Coherence as entropy increment for Tsallis and Renyi entropies

TL;DR

This paper explores the extension of coherence measures using Tsallis and Re9nyi entropies, revealing that Re9nyi-based measures are valid coherence measures while Tsallis-based ones are generally not, and provides continuity estimates.

Contribution

It introduces new coherence measures based on Tsallis and Re9nyi entropies and analyzes their validity and properties, including continuity and monotonicity.

Findings

Re9nyi coherence measures are valid and well-behaved.

Tsallis entropy does not generally produce a genuine coherence monotone.

Continuity estimates are provided for both entropy-based coherence measures.

Abstract

Relative entropy of coherence can be written as an entropy difference of the original state and the incoherent state closest to it when measured by relative entropy. The natural question is, if we generalize this situation to Tsallis or R\'enyi entropies, would it define good coherence measures? In other words, we define a difference between Tsallis entropies of the original state and the incoherent state closest to it when measured by Tsallis relative entropy. Taking R\'enyi entropy instead of the Tsallis entropy, leads to the well-known distance-based R\'enyi coherence, which means this expression defined a good coherence measure. Interestingly, we show that Tsallis entropy does not generate even a genuine coherence monotone, unless it is under a very restrictive class of operations. Additionally, we provide continuity estimates for both Tsallis and R\'enyi coherence expressions. Furthermore, we present two coherence measures based on the closest incoherent state when measures by Tsallis or R\'enyi relative entropy.