$l_{1}$ norm of coherence is not equal to its convex roof quantifier

TL;DR

This paper demonstrates that the $l_{1}$ norm of coherence does not equal its convex roof quantifier, highlighting a fundamental difference in how this measure extends from pure to mixed quantum states.

Contribution

The work proves that the $l_{1}$ norm of coherence is not equal to its convex roof extension, clarifying a key aspect of quantum coherence measures.

Findings

$C_{l_{1}} eq ar{C_{l_{1}}}$ for the convex roof extension

Highlights a fundamental difference in coherence measure extensions

Provides insight into quantum coherence quantification methods

Abstract

Since a rigorous framework for quantifying quantum coherence was established by Baumgratz et al. [T. Baumgratz, M. Cramer, and M. B. Plenio, Phys. Rev. Lett. 113, 140401 (2014)], many coherence measures have been found. For a given coherence measure $C$, extending the values of $C$ on pure states to mixed states by the convex roof construction, we will get a valid coherence measure $\overline{C}$, we call $\overline{C}$ the corresponding convex roof quantifier of $C$. Whether $C=\overline{C}$ for a given coherence measure is an important question. In this work, we show that for the widely used coherence measure, $l_{1}$ norm of coherence $C_{l_{1}}$, it holds that $C_{l_{1}}\neq \overline{C_{l_{1}}}$.