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arXiv:2206.05594 · Physical Review A

Filter functions for the Glauber-Sudarshan P -function regularization

Mani Zartab, Ezad Shojaee, Saleh Rahimi-Keshari

Published May 10, 2023• 0 citations
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Abstract

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 $P$ 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 $P$ 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.

DOI
10.1103/physreva.107.053706

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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.
Nonclassicality filters provide a universal method to visualize the nonclassicality of arbitrary quantum states of light through negativities of a regularized Glauber-Sudarshan PPP function, also denoted as nonclassicality quasiprobability. Such filters are introduced and analyzed for optimizing the experimental certification of nonclassical effects. An analytic filter is constructed which preserves the full information on the quantum state. For balanced homodyne detection, the number of data points is analyzed to get the negativities of the nonclassicality quasiprobability with high statistical significance. The method is applied to different scenarios, such as phase-randomized squeezed vacuum states, single-photon-added thermal states, and heralded state engineering with array detectors. The generalization to visualize quantum correlations of multimode radiation fields is also considered.
Measures of quantum properties are essential to understanding the fundamental differences between quantum and classical systems as well as quantifying resources for quantum technologies. Here, two broad classes of bosonic phase-space functions, which are filtered versions of the Glauber-Sudarshan P function, are compared with regard to their ability to uncover nonclassical effects of light through their negativities. Gaussian filtering of the P function yields the family of s-parametrized quasiprobabilities, while more powerful regularized nonclassicality quasiprobabilities are obtained by non-Gaussian filtering. A method is proposed to directly sample such phase-space functions for the restricted case of phase-independent quantum states from balanced homodyne measurements. This overcomes difficulties of previous approaches that manually append uniformly distributed optical phases to the measured quadrature data. We experimentally demonstrate this technique for heralded single- and two-photon states using balanced homodyne detection with varying efficiency. The s-parametrized quasiprobabilities, which can be directly sampled, are non-negative for detection efficiencies below 0.5. By contrast, we show that significant negativities of non-Gaussian filtered quasiprobabilities uncover nonclassical effects for arbitrarily low efficiencies.
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 propose a quantum optical device to experimentally realize quantum processes, which perform the regularization of the---in general highly singular---Glauber--Sudarshan PPP functions of arbitrary quantum states before their application and/or measurement. This allows us to produce a broad class of nonclassical states with regular PPP functions, also called nonclassicality quasiprobabilities. For this purpose, the input states are combined on highly transmissive beam splitters with specific Gaussian or non-Gaussian classical states. We study both balanced and unbalanced homodyne detections for the direct sampling of the output states of the implemented processes, which requires no further regularization or state-reconstruction. By numerical simulations we demonstrate the feasibility of our approach and we outline the generalization to multimode light.
The algebraic quantification of nonclassicality, which naturally arises from the quantum superposition principle, is related to properties of regular nonclassicality quasiprobabilities. The latter are obtained by non-Gaussian filtering of the Glauber-Sudarshan PPP function. They yield lower bounds for the degree of nonclassicality. We also derive bounds for convex combinations of Gaussian states for certifying quantum non-Gaussianity directly from the experimentally accessible nonclassicality quasiprobabilities. Other quantum-state representations, such as sss-parametrized quasiprobabilities, insufficiently indicate or even fail to directly uncover detailed information on the properties of quantum states. As an example, our approach is applied to multi-photon-added squeezed vacuum states.
The characterization of nonclassicality of quantum states in terms of measurable quadrature distributions is studied, on the basis of recently derived necessary and sufficient conditions for the failure of the Glauber–Sudarshan P -function to be a probability density. When the noise-subtracted quadrature distribution, that is the marginal of the P -function, is not a probability density, the same is true for the Glauber–Sudarshan P -function. Inversely, for a complete characterization of a nonclassical P -function one needs phase-sensitive criteria that connect quadrature distributions for different phases. Even for phase-insensitive quantum states, phase-sensitive nonclassicality conditions are needed, as is illustrated for one-photon added thermal states.