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.
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.
Nonclassical states of quantized light, except for the Gaussian states, also possess non-Gaussian phase-space distributions. Despite several attempts, a unified description of nonclassicality (NC) and non-Gaussianity (NG) of quantum states of light has not been developed as-of-yet. Here, we propose an experimentally verifiable scheme for quantification of NC, in terms of Wehrl entropy, that further leads to the simultaneous quantification of NC and NG. While requiring no optimization, present work recovers earlier results qualitatively as well as explores several new possibilities on the conjugation of nonclassical and non-Gaussian character of quantum states. Moreover, current formalism, due to its possible extension to the finite-dimensional systems, bridges the gap between discrete and continuous variable systems. Our work, thus, becomes crucial in describing NC of quantum processes including open quantum systems as well as understanding the role of NC and NG as resources in several information theoretic tasks processing such as entanglement distillation, quantum network, quantum computation etc.
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.
Abstract This paper aims to stress the role of the Cahill–Glauber quasi-probability densities in defining, detecting, and quantifying the non-classicality of field states in quantum optics. The distance between a given pure state and the set of all pure classical states is called here a geometric degree of non-classicality. As such, we investigate non-classicality of a pure single-mode state of the radiation field by using the coherent states as a reference set of pure classical states. It turns out that any such distance is expressed in terms of the maximal value of the Husimi Q function. As an insightful application we consider the de-Gaussification process produced when preparing a quantum state by adding p photons to a pure Gaussian one. For a coherent-state input, we get an analytic degree of non-classicality which compares interestingly with the previously evaluated entanglement potential. Then we show that addition of a single photon to a squeezed vacuum state causes a considerable enhancement of non-classicality, especially at weak and moderate squeezing of the original state. By contrast, addition of further photons is less effective.
A bosonic state is commonly considered nonclassical (or quantum) if its Glauber-Sudarshan P function is not a classical probability density, which implies that only coherent states and their statistical mixtures are classical. We quantify the nonclassicality of a single qubit, defined by the vacuum and single-photon states, by applying the following four well-known measures of nonclassicality: (1) the nonclassical depth, $\ensuremath{\tau}$, related to the minimal amount of Gaussian noise which changes a nonpositive P function into a positive one; (2) the nonclassical distance D, defined as the Bures distance of a given state to the closest classical state, which is the vacuum for the single-qubit Hilbert space; together with (3) the negativity potential (NP), and (4) concurrence potential, which are the nonclassicality measures corresponding to the entanglement measures (i.e., the negativity and concurrence, respectively) for the state generated by mixing a single-qubit state with the vacuum on a balanced beam splitter. We show that complete statistical mixtures of the vacuum and single-photon states are the most nonclassical single-qubit states regarding the distance D for a fixed value of both the depth $\ensuremath{\tau}$ and NP in the whole range [0,1] of their values, as well as the NP for a given value of $\ensuremath{\tau}$ such that $\ensuremath{\tau}>0.315\phantom{\rule{0.16em}{0ex}}4$. Conversely, pure states are the most nonclassical single-qubit states with respect to $\ensuremath{\tau}$ for a given D, NP versus D, and $\ensuremath{\tau}$ versus NP. We also show the ``relativity'' of these nonclassicality measures by comparing pairs of single-qubit states: if a state is less nonclassical than another state according to some measure then it might be more nonclassical according to another measure. Moreover, we find that the concurrence potential is equal to the nonclassical distance for single-qubit states. This implies an operational interpretation of the nonclassical distance as the potential for the entanglement of formation.
If a single-mode nonclassical light is combined with the vacuum on a beam splitter, then the output state is entangled. As proposed in [Phys. Rev. Lett. 94, 173602 (2005)], by measuring the output-state entanglement for a balanced lossless beam splitter, one can quantify the input-state nonclassicality. These measures of nonclassicality (referred to as entanglement potentials) can be based, in principle, on various entanglement measures, leading to the negativity (NP) and concurrence (CP) potentials, and the potential for the relative entropy of entanglement (REEP). We search for the maximal nonclassicality, which can be achieved by comparing two entanglement measures for arbitrary two-qubit states and those which can be generated from a photon-number qubit via a balanced lossless beam-splitter, where the qubit basis states are the vacuum and single-photon states. Surprisingly, we find that the maximal nonclassicality, measured by the REEP for a given value of the NP, can be increased (if NP<0.527) by using either a tunable beam splitter or by amplitude damping of the output state of the balanced beam splitter. We also show that the maximal nonclassicality, measured by the NP for a given value of the REEP, can be increased by dephasing. The entanglement itself is not increased by these local losses, but the possible ratios of different measures are affected. Moreover, we show that partially-mixed states can be more nonclassical than both pure states and completely-mixed states, by comparing the NP for a given value of the REEP. This implies that not all entanglement measures can be used as entanglement potentials. Alternatively, a single balanced lossless beam splitter is not always transferring the whole nonclassicality of its input state into the entanglement of its output modes. Applying a lossy beam splitter can solve this problem at least for the cases analyzed in this paper.
Continuous-variable beam-splitter (BS)-generated entanglement from single-mode optical states generated by a single nonclassicality (NC)-inducing operation has been found to be immensely important in several information processing tasks. There exists a broader class of optical states, generated from successive action of multiple different NC-inducing operations, which show many intriguing nonclassical properties; however, the BS conversion of the NC for such states remains unexplored. In this work we have critically analyzed the BS-generated entanglement from such nonclassical optical states at input. Here we present a scenario where BS output entanglement becomes nonmonotonic with the input NC parameters, accessible experimentally (e.g., number of photon excitation and squeezing strength), in contrast to the previous results with states comprising a single NC-inducing operation. We explain this counterintuitive feature in terms of the competition between these two NC-inducing operations as manifest in the contours of the Q functions associated with these states.