Ordering states of Tsallis relative $\alpha$-entropies of coherence

TL;DR

This paper investigates how different coherence measures based on Tsallis relative $$-entropies and $l_1$ norm order quantum states, revealing both consistencies and discrepancies across various state types and measures.

Contribution

It provides a detailed comparison of ordering properties of Tsallis relative $$-entropies of coherence and $l_1$ norm for single-qubit states, highlighting their similarities and differences.

Findings

Same ordering for pure states by both measures.

Different ordering for some high-dimensional pure states.

Inconsistencies among special Tsallis measures for mixed states.

Abstract

In this paper, we study the ordering states with Tsallis relative $\alpha$-entropies of coherence and $l_{1}$ norm of coherence for single-qubit states. We show that any Tsallis relative $\alpha$-entropies of coherence and $l_{1}$ norm of coherence give the same ordering for single-qubit pure states. However, they don't generate the same ordering for some high dimensional pure states, even though these states are pure. We also consider three special Tsallis relative $\alpha$-entropies of coherence, such as $C_{1}$ ,$C_{2}$ and $C_{\frac{1}{2}}$, and show any one of these three measures and $C_{l_{1}}$ will not generate the same ordering for single-qubit mixed states. Furthermore, we find that any two of these three special measures generate different ordering for single-qubit mixed states.