Symmetric distinguishability as a quantum resource

TL;DR

This paper develops a resource theory for symmetric distinguishability of quantum sources, establishing its monotonicity under specific operations and linking it to quantum Chernoff divergence for source conversion tasks.

Contribution

It introduces a new resource theory for symmetric distinguishability, proving its monotonicity and connecting it to quantum Chernoff divergence for asymptotic source conversion.

Findings

Quantum Chernoff divergence determines optimal source conversion rate.

Symmetric distinguishability is a monotone under specific quantum operations.

Operational interpretations of the Thompson metric are provided.

Abstract

We develop a resource theory of symmetric distinguishability, the fundamental objects of which are elementary quantum information sources, i.e., sources that emit one of two possible quantum states with given prior probabilities. Such a source can be represented by a classical-quantum state of a composite system $XA$, corresponding to an ensemble of two quantum states, with $X$ being classical and $A$ being quantum. We study the resource theory for two different classes of free operations: $(i)$ ${\rm{CPTP}}_A$, which consists of quantum channels acting only on $A$, and $(ii)$ conditional doubly stochastic (CDS) maps acting on $XA$. We introduce the notion of symmetric distinguishability of an elementary source and prove that it is a monotone under both these classes of free operations. We study the tasks of distillation and dilution of symmetric distinguishability, both in the one-shot and asymptotic regimes. We prove that in the asymptotic regime, the optimal rate of converting one elementary source to another is equal to the ratio of their quantum Chernoff divergences, under both these classes of free operations. This imparts a new operational interpretation to the quantum Chernoff divergence. We also obtain interesting operational interpretations of the Thompson metric, in the context of the dilution of symmetric distinguishability.