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Physical Review A

Comment on “Quantum coherence measures based on Fisher information with applications”

Jianwei Xu

Published May 10, 2022• 0 citations
Publisher

Abstract

In their recent paper [Li et al., Phys. Rev. A 103, 012401 (2021)], the authors proposed two quantities ${C}_{\mathrm{F}}^{D}(\ensuremath{\rho}|\mathcal{K})$ in Eq. (11) and ${C}_{\mathrm{F}}^{C}(\ensuremath{\rho}|\mathcal{K})$ in Eq. (12). The authors proved that ${C}_{\mathrm{F}}^{C}(\ensuremath{\rho}|\mathcal{K})$ is a valid coherence measure in Theorem 5 and proved that ${C}_{\mathrm{F}}^{D}(\ensuremath{\rho}|\mathcal{K})={C}_{\mathrm{F}}^{C}(\ensuremath{\rho}|\mathcal{K})$ in Proposition 4. In this Comment, we show that the proof of Proposition 4 is not justified, that is, ${C}_{\mathrm{F}}^{D}(\ensuremath{\rho}|\mathcal{K})={C}_{\mathrm{F}}^{C}(\ensuremath{\rho}|\mathcal{K})$ is not true, in general. Consequently, whether ${C}_{\mathrm{F}}^{D}(\ensuremath{\rho}|\mathcal{K})$ is a valid coherence measure is still an open problem.

DOI
10.1103/physreva.105.056401

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In their recent paper [Li et al., Phys. Rev. A 103, 012401 (2021)], the authors proposed two quantities CFD(\ensuremathρ∣K){C}_{\mathrm{F}}^{D}(\ensuremath{\rho}|\mathcal{K})CFD​(\ensuremathρ∣K) in Eq. (11) and CFC(\ensuremathρ∣K){C}_{\mathrm{F}}^{C}(\ensuremath{\rho}|\mathcal{K})CFC​(\ensuremathρ∣K) in Eq. (12). The authors proved that CFC(\ensuremathρ∣K){C}_{\mathrm{F}}^{C}(\ensuremath{\rho}|\mathcal{K})CFC​(\ensuremathρ∣K) is a valid coherence measure in Theorem 5 and proved that CFD(\ensuremathρ∣K)=CFC(\ensuremathρ∣K){C}_{\mathrm{F}}^{D}(\ensuremath{\rho}|\mathcal{K})={C}_{\mathrm{F}}^{C}(\ensuremath{\rho}|\mathcal{K})CFD​(\ensuremathρ∣K)=CFC​(\ensuremathρ∣K) in Proposition 4. In this Comment, we show that the proof of Proposition 4 is not justified, that is, CFD(\ensuremathρ∣K)=CFC(\ensuremathρ∣K){C}_{\mathrm{F}}^{D}(\ensuremath{\rho}|\mathcal{K})={C}_{\mathrm{F}}^{C}(\ensuremath{\rho}|\mathcal{K})CFD​(\ensuremathρ∣K)=CFC​(\ensuremathρ∣K) is not true, in general. Consequently, whether CFD(\ensuremathρ∣K){C}_{\mathrm{F}}^{D}(\ensuremath{\rho}|\mathcal{K})CFD​(\ensuremathρ∣K) is a valid coherence measure is still an open problem.
The notion of measurement is of both foundational and instrumental significance in quantum mechanics, and coherence destroyed by measurements (decoherence) lies at the very heart of quantum to classical transition. Qualitative aspects of this spirit have been widely recognized and analyzed ever since the inception of quantum theory. However, axiomatic and quantitative investigations of coherence are attracting great interest only recently with several figures of merit for coherence introduced [Baumgratz, Cramer, and Plenio, Phys. Rev. Lett. 113, 140401 (2014)]. While these resource theoretic approaches have many appealing and intuitive features, they rely crucially on various notions of incoherent operations which are sophisticated, subtle, and not uniquely defined, as have been critically assessed [Chitambar and Gour, Phys. Rev. Lett. 117, 030401 (2016)]. In this paper, we elaborate on the idea that coherence and quantum uncertainty are dual viewpoints of the same quantum substrate, and address coherence quantification by identifying coherence of a state (with respect to a measurement) with quantum uncertainty of a measurement (with respect to a state). Consequently, coherence measures may be set into correspondence with measures of quantum uncertainty. In particular, we take average quantum Fisher information as a measure of quantum uncertainty, and introduce the corresponding measure of coherence, which is demonstrated to exhibit desirable properties. Implications for interpreting quantum purity as maximal coherence, and quantum discord as minimal coherence, are illustrated.
The author of the Comment [Phys. Rev. A 104, 016401 (2021)] pointed out the missing part in the proof of Theorem 20 in our work [L. Qian et al., Phys. Rev. A 99, 032312 (2019)], and presented a sufficient and necessary condition for the separability of completely symmetric (CS) states in the two-qutrit system. While being technically correct, the proposed example is still a separable CS state according to Ha's sufficient and necessary condition. We provide another proof to show that every bipartite CS state of rank five is separable. This result bridges the gap appearing in our previous proof. It turns out that every two-qutrit CS state is separable, and thus making our conclusion (Theorem 20 of our work) still valid.
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In the preceding Comment it has been claimed that an entangled-based quantum key distribution protocol proposed by Zhang, Li, and Gao [Phys. Rev. A 64, 024302 (2001)] and its generalization to ddd-level systems by Karimipour, Bahraminasab, and Bagherinezhad [Phys. Rev. A 65, 052331 (2002)] are insecure against an attack devised by the authors of the Comment. We invalidate the arguments of the Comment and show that the protocols are still secure.