Ordering states with coherence measures

TL;DR

This paper investigates whether different quantum coherence measures agree on the ordering of quantum states, finding they can produce different orderings, which impacts how coherence is quantified in quantum information.

Contribution

It demonstrates that common coherence measures, including the $l_1$ norm and relative entropy, can assign different orderings to quantum states, revealing inconsistencies in coherence quantification.

Findings

Different coherence measures can produce different state orderings.

The $l_1$ norm and relative entropy of coherence do not always agree.

The formation of coherence and coherence of formation also yield different orderings.

Abstract

The quantification of quantum coherence has attracted a growing attention, and based on various physical contexts, several coherence measures have been put forward. An interesting question is whether these coherence measures give the same ordering when they are used to quantify the coherence of quantum states. In this paper, we consider the two well-known coherence measures, the $l_1$ norm of coherence and the relative entropy of coherence, to show that there are the states for which the two measures give a different ordering. Our analysis can be extended to other coherence measures, and as an illustration of the extension we further consider the formation of coherence to show that the $l_1$ norm of coherence and the formation of coherence, as well as the relative entropy of coherence and the coherence of formation, do not give the same ordering too.