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Frontiers in Physics

On Non-Convexity of the Nonclassicality Measure via Operator Ordering Sensitivity

Shuangshuang Fu, Shunlong Luo, Yue Zhang

Published Jul 8, 2022• 2 citations
Publisher

Abstract

In quantum optics, nonclassicality in the sense of Glauber-Sudarshan is a valuable resource related to the quantum aspect of photons. A desirable and intuitive requirement for a consistent measure of nonclassicality is convexity: Classical mixing should not increase nonclassicality. We show that the recently introduced nonclassicality measure [Phys. Rev. Lett. 122, 080402 (2019)] is not convex. This nonclassicality measure is defined via operator ordering sensitivity, which is an interesting and significant probe (witness) of nonclassicality without convexity but can be intrinsically connected to the convex Wigner-Yanase skew information [Proc. Nat. Acad. Sci. United States 49, 910 (1963)] via the square root operation on quantum states. Motivated by the Wigner-Yanase skew information, we also propose a faithful measure of nonclassicality, although it cannot be readily computed, it is convex.

DOI
10.3389/fphy.2022.955786

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In quantum optics, nonclassicality in the sense of Glauber-Sudarshan is a valuable resource related to the quantum aspect of photons. A desirable and intuitive requirement for a consistent measure of nonclassicality is convexity: Classical mixing should not increase nonclassicality. We show that the recently introduced nonclassicality measure [Phys. Rev. Lett. 122, 080402 (2019)] is not convex. This nonclassicality measure is defined via operator ordering sensitivity, which is an interesting and significant probe (witness) of nonclassicality without convexity but can be intrinsically connected to the convex Wigner-Yanase skew information [Proc. Nat. Acad. Sci. United States 49, 910 (1963)] via the square root operation on quantum states. Motivated by the Wigner-Yanase skew information, we also propose a faithful measure of nonclassicality, although it cannot be readily computed, it is convex.
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.
Nonclassicality of optical states, as a key characteristic of bosonic fields, is a valuable resource for quantum information processing. We investigate the generation of nonclassicality in quantum processes from a quantitative perspective, introduce three information-theoretic measures of nonclassicality for quantum-optical processes based on the Wigner–Yanase skew information and coherent states, and illustrate their physical significance through several well-known single-mode quantum processes.
Nonclassicality of light has been studied, both extensively and intensively, in terms of optical coherent states, and various quantifiers for nonclassicality of light have been introduced in recent decades. In contrast, the corresponding quantification issue for atomic spin systems is relatively less studied, though there has been some remarkable progress. In this paper, by virtue of the Wigner-Yanase skew information and working in the framework of atomic coherent states, we introduce a quantifier for nonclassicality of atomic spin states from an information-theoretic perspective and reveal its fundamental properties. This quantifier leads to a simple yet powerful criterion for detecting nonclassicality via convexity of the skew information. This criterion is sufficient, but not necessary. The concept is illustrated through several paradigms. We further elucidate some intrinsic relations between nonclassicality and entanglement and indicate that the nonclassicality quantifier may be transferred to an entanglement quantifier.
Quantifying nonclassicality of a bosonic mode is an important but challenging task in quantum optics. Recently, a nonclassicality measure based on the concept of operational resource theory has been proposed [W. Ge, K. Jacobs, S. Asiri, M. Foss-Feig, and M. S. Zubairy, Phys. Rev. Res. 2, 023400 (2020)], which shows several crucial properties as a resource for quantum metrology. Here we apply the measure to evaluate and categorize different classes of nonclassical states. We discover a class of states that can achieve the maximum nonclassicality in the asymptotic limit of a large mean number of excitations. These states can provide the same sensitivity in place of a squeezed vacuum in a sensing scheme, e.g., the laser interferometer gravitational-wave observatory (LIGO) experiments. We also discover that the nonclassicality of certain states can be greatly improved with a single-photon addition. In addition, we explore some examples on how to evaluate analytically the nonclassicality of mixed states, which are in general difficult to calculate due to the convex roof nature of the measure. Our results will be useful for preparing and utilizing nonclassical states for applications in precision-sensing tasks, such as quadrature sensing and phase sensing in a Mach-Zehnder interferometer.
We propose a convenient and easily computable nonclassicality quantifier for bosonic field states based on the Wigner-Yanase skew information. The proposed nonclassicality quantifier is reflected by the quantum interaction between the maximum phase angle of the homodyne rotated quadrature operator and the bosonic field states. If the value of the nonclassical quantifier is greater than one-half, the state is nonclassical, and the quantifier is one-half for the pure classical. It is worth mentioning that an increase in the strength of nonclassicality inducing operations, such as squeezing and photon addition, leads to an enhancement of the nonclassicality in the quantum state. By computing some well-known nonclassical states and summarizing their existence features, we have confirmed the validity of our proposed nonclassicality quantifier. We have shown that in a range of values of the nonclassicality of the Gaussian state, revealing the sufficiency of the quantifier. We also compare it with two common nonclassical quantifiers to illustrate its advantages.
We introduce a new distance-based measure for the nonclassicality of the states of a bosonic field, which outperforms the existing such measures in several ways. We define for that purpose the operator ordering sensitivity of the state which evaluates the sensitivity to operator ordering of the Renyi entropy of its quasiprobabilities and which measures the oscillations in its Wigner function. Through a sharp control on the operator ordering sensitivity of classical states we obtain a precise geometric image of their location in the density matrix space allowing us to introduce a distance-based measure of nonclassicality. We analyze the link between this nonclassicality measure and a recently introduced quantum macroscopicity measure, showing how the two notions are distinct.