Defining quantum divergences via convex optimization

TL;DR

This paper introduces a new quantum Rényi divergence defined via convex optimization, enabling efficient computation, chain rule properties, and improved bounds on quantum channel capacities, with applications to channel discrimination.

Contribution

It defines a new quantum Rényi divergence with convex optimization, linking it to existing divergences and enabling new operational and computational results.

Findings

Established a semidefinite programming representation for the divergence.

Proved a chain rule property for the sandwiched Rényi divergence.

Derived improved bounds on quantum channel capacities.

Abstract

We introduce a new quantum R\'enyi divergence $D^{\#}_{\alpha}$ for $\alpha \in (1,\infty)$ defined in terms of a convex optimization program. This divergence has several desirable computational and operational properties such as an efficient semidefinite programming representation for states and channels, and a chain rule property. An important property of this new divergence is that its regularization is equal to the sandwiched (also known as the minimal) quantum R\'enyi divergence. This allows us to prove several results. First, we use it to get a converging hierarchy of upper bounds on the regularized sandwiched $\alpha$-R\'enyi divergence between quantum channels for $\alpha > 1$. Second it allows us to prove a chain rule property for the sandwiched $\alpha$-R\'enyi divergence for $\alpha > 1$ which we use to characterize the strong converse exponent for channel discrimination. Finally it allows us to get improved bounds on quantum channel capacities.