Examining the validity of Schatten-$p$-norm-based functionals as coherence measures

Abstract

It has been asked by different authors whether the two classes of Schatten-$p$-norm-based functionals $C_p(\rho )={\min }_{\sigma \in \mathcal{I}}||\rho -\sigma |{|}_p$ and ${\tilde{C}}_p(\rho )=\Vert \rho -\Delta \rho {\Vert }_p$ with $p\ge 1$ are valid coherence measures under incoherent operations, strictly incoherent operations, and genuinely incoherent operations, respectively, where $\mathcal{I}$ is the set of incoherent states and $\Delta \rho$ is the diagonal part of density operator $\rho$. Of these questions, all we know is that $C_p(\rho )$ is not a valid coherence measure under incoherent operations and strictly incoherent operations, but all other aspects remain open. In this paper, we prove that (1) ${\tilde{C}}_1(\rho )$ is a valid coherence measure under both strictly incoherent operations and genuinely incoherent operations but not a valid coherence measure under incoherent operations, (2) $C_1(\rho )$ is not a valid coherence measure even under genuinely incoherent operations, and (3) neither $C_{p>1}(\rho )$ nor ${\tilde{C}}_{p>1}(\rho )$ is a valid coherence measure under any of the three sets of operations. This paper not only provides a thorough examination on the validity of taking $C_p(\rho )$ and ${\tilde{C}}_p(\rho )$ as coherence measures, but also finds an example that fulfills the monotonicity under strictly incoherent operations but violates it under incoherent operations.

8 pages

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