Accelerated optimization of measured relative entropies

TL;DR

This paper develops efficient gradient-based algorithms for computing measured quantum relative entropies, leveraging their convexity properties to improve speed and memory efficiency over previous semi-definite programming methods.

Contribution

It establishes the convexity and smoothness of the variational formulas for measured quantum relative entropies, enabling the use of accelerated gradient methods for their computation.

Findings

Algorithms are more memory efficient than previous semi-definite optimization methods.

For well-conditioned states, the algorithms are significantly faster.

The methods achieve arbitrary precision in calculating measured relative entropies.

Abstract

The measured relative entropy and measured R\'enyi relative entropy are quantifiers of the distinguishability of two quantum states $\rho$ and $\sigma$. They are defined as the maximum classical relative entropy or R\'enyi relative entropy realizable by performing a measurement on $\rho$ and $\sigma$, and they have interpretations in terms of asymptotic quantum hypothesis testing. Crucially, they can be rewritten in terms of variational formulas involving the optimization of a concave or convex objective function over the set of positive definite operators. In this paper, we establish foundational properties of these objective functions by analyzing their matrix gradients and Hessian superoperators; namely, we prove that these objective functions are $\beta$-smooth and $\gamma$-strongly convex / concave, where $\beta$ and $\gamma$ depend on the max-relative entropies of $\rho$ and $\sigma$. A practical consequence of these properties is that we can conduct Nesterov accelerated projected gradient descent / ascent, a well known classical optimization technique, to calculate the measured relative entropy and measured R\'enyi relative entropy to arbitrary precision. These algorithms are generally more memory efficient than our previous algorithms based on semi-definite optimization [Huang and Wilde, arXiv:2406.19060], and for well conditioned states $\rho$ and $\sigma$, these algorithms are notably faster.