introduces a novel approach to approximate the matrix logarithm with semidefinite programs, facilitating practical optimization in quantum information theory.

## Key findings

- Allows use of standard SDP solvers for log-based convex problems
- Provides small-size semidefinite representations for operator concave functions
- Extends to other operator concave and monotone functions

## 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 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.