# Resource theory of asymmetric distinguishability

## TL;DR

This paper develops a comprehensive resource theory for asymmetric quantum distinguishability, establishing fundamental limits and conversions between quantum states and distinguishability bits using quantum relative entropy.

## Contribution

It introduces bits of asymmetric distinguishability as a new resource, proves the reversibility of the theory in the asymptotic limit, and characterizes conversion rates via quantum relative entropy.

## Key findings

- Quantum relative entropy determines asymptotic conversion rates.
- One-shot distinguishability measures relate to min- and max-relative entropies.
- The theory provides a complete characterization of quantum state transformations.

## Abstract

This paper systematically develops the resource theory of asymmetric distinguishability, as initiated roughly a decade ago [K. Matsumoto, arXiv:1010.1030 (2010)]. The key constituents of this resource theory are quantum boxes, consisting of a pair of quantum states, which can be manipulated for free by means of an arbitrary quantum channel. We introduce bits of asymmetric distinguishability as the basic currency in this resource theory, and we prove that it is a reversible resource theory in the asymptotic limit, with the quantum relative entropy being the fundamental rate of resource interconversion. The distillable distinguishability is the optimal rate at which a quantum box consisting of independent and identically distributed (i.i.d.) states can be converted to bits of asymmetric distinguishability, and the distinguishability cost is the optimal rate for the reverse transformation. Both of these quantities are equal to the quantum relative entropy. The exact one-shot distillable distinguishability is equal to the min-relative entropy, and the exact one-shot distinguishability cost is equal to the max-relative entropy. Generalizing these results, the approximate one-shot distillable distinguishability is equal to the smooth min-relative entropy, and the approximate one-shot distinguishability cost is equal to the smooth max-relative entropy. As a notable application of the former results, we prove that the optimal rate of asymptotic conversion from a pair of i.i.d. quantum states to another pair of i.i.d. quantum states is fully characterized by the ratio of their quantum relative entropies.