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

• URL: http://arxiv.org/abs/2009.05895v1
• Date: Sun, 13 Sep 2020 01:42:00 GMT
• Title: Examining the validity of Schatten-$p$-norm-based functionals as coherence measures
• Authors: Xiao-Dan Cui, C. L. Liu, D. M. Tong
• Abstract summary: It has been asked whether the two classes of Schatten-$p$-norm-based functionals $C_p(rho)=min_sigmainmathcalI||rho-sigma_p$ and $ tildeC_p(rho)= |rho-Deltarho|_p$ with $pgeq 1$ are valid coherence measures under incoherent operations, strictly incoherent operations, and genuinely incoherent operations. We prove that
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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)= \|\rho-\Delta\rho\|_{p}$ with $p\geq 1$ 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.

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