Maximal Coherence and the Resource Theory of Purity

TL;DR

This paper establishes a fundamental link between the resource theories of coherence and purity, showing that purity represents the maximum coherence achievable by unitaries and serves as a basic resource for quantum information tasks.

Contribution

It introduces a universal family of maximally coherent mixed states and connects purity with coherence, entanglement, and discord in a unified framework.

Findings

Maximal coherence equals the distance-based purity quantifier.

Purity bounds the maximum entanglement and discord generated by unitaries.

Maximal coherence can be exactly evaluated for distance-based measures.

Abstract

The resource theory of quantum coherence studies the off-diagonal elements of a density matrix in a distinguished basis, whereas the resource theory of purity studies all deviations from the maximally mixed state. We establish a direct connection between the two resource theories, by identifying purity as the maximal coherence which is achievable by unitary operations. The states that saturate this maximum identify a universal family of maximally coherent mixed states. These states are optimal resources under maximally incoherent operations, and thus independent of the way coherence is quantified. For all distance-based coherence quantifiers the maximal coherence can be evaluated exactly, and is shown to coincide with the corresponding distance-based purity quantifier. We further show that purity bounds the maximal amount of entanglement and discord that can be generated by unitary operations, thus demonstrating that purity is the most elementary resource for quantum information processing.