# Ordering states with various coherence measures

## TL;DR

This paper investigates how different coherence measures in quantum physics order quantum states, revealing that they often produce different orderings, especially in higher dimensions, and clarifies their relationships for single-qubit states.

## Contribution

The study compares four key coherence measures, showing their ordering differences in qudit states and clarifying their relationships in single-qubit states.

## Key findings

- Different coherence measures produce distinct orderings for qudit states.
- For single-qubit states, some measures agree on ordering, while others differ.
- The paper partially answers an open question about measure-based state ordering.

## Abstract

Quantum coherence is one of the most significant theories in quantum physics. Ordering states with various coherence measures is an intriguing task in quantification theory of coherence. In this paper, we study this problem by use of four important coherence measures -- the $l_1$ norm of coherence, the relative entropy of coherence, the geometric measure of coherence and the modified trace distance measure of coherence. We show that each pair of these measures give a different ordering of qudit states when $d\geq 3$. However, for single-qubit states, the $l_1$ norm of coherence and the geometric coherence provide the same ordering. We also show that the relative entropy of coherence and the geometric coherence give a different ordering for single-qubit states. Then we partially answer the open question proposed in [Quantum Inf. Process. 15, 4189 (2016)] whether all the coherence measures give a different ordering of states.