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arXiv:1605.04425 · Physical Review A

Characterizing maximally singular phase-space distributions

Jan Sperling

Published Jul 7, 2016• 35 citations
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Abstract

Phase-space distributions are widely applied in quantum optics to access the nonclassical features of radiations fields. In particular, the inability to interpret the Glauber-Sudarshan distribution in terms of a classical probability density is the fundamental benchmark for quantum light. However, this phase-space distribution cannot be directly reconstructed for arbitrary states, because of its singular behavior. In this work, we perform a characterization of the Glauber-Sudarshan representation in terms of distribution theory. We address important features of such distributions: (i) the maximal degree of their singularities is studied, (ii) the ambiguity of representation is shown, and (iii) their dual space for nonclassicality tests is specified. In this view, we reconsider the methods for regularizing the Glauber-Sudarshan distribution for verifying its nonclassicality. This treatment is supported with comprehensive examples and counterexamples.

DOI
10.1103/physreva.94.013814

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Phase-space distributions are widely applied in quantum optics to access the nonclassical features of radiations fields. In particular, the inability to interpret the Glauber-Sudarshan distribution in terms of a classical probability density is the fundamental benchmark for quantum light. However, this phase-space distribution cannot be directly reconstructed for arbitrary states, because of its singular behavior. In this work, we perform a characterization of the Glauber-Sudarshan representation in terms of distribution theory. We address important features of such distributions: (i) the maximal degree of their singularities is studied, (ii) the ambiguity of representation is shown, and (iii) their dual space for nonclassicality tests is specified. In this view, we reconsider the methods for regularizing the Glauber-Sudarshan distribution for verifying its nonclassicality. This treatment is supported with comprehensive examples and counterexamples.
We examine the notion of nonclassicality in terms of quasiprobability distributions. In particular, we not only ask if a specific quasiprobability can be interpreted as a classical probability density, but we also require that characteristic features of classical electrodynamics are resembled. We show that the only quasiprobabilities that correctly describe the superposition principle of classical electromagnetic fields are the sss-parametrized quasiprobabilities. Furthermore, the Glauber-Sudarshan PPP function is the only quantum-mechanical quasiprobability that is transformed at a classical attenuator in the same way as a classical probability distribution. This result strengthens the definition of nonclassicality in terms of the PPP function, in contrast to possible definitions in terms of other quasiprobabilities.
From the Wigner function to the sss-ordered phase-space distribution via a Gaussian noise channel
Various phase-space distributions, from the celebrated Wigner function, to the Husimi QQQ function and the Glauber–Sudarshan PPP distribution, have played an interesting and important role in the phase-space formulation of quantum mechanics in general, and quantum optics in particular. A unified approach to all these distributions based on the notion of the sss -ordered phase-space distribution was introduced by Cahill and Glauber. With the intention of illuminating the physical meaning of the parameter sss , we interpret the sss -ordered phase-space distribution as the Wigner function of a state under the Gaussian noise channel, and thus reveal an intrinsic connection between the sss -ordered phase-space distribution and the Gaussian noise channel, which yields a physical insight into the sss -ordered phase-space distribution. In this connection, the parameter −s/2-s/2−s/2 (rather than the original sss ) acquires the role of the noise occurring in the Gaussian noise channel. An alternative representation of the Gaussian noise channel as the scaling-measurement preparation in a coherent states is illuminated. Furthermore, by exploiting the freedom in the parameter sss , we introduce a computable and experimentally testable quantifier for optical nonclassicality, reveal its basic properties, and illustrate it by typical examples. A simple and convenient criterion for optical nonclassicality in terms of the sss -ordered phase-space distribution is derived.
The phase-space quasiprobability distribution formalism for representing quantum states provides practical tools for various applications in quantum optics such as identifying the nonclassicality of quantum states. We study filter functions that are introduced to regularize the Glauber-Sudarshan PPP function. We show that the quantum map associated with a filter function is completely positive and trace preserving and hence physically realizable if and only if the Fourier transform of this function is a probability density distribution. We also derive a lower bound on the fidelity between the input and output states of a physical quantum filtering map. Therefore, based on these results, we show that any quantum state can be approximated, to arbitrary accuracy, by a quantum state with a regular Glauber-Sudarshan PPP function. We propose applications of our results for estimating the output state of an unknown quantum process and estimating the outcome probabilities of quantum measurements.
If the phase space-based Glauber-Sudarshan distribution, PϱP_{\varrho}Pϱ​, has negative values the quantum state, ϱ\varrhoϱ, it describes is nonclassical. Due to PPP's singular behavior, this simple criterion is impractical to use. Recent work has presented a general, sensitive, and noise-tolerant certification functional for the detection of nonclassical behavior of quantum states ϱ\varrhoϱ. There, it was shown that when this functional takes on negative values somewhere in phase space, ξP(x,p)<0ξ_{P}(x,p) < 0ξP​(x,p)<0, this is sufficient to certify the nonclassicality of a state. Here we give examples where this certification fails. We investigate states which are known to be nonclassical but the certification function is nonnegative, ξP(x,p)≥0ξ_{P}(x,p) \geq 0ξP​(x,p)≥0, everywhere in phase space. We generalize ξξξ, giving it an appealing form, S\cal SS, which allows for slight improvements in certification, but S\cal SS also fails for mixed very weakly nonclassical states. More important than a slight improvement in sensitivity of S\cal SS over ξξξ is that we showed how very sensitive ξξξ and S\cal SS are, and more important still is our simple derivation of S\cal SS allowing us to generalize ξξξ and S\cal SS to multiple modes.
General quasiprobabilities are introduced to visualize time-dependent quantum correlations of light in phase space. They are based on the generalization of the Glauber-Sudarshan PPP function to a time-dependent PPP functional [W. Vogel, Phys. Rev. Lett. 100, 013605 (2008)], which fully describes temporal correlations of radiation fields on the basis of continuous phase-space distributions. This approach is nontrivial, as the PPP functional itself is highly singular for many quantum states and nonlinear processes. In general, it yields neither a well-behaved nor an experimentally accessible description of quantum stochastic processes. Our regularized version of this multitime-dependent quasiprobability is a smooth function and applies to stronger divergences compared to the single-time and multimode scenario. The technique is used to characterize an optical parametric process with frequency mismatch and a strongly nonlinear evolution of the quantized center-of-mass motion of a trapped ion. A measurement scheme, together with a sampling approach, is provided which yields direct experimental access to the regularized PPP functional from measured data.
In this paper, we use the characteristic function, i.e., the Fourier transform of the Glauber-Sudarshan phase-space distribution, to determine the degree of nonclassicality of a given state. This degree of nonclassicality quantifies the nonclassicality in terms of quantum superpositions. We demonstrate two ways to exactly find or to lower-bound the degree of nonclassicality by studying the properties of the characteristic functions. The developed criteria are applied to two examples of squeezed states undergoing a classical mixing or a nonclassical superposition with vacuum.