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arXiv:2406.01717 · Physical Review A

Quantifying nonclassicality of mixed Fock states

Spencer Rogers, Tommy Muth, Wenchao Ge

Published Oct 1, 2024• 1 citation
arXivHTMLPDF

Abstract

Nonclassical states of bosonic modes are important resources for quantum-enhanced technologies. However, quantifying nonclassicality of these states, in particular mixed states, can be a challenge. Here we present results of quantifying the nonclassicality of a bosonic mode in a mixed Fock state via the operational resource theory measure [Phys. Rev. Res. 2, 023400 (2020)], which relates nonclassicality to metrological advantage. Generally speaking, evaluating a resource-theoretic measure for mixed states is challenging, since it involves finding a convex roof. However, we show that our problem can be reduced to a linear programming problem. By analyzing the results of numerical optimization, we are able to extract analytical results for the case where three or four neighboring Fock states have nonzero population. Interestingly, we find that such a mode can be in distinct phases, depending on the populations. Finally, we demonstrate how our method is generalizable to density matrices of higher ranks. Our findings suggest a viable method for evaluating nonclassicality of arbitrary mixed bosonic states and potentially for solving other convex roof optimization problems.

DOI
10.1103/physreva.110.042602

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Nonclassical states of bosonic modes are important resources for quantum-enhanced technologies. However, quantifying nonclassicality of these states, in particular mixed states, can be a challenge. Here we present results of quantifying the nonclassicality of a bosonic mode in a mixed Fock state via the operational resource theory measure [Phys. Rev. Res. 2, 023400 (2020)], which relates nonclassicality to metrological advantage. Generally speaking, evaluating a resource-theoretic measure for mixed states is challenging, since it involves finding a convex roof. However, we show that our problem can be reduced to a linear programming problem. By analyzing the results of numerical optimization, we are able to extract analytical results for the case where three or four neighboring Fock states have nonzero population. Interestingly, we find that such a mode can be in distinct phases, depending on the populations. Finally, we demonstrate how our method is generalizable to density matrices of higher ranks. Our findings suggest a viable method for evaluating nonclassicality of arbitrary mixed bosonic states and potentially for solving other convex roof optimization problems.
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.
The operational resource theory (ORT) measure is a nonclassicality measure for bosonic states, notable for its resource-theoretic properties and connection to metrology. However, it can be difficult to evaluate, being linked to an optimization problem for mixed states. Here, we present ORT measure calculations for mixed states with photon-number coherence. We give exact formulas governing the ORT measure of a broad class of rank-two mixed states, and numerical solutions for some higher-rank states. We also compare the nonclassicality of these states to their metrological power, thus showing in what regimes the metrological power manages to saturate the ORT bound. Throughout, we consider the role of coherence. In particular, we show that nonclassicality and metrological power never increase under bosonic dephasing, but may plateau in a manner similar to entanglement sudden death. Nevertheless, lowering photon-number coherence more freely can sometimes yield more nonclassical and metrologically useful states.
Nonclassicality in bosonic systems can be certified directly from occupation probabilities without full state reconstruction. To complete a library of such criteria we introduce a numerical framework that constructs optimal linear nonclassicality witnesses from arbitrary subsets of measured occupation probabilities by solving a semi-infinite linear program. The method contains known analytical criteria as special cases while generally detecting substantially larger nonclassical regions. We provide an open-source implementation of the numerical optimization together with the analytical criteria discussed here, enabling direct application of the framework to measured occupation statistics across bosonic platforms. As a case study, we apply it to the resonant Jaynes--Cummings model with thermal states and reveal nonclassicality at short interaction times and close to thermal balance, where detection of nonclassicality is otherwise challenging.
Abstract Nonclassical states of a quantized light are described in terms of Glauber–Sudarshan distribution, which is not a genuine classical probability distribution. Despite several attempts, defining a uniform measure of nonclassicality (NC) for the single-mode quantum states of light remains an open task. In our previous work Bose and Sanjay Kumar (2017 Phys. Rev . A 95 012330), we have shown that the existing well-known measures fail to quantify the NC of single-mode states that are generated under multiple NC-inducing operations. In this paper, we propose a new quantification of NC for the single-mode quantum states as the difference between the total Wehrl entropy of the state and the maximum Wehrl entropy arising due to its classical characteristics. The classical reference state is chosen at the same randomness, given by the von-Neumann entropy, as the state itself. We further suggest operational interpretation of the proposed measure in terms of phase-space sampling entropy, as well as potential to generate entanglement in the case of pure states. We obtain analytic expressions of NC for pure states and Gaussian mixed states. We show that, along with the states generated under single NC-inducing operations as well as the broader class of states that are generated under multiple NC-inducing operations, our quantification enumerates the NC consistently. However, the evaluation of NC for the non-Gaussian mixed states is subject to extensive numerical computation that lies beyond the scope of the current work. We finally conclude with certain open questions.
We establish the nonclassicality of continuous-variable states as a resource for quantum metrology. Based on the quantum Fisher information of multimode quadratures, we introduce the metrological power as a measure of nonclassicality with a concrete operational meaning of displacement sensitivity beyond the classical limit. This measure belongs to the resource theory of nonclassicality, which is nonincreasing under linear optical elements. Our Letter reveals that a single copy, highly nonclassical quantum state is intrinsically advantageous when compared to multiple copies of a quantum state with moderate nonclassicality. This suggests that metrological power is related to the degree of quantum macroscopicity. Finally, we demonstrate that metrological resources useful for nonclassical displacement sensing tasks can be always converted into a useful resource state for phase sensitivity beyond the classical limit.