# Notes on modified trace distance measure of coherence

## TL;DR

This paper analyzes a modified trace distance measure of quantum coherence, providing analytical formulas for certain states and exploring its relation to other coherence measures and mixedness.

## Contribution

It establishes the equivalence of the modified trace distance and $l_1$-norm of coherence for single-qubit states and derives formulas for maximally coherent mixed states.

## Key findings

- Modified trace distance equals $l_1$-norm for single-qubit states.
- Analytical formula for maximally coherent mixed states.
- Trade-off relation between coherence and mixedness.

## Abstract

We investigate the modified trace distance measure of coherence recently introduced in [Phys. Rev. A 94, 060302(R) (2016)]. We show that for any single-qubit state, the modified trace norm of coherence is equal to the $l_{1}$-norm of coherence. For any $d$-dimensional quantum system, an analytical formula of this measure for a class of maximally coherent mixed states is provided. The trade-off relation between the coherence quantified by the new measure and the mixedness quantified by the trace norm is also discussed. Furthermore, we explore the relation between the modified trace distance measure of coherence and other measures such as the $l_{1}$-norm of coherence and the geometric measure of coherence.