The modified trace distance of coherence is constant on most pure states

TL;DR

This paper analyzes a modified trace distance measure of quantum coherence, providing an explicit formula for pure states and revealing its limitations due to being maximal on most pure states, thus questioning its usefulness.

Contribution

The paper derives an explicit formula for the modified trace distance of coherence on pure states and demonstrates its limitations as a coherence measure.

Findings

The modified trace distance of coherence is maximal on almost all pure states.

It is not a practical measure of coherence due to its constant maximal value.

The formula clarifies the behavior of the measure across pure states.

Abstract

Recently, the much-used trace distance of coherence was shown to not be a proper measure of coherence, so a modification of it was proposed. We derive an explicit formula for this modified trace distance of coherence on pure states. Our formula shows that, despite satisfying the axioms of proper coherence measures, it is likely not a good measure to use, since it is maximal (equal to 1) on all except for an exponentially-small (in the dimension of the space) fraction of pure states.