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SIAM Journal on Mathematical Analysis

A Note on Hudson’s Theorem about Functions with Nonnegative Wigner Distributions

Augustus J. E. M. Janssen

Published Jan 1, 1984• 29 citations
Publisher

Abstract

We show that a (generalized) function f has a nonnegative Wigner distribution $W(f,f)$ if and only if f is a Gauss function (possibly degenerate). We prove, more generally, that the convolution of $W(f,f)$ 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 $\exp ( - \pi \alpha t^2 + 2\pi \beta t + \gamma )$ with $\alpha $, $\beta $, $\gamma $ complex, $\operatorname{Re} \alpha = 0$. The main tool used is Moyal’s formula for the Wigner distribution together with Bargmann’s integral transform.

DOI
10.1137/0515014

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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 α\alpha α, β\beta β, γ\gamma γ 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.
The celebrated Hudson theorem states that the Gaussian functions in Rd\mathbb{R}^dRd are the only functions whose Wigner distribution is everywhere positive. Motivated by quantum information theory, D. Gross proved an analogous result on the Abelian group Zdn\mathbb{Z}_d^nZdn​, for ddd odd - corresponding to a system of nnn qudits - showing that the Wigner distribution is nonnegative only for the so-called stabilizer states. Extending this result to the thermodynamic limit of finite-dimensional systems naturally leads us to consider general 222-regular LCA groups that possess a compact open subgroup, where the issue of the positivity of the Wigner distribution is currently an open problem. We provide a complete solution to this question by showing that if the map x↦2xx\mapsto 2xx↦2x is measure-preserving, the functions whose Wigner distribution is nonnegative are exactly the subcharacters of second degree, up to translation and multiplication by a constant. Instead, if the above map is not measure-preserving, the Wigner distribution always takes negative values. We discuss in detail the particular case of infinite sums of discrete groups and infinite products of compact groups, which correspond precisely to infinite quantum spin systems. Further examples include nnn-adic systems, where n≥2n\geq 2n≥2 is an arbitrary integer (not necessarily a prime), as well as solenoid groups.
The Fourier–Wigner transform of two functions is first defined and is given an alternative expression. The Wigner transform is then defined to be the Fourier transform of the Fourier–Wigner transform. The Moyal identity, which is an analog of the Plancherel formula, is then proved for the Wigner transform. The Weyl transform with a given symbol is then defined in terms of the Wigner transform. Another formula for the Weyl transform with a given symbol is then proved and used to establish the fact that the product of two Weyl transforms is also a Weyl transform with symbol given by a twisted convolution of the two given symbols.
There is a theorem of Wigner that states that phase-plane distribution functions involving the state bilinearly and having correct marginals must take negative values for certain states. The purpose of this paper is to support the statement that these phase-plane distribution functions are for hardly any state everywhere non-negative. In particular, it is shown that for certain generalized Wigner distribution functions there are no smooth states (except the Gaussians for the Wigner distribution function itself) whose distribution function is everywhere non-negative. This class of Wigner-type distribution functions contains the Margenau–Hill distribution. Furthermore, the argument used in the proof of Wigner’s theorem is augmented to show that under mild conditions one can find for any two states f, g with non-negative distribution functions a linear combination h of f and g whose distribution function takes negative values, unless f and g are proportional.
Using the Wigner-Vlasov formalism, an exact three-dimensional solution of the Schr\"odinger equation for a scalar particle in an electromagnetic field is constructed. Electric and magnetic fields are nonuniform. According to the exact expression for the wave function, the search for two types of Wigner function is conducted. The first function is the usual Wigner function with a kinetic momentum. The second Wigner function is constructed on the basis of the Weyl-Stratonovich transform [Javanainen et al., Phys. Rev. A 35, 2791 (1987); Nedjalkov et al., Phys. Rev. B 99, 014423 (2019)]. It turns out that the second function, unlike the first one, has areas of negative values for wave functions with the Gaussian distribution (Hudson's theorem). An example of an electromagnetic quantum system described by a non-Gaussian wave function has successfully been found. The second Wigner function is positive over the whole phase space for the non-Gaussian wave function. This result is analogous to the Hudson theorem for the gauge invariant Wigner function. On the one hand, knowing the Wigner functions allows one to find the distribution of the mean momentum vector field and the energy spectrum of the quantum system. On the other hand, within the framework of the Wigner-Vlasov formalism, the mean momentum distribution and the magnitude of the energy are initially known. Consequently, the mean momentum distributions and energy values obtained according to the Wigner functions can be compared with the exact momentum distribution and energy values. This paper presents this comparison and describes the differences. The Vlasov-Moyal approximation of average acceleration flow has been built in phase space for a quantum system with an electromagnetic field. The obtained approximation makes it possible to cut the Vlasov chain off at the second equation and also to analyze the Boltzmann H-function evolution. By averaging the Vlasov-Moyal approximation over momentum space we can derive an expression for the electromagnetic force of the classical system. This averaging makes the high-order quantum terms disappear from the ``motion equation.''