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arXiv:1201.0861 · Physical Review A

Moments of nonclassicality quasiprobabilities

Saleh Rahimi-Keshari, T. Kiesel, W. Vogel

Published Apr 18, 2012• 6 citations
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Abstract

A method is introduced for the verification of nonclassicality in terms of moments of nonclassicality quasiprobability distributions. The latter are easily obtained from experimental data and will be denoted as nonclassicality moments. Their relation to normally ordered moments is derived, which enables us to verify nonclassicality by using well established criteria. Alternatively, nonclassicality criteria are directly formulated in terms of nonclassicality moments. The latter converge in proper limits to the usually used criteria, as is illustrated for squeezing and sub-Poissonian photon statistics. Our theory also yields expectation values of any observable in terms of nonclassicality moments.

DOI
10.1103/physreva.85.043827

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A method is introduced for the verification of nonclassicality in terms of moments of nonclassicality quasiprobability distributions. The latter are easily obtained from experimental data and will be denoted as nonclassicality moments. Their relation to normally ordered moments is derived, which enables us to verify nonclassicality by using well established criteria. Alternatively, nonclassicality criteria are directly formulated in terms of nonclassicality moments. The latter converge in proper limits to the usually used criteria, as is illustrated for squeezing and sub-Poissonian photon statistics. Our theory also yields expectation values of any observable in terms of nonclassicality moments.
Nonclassicality was introduced to examine if experimental observations can be explained with a classical theory. Although the formal definition of nonclassicality is known for several decades, there has been a lack of simple and complete criteria to check nonclassicality. In this work, some criteria are compared. Then, so-called nonclassicality quasiprobabilities are introduced, which allow a full examination of nonclassicality for any quantum state. Finally, some issues of the relation between nonclassicality and entanglement are discussed. All concepts are illustrated with experimental data.
Practically applicable criteria for the nonclassicality of quantum states are formulated in terms of different types of moments. For this purpose the moments of the creation and annihilation operators, of two quadratures, and of a quadrature and the photon number operator turn out to be useful. It is shown that all the required moments can be determined by homodyne correlation measurements. An example of a nonclassical effect that is easily characterized by our methods is amplitude-squared squeezing.
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Criteria for the nonclassicality of a quantum state of the harmonic oscillator are formulated in terms of normally ordered moments of quadratures. Inequalities are obtained that are equivalent to the failure of the Glauber-Sudarshan PPP function to be a probability density. A possibility to measure the required moments is proposed.
The nonclassical nature of light from a degenerate parametric oscillator is discussed in terms of criteria based on direct photon counting probabilities of light. These criteria do not rely on phase sensitive detection and reveal the nonclassical nature of the field where the conventional criteria based on moments of light intensity (like sub-Poissonian statistics or anti-bunching) fail. We also consider the effect of detector efficiency on the signature of nonclassicality and find that under certain circumstances inefficient detection can enhance nonclassical behavior.
Quasiprobability distributions offer a useful way of describing nonclassical features of quantum systems through their departure from classical probability theory. In this work, we investigate the quantification of nonclassicality associated with the Margenau--Hill quasiprobability (MHQ) distribution using its moments, without requiring reconstruction of the full distribution. First we introduce the logarithmic MHQ negativity as a quantifier of nonclassicality, and then derive a hierarchy of rigorous lower bounds in terms of low-order moments. Within the resulting hierarchy, the fourth-moment bound is the strongest among the bounds based on even moments. For qubits, we further derive a tight upper bound on the exact MHQ negativity and identify the corresponding extremal state. We also obtain an optimal positivity threshold based on the fourth moment for arbitrary pairs of qubit observables. Finally, we show that the relevant moments admit exact multicopy representations, providing a route to their estimation using interferometry or classical shadow techniques. Our results establish a moment-based framework for extracting quantitative information about MHQ negativity from a finite set of low-order observables.