Semidefinite optimization of the quantum relative entropy of channels

TL;DR

This paper presents a practical method for computing the quantum relative entropy of channels by discretizing and linearizing the integral representation, enabling efficient bounds and maximization.

Contribution

It extends previous state-relative entropy optimization techniques to channels, providing a discretized linearization approach for practical computation.

Findings

Provides a discretized linearization method for quantum channel relative entropy

Enables computation of upper and lower bounds with arbitrary precision

Extends state relative entropy optimization to quantum channels

Abstract

This paper introduces a method for calculating the quantum relative entropy of channels, an essential quantity in quantum channel discrimination and resource theories of quantum channels. By building on recent developments in the optimization of relative entropy for quantum states [Ko{\ss}mann and Schwonnek, arXiv:2404.17016], we introduce a discretized linearization of the integral representation for the relative entropy of states, enabling us to handle maximization tasks for the relative entropy of channels. Our approach here extends previous work on minimizing relative entropy to the more complicated domain of maximization. It also provides efficiently computable upper and lower bounds that sandwich the true value with any desired precision, leading to a practical method for computing the relative entropy of channels.