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Journal of Mathematical Physics

Conditions for the convolution of two Wigner distributions to be itself a Wigner distribution

Francis J. Narcowich

Published Sep 1, 1988• 46 citations
Publisher

Abstract

The convolution of two Wigner distribution functions (WDF’s) is always non-negative, but not always a WDF, as Jagannathan et al. [Phys. Lett. A 120, 161 (1987)] have shown. In this paper conditions are given that are sufficient, and probably necessary, for such a convolution to be a WDF, and a new characterization of Gaussian WDF’s is obtained as a by-product.

DOI
10.1063/1.527861

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The convolution of two Wigner distribution functions (WDF’s) is always non-negative, but not always a WDF, as Jagannathan et al. [Phys. Lett. A 120, 161 (1987)] have shown. In this paper conditions are given that are sufficient, and probably necessary, for such a convolution to be a WDF, and a new characterization of Gaussian WDF’s is obtained as a by-product.
We discuss two sets of conditions that are necessary and sufficient for a function defined on phase space to be a Wigner distribution function (WDF). The first set is well known and involves the function itself; the second set is less familiar and involves the function's ``symplectic'' Fourier transform. After explaining why these two sets are equivalent, we explore some properties and applications of the second one. Among other things, we show that that set includes the position-momentum uncertainty relations as a special case, and in doing so we give a new derivation of them. This derivation itself serves as the starting point for the discussion of a quantum-mechanical moment problem. It also enables us to exhibit a real-valued phase-space function that obeys the uncertainty relations but that is not a WDF.
A coarse-grained Wigner distribution ${p}_{W}(x,\ensuremath{\mu})$ obeying positivity derives out of information-theoretic considerations. Let $p(x,\ensuremath{\mu})$ be the unknown joint probability density function (PDF) on position and momentum fluctuations xxx, $\ensuremath{\mu}$ for a particle in a pure state $\ensuremath{\psi}(x)$. Suppose that the phase part $\ensuremath{\Psi}(x,z)$ of its Fourier transform ${T}_{F}[p(x,\ensuremath{\mu})]\ensuremath{\equiv}\ensuremath{\mid}G(x,z)\ensuremath{\mid}\mathrm{exp}[i\ensuremath{\Psi}(x,z)]$ is constructed as a hologram. (Such a hologram is often used in heterodyne interferometry.) Consider a particle randomly illuminating this phase hologram. Let its two position coordinates be measured. Require that the measurements contain an extreme amount of Fisher information about true position, through variation of the phase function $\ensuremath{\Psi}(x,z)$. The extremum solution gives an output PDF $p(x,\ensuremath{\mu})$ that is the convolution of the Wigner ${p}_{W}(x,\ensuremath{\mu})$ with an instrument function defining uncertainty in either position xxx or momentum $\ensuremath{\mu}$. The convolution arises naturally out of the approach, and is one dimensional, in comparison with the ad hoc two-dimensional convolutions usually proposed for coarse graining purposes. The output obeys positivity, as required of a PDF, if the one-dimensional instrument function is sufficiently wide. The result holds for a large class of systems: those whose amplitudes $\ensuremath{\psi}(x)$ are the same at their boundaries [examples: states $\ensuremath{\psi}(x)$ with positive parity; with periodic boundary conditions; free particle trapped in a box].
We show that a (generalized) function f has a nonnegative Wigner distribution W(f,f)W(f,f)W(f,f) if and only if f is a Gauss function (possibly degenerate). We prove, more generally, that the convolution of W(f,f)W(f,f)W(f,f) with certain Gauss functions is nonnegative if and only if f is of the special type mentioned. As a consequence we have that the only (generalized) functions whose Wigner distributions are concentrated on a curve of a particular type are delta functions or exponentials exp⁡(−παt2+2πβt+γ)\exp ( - \pi \alpha t^2 + 2\pi \beta t + \gamma )exp(−παt2+2πβt+γ) with $\alpha $, $\beta $, $\gamma $ complex, Re⁡α=0\operatorname{Re} \alpha = 0Reα=0. The main tool used is Moyal’s formula for the Wigner distribution together with Bargmann’s integral transform.
We examine the affine Wigner distribution from a quantization perspective with an emphasis on the underlying group structure. One of our main results expresses the scalogram as (affine) convolution of affine Wigner distributions. We strive to unite the literature on affine Wigner distributions and we provide the connection to the Mellin transform in a rigorous manner. Moreover, we present an affine ambiguity function and show how this can be used to illuminate properties of the affine Wigner distribution. In contrast with the usual Wigner distribution, we demonstrate that the affine Wigner distribution is never an analytic function. Our approach naturally leads to several applications, one of which is an approximation problem for the affine Wigner distribution. We show that the deviation for a symbol to be an affine Wigner distribution can be expressed purely in terms of intrinsic operator-related properties of the symbol. Finally, we present a positivity conjecture regarding the non-negativity of the affine Wigner distribution.
We study the properties of the discrete Wigner distribution for two qubits introduced by Wotters. In particular, we analyse the entanglement properties within the Wigner distribution picture by considering the negativity of the Wigner function (WF) and the correlations of the marginal distribution. We show that a state is entangled if at least one among the values assumed by the corresponding discrete WF is smaller than a certain critical (negative) value. Then, based on the partial transposition criterion, we establish a relation between the separability of a density matrix and the non-negativity of the WFs relevant both to such a density matrix and to the partially transposed thereof. Finally, we derive a simple inequality—involving the covariance-matrix elements of a given WF—which appears to provide a separability criterion stronger than the one based on the local uncertainty relations.