# Efficient optimization of the quantum relative entropy

## TL;DR

This paper introduces a unified semidefinite programming framework to efficiently compute quantum information measures based on the quantum relative entropy, enabling new numerical insights and counterexamples in quantum information theory.

## Contribution

It develops a general numerical method for quantum relative entropy optimization using semidefinite programming, applicable to various quantum information measures.

## Key findings

- Provides a practical computational tool for quantum relative entropy measures.
- Generates numerical counterexamples for a proposed lower bound on quantum conditional mutual information.
- Demonstrates the method's effectiveness with applications to quantum channel capacities and entanglement measures.

## Abstract

Many quantum information measures can be written as an optimization of the quantum relative entropy between sets of states. For example, the relative entropy of entanglement of a state is the minimum relative entropy to the set of separable states. The various capacities of quantum channels can also be written in this way. We propose a unified framework to numerically compute these quantities using off-the-shelf semidefinite programming solvers, exploiting the approximation method proposed in [Fawzi, Saunderson, Parrilo, Semidefinite approximations of the matrix logarithm, arXiv:1705.00812]. As a notable application, this method allows us to provide numerical counterexamples for a proposed lower bound on the quantum conditional mutual information in terms of the relative entropy of recovery.