Quantifying the coherence of pure quantum states

TL;DR

This paper analyzes measures of quantum coherence for pure states, providing explicit characterizations, identifying maximally coherent states, and relating coherence to entanglement measures.

Contribution

It offers new explicit characterizations of coherence measures for pure states and links coherence to entanglement through trace distance.

Findings

Maximally coherent states are identified as those maximizing trace distance of coherence.

Trace distance of entanglement coincides with trace distance of coherence for pure states.

The $ ext{l}_1$ measure of coherence is never smaller than the relative entropy of coherence for pure states.

Abstract

In recent years, several measures have been proposed for characterizing the coherence of a given quantum state. We derive several results that illuminate how these measures behave when restricted to pure states. Notably, we present an explicit characterization of the closest incoherent state to a given pure state under the trace distance measure of coherence. We then use this result to show that the states maximizing the trace distance of coherence are exactly the maximally coherent states. We define the trace distance of entanglement and show that it coincides with the trace distance of coherence for pure states. Finally, we give an alternate proof to a recent result that the $\ell_1$ measure of coherence of a pure state is never smaller than its relative entropy of coherence.