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Journal of Modern Optics

Wigner quasi-distributions for arbitrary operators

Leon W. Cohen

Published Nov 1, 2004• 6 citations
Publisher

Abstract

We present methods for constructing quasi-distributions for two arbitrary operators. Among the methods are the characteristic function, the local autocorrelation, windowing, and unitary transformation approaches. We also consider the first conditional moment of the distribution and interpret it as a local value.
DOI
10.1080/09500340408231835

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Methods are presented to generate conditional moments for all quasi-probability distributions, and explicit expressions for all conditional moments are given. In analogy to the generalized characteristic function operator, which generates all joint quasi-distribution functions, the conditional characteristic function operator which generates all conditional quasi-distributions is defined. This allows one to introduce the concept of conditional correspondence rules; formulations are given for all such rules.
Abstract We derive a general operator transformation between any two functions and we show that the operator transformation of quantum quasi-distributions is a special case. We also show that our approach generalizes the Edgeworth series of standard probability theory, and results in a unified approach for the transformation of probability densities. Our approach generalizes the classical Edgeworth series in two ways. It relates two arbitrary distributions rather than one of them being a Gaussian and secondly we generalize to an arbitrary Hermitian operator rather than the differentiation operator which appears in the Edgeworth formulation. We also consider applying the transformation to the same distribution but at different times.
We study a generalization of the Wigner function to arbitrary tuples of hermitian operators, which is a distribution uniquely characterized by the property that the marginals for all linear combinations of the given operators agree with the quantum mechanical distributions. Its role as a joint quasi-probability distribution is underlined by the property that its support always lies in the set of expectation value tuples of the operators. We characterize the set of singularities and positivity, and provide some basic examples.
The language of statistical moments and covariance matrices we introduced in the previous chapter offers a compact and efficient formalism to deal with Gaussian states. A much more general approach may be taken to describe any quantum state in a setting which is reminiscent of classical phase space. This is the framework of quantum characteristic functions and quasi-probability distributions, which goes back to seminal work by Wigner on quantum corrections to classical statistical mechanics, and bloomed in the sixties with the rise of theoretical quantum optics and the emergence of a general unifying picture. Conceptually, the phase space description of quantum states hinges on the completeness of the set of displacement operators, which we shall prove in the form of the Fourier–Weyl relation between density matrices and characteristic functions. This relationship will constitute the bridge between phase space and Hilbert space descriptions, which will be useful in several applications in the context of quantum information. It will be convenient to handle most of the proofs and mathematical arguments concerning characteristic functions and quasi-probability distributions on a single mode of the system. Because displacement operators of multimode systems are just tensor products of local displacement operators, the extension of the formalism to systems with many degrees of freedom will be straightforward. Nevertheless, we shall always take care of explicitly linking the single-mode formulae which will appear in this chapter to the general multimode description adopted in the previous one.
We compare the methods of Cohen (1971) and of Baraniuk and Jones (see Proc. IEEE ICASSP-93, p.320-323, 1993) for obtaining joint distributions for arbitrary variables. We show that the two methods produce identical results for variables whose associated operators are obtained via a unitary transformation of the time and frequency operators. In addition, we generalize this result and show that all pairs of variables connected by a unitary transformation have joint distributions that are identical in functional form. This result shows how pairs of variables can be grouped into classes whose joint distributions are functionally equivalent and, therefore, share equivalent properties.>