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 ${C}_{p}(\ensuremath{\rho})={min}_{\ensuremath{\sigma}\ensuremath{\in}\mathcal{I}}{\ensuremath{\parallel}\ensuremath{\rho}\ensuremath{-}\ensuremath{\sigma}\ensuremath{\parallel}}_{p}$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{C}}_{p}(\ensuremath{\rho})={\ensuremath{\parallel}\ensuremath{\rho}\ensuremath{-}\mathrm{\ensuremath{\Delta}}\ensuremath{\rho}\ensuremath{\parallel}}_{p}$ with $p\ensuremath{\ge}1$ 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 $\mathrm{\ensuremath{\Delta}}\ensuremath{\rho}$ is the diagonal part of density operator $\ensuremath{\rho}$. Of these questions, all we know is that ${C}_{p}(\ensuremath{\rho})$ 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) ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{C}}_{1}(\ensuremath{\rho})$ is a valid coherence measure under both strictly incoherent operations and genuinely incoherent operations but not a valid coherence measure under incoherent operations, (2) ${C}_{1}(\ensuremath{\rho})$ is not a valid coherence measure even under genuinely incoherent operations, and (3) neither ${C}_{p>1}(\ensuremath{\rho})$ nor ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{C}}_{p>1}(\ensuremath{\rho})$ 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 ${C}_{p}(\ensuremath{\rho})$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{C}}_{p}(\ensuremath{\rho})$ as coherence measures, but also finds an example that fulfills the monotonicity under strictly incoherent operations but violates it under incoherent operations.