Additivity of quantum relative entropies as a single-copy criterion

TL;DR

This paper identifies conditions under which quantum relative entropies are additive at the single-copy level, enabling efficient characterization of various quantum information tasks without regularization.

Contribution

It establishes necessary and sufficient conditions for the additivity of quantum relative entropies, simplifying the analysis of several quantum information problems.

Findings

Regularization is unnecessary if a single-copy optimizer meets certain criteria.

Derived the Stein, Chernoff, and Hoeffding exponents for key quantum hypothesis testing problems.

Provided partial results on the strong converse exponent for these tasks.

Abstract

The fundamental goal of information theory is to characterize complex operational tasks using efficiently computable information quantities, Shannon's capacity formula being the prime example of this. However, many tasks in quantum information can only be characterized by regularized entropic measures that are often not known to be computable and for which efficient approximations are scarce. It is thus of fundamental importance to understand when regularization is not needed, opening the door to an efficiently computable characterization based on additive quantities. Here, we demonstrate that for a large class of problems, the question of whether regularization is needed or not can be determined at the single-copy level. Specifically, we demonstrate that regularization of the Umegaki relative entropy, along with related quantities such as the Petz and sandwiched relative entropies, is not needed if and only if a single-copy optimizer satisfies a certain property. These problems include hypothesis testing with arbitrarily varying hypotheses as well as quantum resource theories used to derive fundamental bounds for entanglement and magic state distillation. We derive the Stein, Chernoff, and Hoeffding exponents for these problems and establish necessary and sufficient conditions for their additivity, while also presenting partial results for the strong converse exponent.