### 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.

Original language | English |
---|---|

Pages (from-to) | 259-296 |

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

### Cite this

*Foundations of Computational Mathematics*,

*19*(2), 259-296. https://doi.org/10.1007/s10208-018-9385-0

}

*Foundations of Computational Mathematics*, vol. 19, no. 2, pp. 259-296. https://doi.org/10.1007/s10208-018-9385-0

**Semidefinite approximations of the Matrix Logarithm.** / Fawzi, Hamza; Saunderson, James; Parrilo, Pablo A.

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - Semidefinite approximations of the Matrix Logarithm

AU - Fawzi, Hamza

AU - Saunderson, James

AU - Parrilo, Pablo A.

PY - 2019/4/1

Y1 - 2019/4/1

N2 - 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.

AB - 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.

KW - Convex optimization

KW - Matrix concavity

KW - Quantum relative entropy

UR - http://www.scopus.com/inward/record.url?scp=85044219270&partnerID=8YFLogxK

U2 - 10.1007/s10208-018-9385-0

DO - 10.1007/s10208-018-9385-0

M3 - Article

VL - 19

SP - 259

EP - 296

JO - Foundations of Computational Mathematics

JF - Foundations of Computational Mathematics

SN - 1615-3375

IS - 2

ER -