Trace-distance measure of coherence

TL;DR

This paper investigates the trace distance measure of quantum coherence, demonstrating its strong monotonicity for certain states, providing explicit formulas, and comparing it with other coherence measures.

Contribution

It establishes the strong monotonicity of trace distance coherence for qubit and X states, and derives relations and inequalities between different coherence measures.

Findings

Trace distance coherence is a strong monotone for qubit and X states.

Explicit expression for pure states and a semi-definite program for arbitrary states.

Inequality connecting l1-norm and relative entropy measures of coherence.

Abstract

We show that trace distance measure of coherence is a strong monotone for all qubit and, so called, $X$ states. An expression for the trace distance coherence for all pure states and a semi definite program for arbitrary states is provided. We also explore the relation between $l_1$-norm and relative entropy based measures of coherence, and give a sharp inequality connecting the two. In addition, it is shown that both $l_p$-norm- and Schatten-$p$-norm-based measures violate the (strong) monotonicity for all $p\in(1,\infty)$.