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Journal of Physics A Mathematical and Theoretical

Quantifying asymmetry via generalized Wigner–Yanase–Dyson skew information

Yuan Sun, Nan Li

Published Jun 3, 2021• 7 citations
Publisher

Abstract

Abstract Asymmetry is a useful physical resource which quantifies the extent to which a quantum state breaks a symmetry. Based on the generalization of Wigner–Yanase–Dyson skew information of quantum states for any operator (not necessarily Hermitian), we introduce an asymmetry measure of a quantum state with respect to a compact Lie group as the average skew information of this state for each operator in its unitary representation and prove that it meets the requirements for the resource theory of asymmetry. For several significant symmetry groups, such as the group U (1), the full unitary group, the local unitary group, the group UU , the group U U ̄ , and the orthogonal group OO , we calculate the asymmetry of quantum states. Furthermore, we compare the asymmetry measure of quantum states with respect to a compact Lie group with the asymmetry measure of quantum states with respect to the corresponding Lie algebras introduced in (2020 Europhys. Lett. 130 30004 ) and illustrate that they are closely related, although they capture different aspects of asymmetry of quantum states.

DOI
10.1088/1751-8121/ac07ec

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Abstract Asymmetry is a useful physical resource which quantifies the extent to which a quantum state breaks a symmetry. Based on the generalization of Wigner–Yanase–Dyson skew information of quantum states for any operator (not necessarily Hermitian), we introduce an asymmetry measure of a quantum state with respect to a compact Lie group as the average skew information of this state for each operator in its unitary representation and prove that it meets the requirements for the resource theory of asymmetry. For several significant symmetry groups, such as the group U (1), the full unitary group, the local unitary group, the group UU , the group U U ̄ , and the orthogonal group OO , we calculate the asymmetry of quantum states. Furthermore, we compare the asymmetry measure of quantum states with respect to a compact Lie group with the asymmetry measure of quantum states with respect to the corresponding Lie algebras introduced in (2020 Europhys. Lett. 130 30004 ) and illustrate that they are closely related, although they capture different aspects of asymmetry of quantum states.
Symmetry, as well as the dual concept of asymmetry, are essential, powerful and ubiquitous in physics and nature. Motivated by the original Wigner-Yanase skew information and its extension by Dyson ( Wigner E. P. and Yanase M. M., Proc. Natl. Acad. Sci. U.S.A. , 49 (1963) 910), which quantifies the information content of a quantum state skew to a conserved observable and may be reinterpreted as asymmetry of a state with respect to an observable, we address the issue of quantifying asymmetry and symmetry of states with respect to Lie groups and Lie algebras. We elucidate that correlations arise naturally from the difference between asymmetries of global and local states. This provides a fundamental insight into correlations. The idea is further illustrated by several examples.
The Wigner-Yanase skew information was proposed to quantify the information contained in quantum states with respect to a conserved additive quantity, and it was later extended to the Wigner-Yanase-Dyson skew informations. Recently, the Wigner-Yanase-Dyson skew informations have been recognized as valid resource measures for the resource theory of asymmetry, and their properties have been investigated from a resource-theoretic perspective. The Wigner-Yanse-Dyson skew informations have been further generalized to a class called metric-adjusted skew informations, and this general family of skew informations have also been found to be valid asymmetry monotones. Here, we analyze this general family of the skew informations from an operational point of view by utilizing the fact that they are valid asymmetry resource monotones. We show that such an approach allows for clear physical meanings as well as simple proofs of some of the basic properties of the skew informations. Notably, we constructively prove that any type of skew information cannot be superadditive, where the violation of the superadditivity had been only known for a specific class of skew informations with numerical counterexamples. We further show a weaker version of superadditivity relation applicable to the general class of the skew informations, which proves a conjecture made for the Wigner-Yanase skew information as a special case. We finally discuss an application of our results for a situation where quantum clocks are distributed to multiple parties.
Skew information was introduced by Wigner and Yanase in the context of quantum measurements. It was later recognized that skew information in addition to its original significance as a measure of the information content of states admits several interpretations of a more physical and information theory nature. This quantity has now found many applications in quantum information. An intriguing and subtle feature of skew information is the involvement of the square root of quantum states (density operators). Comparing skew information with some of its natural modifications reveals mathematical and also physical reasons for using the square root. We further use skew information to quantify the asymmetry of quantum states with respect to group representations.
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.
Identifying the classical-quantum divide and quantifying nonclassicality (quantumness) are fundamental issues in quantum exploration. By use of the Wigner-Yanase skew information [E. P. Wigner and M. M. Yanase, Proc. Natl. Acad. Sci. USA 49, 910 (1963)], which is a version of quantum Fisher information with a well-established interpretation as quantum uncertainty, we introduce a simple and effective quantifier for nonclassicality of bosonic field states. This is achieved by simultaneously yet approximately measuring the canonical conjugate quadrature pair (equivalently, measuring the non-Hermitian annihilation operator in a generalized sense) from an information-theoretic perspective. This quantifier, apart from capturing the idea of nonclassicality as quantum uncertainty in a succinct way, is mathematically computable, physically intuitive, and possesses operational significance. We reveal its fundamental properties and illustrate its physical applications in quantifying nonclassicality of light. We compare it with various existent quantifiers for nonclassicality and illustrate its merits. A related quantifier for classicality, which is in some sense dual to nonclassicality, is also introduced. This pair of classicality and nonclassicality provides insight into the classical-quantum interplay.
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.