$l_1$ norm of coherence is not equal to its convex roof quantifier

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 $C_{l_1}\ne \overline{C_{l_1}}$.

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Cited by in corpus (3)

• Estimating quantum coherence by noncommutativity of any observable and its incoherent part
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• Quantifying the phase of quantum states