Coherent-induced state ordering with fixed mixedness

TL;DR

This paper investigates how different coherence measures order quantum states with fixed mixedness, showing they agree under certain conditions and analyzing how this ordering evolves under Markovian channels.

Contribution

It demonstrates the equivalence of coherence-induced state ordering by multiple measures for specific quantum states and explores the dynamics of this ordering under noise channels.

Findings

Coherence measures agree for single-qubit states with fixed mixedness.

Measures generate the same ordering for high-dimensional pure states satisfying majorization.

Phase damping preserves, while amplitude damping can alter state ordering.

Abstract

In this paper, we study coherence-induced state ordering with Tsallis relative entropy of coherence, relative entropy of coherence and $l_{1}$ norm of coherence. Firstly, we show that these measures give the same ordering for single-qubit states with a fixed mixedness or a fixed length along the direction $\sigma_{z}$. Secondly, we consider some special cases of high dimensional states, we show that these measures generate the same ordering for the set of high dimensional pure states if any two states of the set satisfy majorization relation. Moreover, these three measures generate the same ordering for all $X$ states with a fixed mixedness. Finally, we discuss dynamics of coherence-induced state ordering under Markovian channels. We find phase damping channel don't change the coherence-induced state ordering for some single-qubit states with fixed mixedness, instead amplitude damping channel change the coherence-induced ordering even though for single-qubit states with fixed mixedness.