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Physical Review A

Quantifying coherence relative to channels via metric-adjusted skew information

Yuan Sun, Nan Li, Shunlong Luo

Published Jul 27, 2022• 37 citations
Publisher

Abstract

In terms of the metric-adjusted skew information (an important and versatile class of quantum Fisher information), which generalizes the seminal Wigner-Yanase skew information arising naturally from the study of quantum measurement, we propose a family of coherence measures of states relative to quantum channels, and reveal their basic properties such as unitary covariance, convexity, and monotonicity. Furthermore, we evaluate these coherence measures of states relative to several prototypical quantum channels, and make a comparative study for this family of coherence measures with relative entropy of coherence. This provides a general approach to coherence of states relative to quantum channels, which also captures decoherence on the states caused by quantum channels and asymmetry of states relative to quantum channels.

DOI
10.1103/physreva.106.012436

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In terms of the metric-adjusted skew information (an important and versatile class of quantum Fisher information), which generalizes the seminal Wigner-Yanase skew information arising naturally from the study of quantum measurement, we propose a family of coherence measures of states relative to quantum channels, and reveal their basic properties such as unitary covariance, convexity, and monotonicity. Furthermore, we evaluate these coherence measures of states relative to several prototypical quantum channels, and make a comparative study for this family of coherence measures with relative entropy of coherence. This provides a general approach to coherence of states relative to quantum channels, which also captures decoherence on the states caused by quantum channels and asymmetry of states relative to quantum channels.
In this paper, we introduce a family of correlation measures relative to a local channel in terms of the metric-adjusted skew information for a global state and reveal their fundamental properties. We quantify the correlation of quantum states with respect to several typical channels, including unitary channels, twirling channels, and weak measurements, and illustrate that the correlation with respect to the twirling channel induced by the fully unitary group is consistent with that relative to the completely depolarizing channel. In particular, we further evaluate the correlations of the Bell diagonal states, Werner states, and isotropic states and make a comparative study for this family of correlation measures with different operator monotone functions of the two-qubit Werner states and isotropic states, respectively.
Quantum coherence is an important feature that distinguishes quantum mechanics from classical physics. In this paper, we study the coherence of a quantum state with respect to a quantum channel, and introduce a coherence measure in terms of the generalized Wigner–Yanase–Dyson skew information. The basic properties and operational significance of the measure are expounded in detail, which shows that it is indeed a veritable quantity and satisfies many desirable conditions intuitively required for the coherence measures. We illustrate the significance of characterizing channels from the coherence perspective through some examples. Finally, we consider the average coherence of a state based on the generalized Wigner–Yanase–Dyson skew information with respect to any complete mutually unbiased bases in prime power dimensional systems, as well as with respect to all orthonormal bases, and show that these two averages are equivalent since they both can be seen as the coherence of the quantum state with respect to the depolarizing quantum channel.
Nonclassical correlations are significant physical resources with extensive applications in quantum information processing. We introduce the modified Wigner-Yanase-Dyson skew information of a quantum state relative to a quantum channel, and a quantitative measure of quantum correlations. Their basic properties are explored in detail. Through a specific example, we also compare our correlations measure with the existing one. Moreover, the correlations relative to various channels including the von Neumann measurements, the unitary channels and the twirling channels are analyzed.
Uncertainty principle is one of the most essential features of quantum mechanics, and it reveals the intrinsic difference that distinguishes the quantum world from the classical world. In this paper, we focus on the uncertainty relations based on metric-adjusted skew information. By making full use of the norm property, we establish an uncertainty relation for arbitrary finite nnn observables based on metric-adjusted skew information, and we give two new uncertainty relations for arbitrary finite NNN quantum channels based on metric-adjusted skew information. For arbitrary two observables and two channels, the uncertainty relations we give are not only better than the uncertainty relations detailed in [Quantum Inf. Process. 20, 72 (2021)], but also are the equations. The equation of the uncertainty relation is more accurate than the inequation of the usual uncertainty relation, which has important advantages in the application of quantum information technology, such as quantum communication and the quantum precision measurement. Meanwhile, our results are suitable to Wigner-Yanase-Dyson skew information that is a special metric-adjusted skew information and Wigner-Yanase skew information that is a special Wigner-Yanase-Dyson skew information. Some examples about Wigner-Yanase-Dyson skew information are given and show that the new lower bounds are tighter. The results play an important role in quantum information processing in this paper.
We extend the concept of Wigner-Yanase-Dyson skew information to something we call "metric adjusted skew information" (of a state with respect to a conserved observable). This "skew information" is intended to be a non-negative quantity bounded by the variance (of an observable in a state) that vanishes for observables commuting with the state. We show that the skew information is a convex function on the manifold of states. It also satisfies other requirements, proposed by Wigner and Yanase, for an effective measure-of-information content of a state relative to a conserved observable. We establish a connection between the geometrical formulation of quantum statistics as proposed by Chentsov and Morozova and measures of quantum information as introduced by Wigner and Yanase and extended in this article. We show that the set of normalized Morozova-Chentsov functions describing the possible quantum statistics is a Bauer simplex and determine its extreme points. We determine a particularly simple skew information, the "lambda-skew information," parametrized by a lambda in (0, 1], and show that the convex cone this family generates coincides with the set of all metric adjusted skew informations.
In order to quantify the information content of quantum states in the presence of conserved quantities, Wigner and Yanase introduced the notion of skew information [Proc. Natl. Acad. Sci. USA 49, 910, (1963)], which was later identified as a paradigmatic version of quantum Fisher information. The skew information is quite different from, yet deeply connected with, the ubiquitous quantum entropies and has important applications in quantum theory. In this paper, pursuing further the original idea of Wigner and Yanase, we propose a measure for correlations in terms of the skew information, investigate its fundamental properties, and elucidate its characteristics. An appealing feature of this measure for correlations is that its evaluation does not involve any optimization, in sharp contrast to the entanglement and discord measures, and can be straightforwardly calculated. In particular, the algorithm and explicit formulas for general bipartite states are prescribed, and simple analytical expressions for some special states, including arbitrary bipartite pure states, the Bell-diagonal states, and some highly symmetric states such as the Werner states and the isotropic states, are obtained.