Semidefinite Approximations of the Matrix Logarithm

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 informatio...

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Bibliographic Details
Published in:Foundations of computational mathematics Vol. 19; no. 2; pp. 259 - 296
Main Authors: Fawzi, Hamza, Saunderson, James, Parrilo, Pablo A.
Format: Journal Article
Language:English
Published: New York Springer US 01.04.2019
Springer Nature B.V
Subjects:
Applications of Mathematics
Computer Science
Concavity
Convexity
Economics
Entropy
Information theory
Linear and Multilinear Algebras
Math Applications in Computer Science
Mathematical analysis
Mathematics
Mathematics and Statistics
Matrix methods
Matrix Theory
Numerical Analysis
Operators (mathematics)
Optimization
Quantum phenomena
Solvers
52A41
Convex optimization
47A63
90C22
Matrix concavity
Quantum relative entropy
ISSN:1615-3375, 1615-3383
Online Access:Get full text
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Summary: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.
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ISSN:1615-3375
1615-3383
DOI:10.1007/s10208-018-9385-0