Semidefinite Approximations of the Matrix Logarithm

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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.

Keywords

Convex optimization Matrix concavity Quantum relative entropy 

Mathematics Subject Classification

90C22 52A41 47A63 

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Copyright information

© SFoCM 2018

Authors and Affiliations

  • Hamza Fawzi
    • 1
  • James Saunderson
    • 2
  • Pablo A. Parrilo
    • 3
  1. 1.Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeCambridgeUK
  2. 2.Department of Electrical and Computer Systems EngineeringMonash UniversityVictoriaAustralia
  3. 3.Laboratory for Information and Decision Systems, Department of Electrical Engineering and Computer ScienceMassachusetts Institute of TechnologyCambridgeUSA

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