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Bulletin of the London Mathematical Society

Hudson's theorem for τ-Wigner transforms

Paolo Boggiatto, Giuseppe De Donno, Alessandro Oliaro

Published Jul 16, 2013• 11 citations
Publisher

Abstract

In this paper, after introducing a natural generalization of the classical Wigner transform, namely the τ-Wigner transforms, depending on the parameter τ∈[0, 1], we study the problem of its positivity. In particular, we prove two theorems of Hudson-type considering the action of the τ-Wigner transforms on functions and on distributions, respectively. We give then an application of our results concerning Weyl and localization pseudo-differential operators.

DOI
10.1112/blms/bdt038

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In this paper, after introducing a natural generalization of the classical Wigner transform, namely the τ-Wigner transforms, depending on the parameter τ∈[0, 1], we study the problem of its positivity. In particular, we prove two theorems of Hudson-type considering the action of the τ-Wigner transforms on functions and on distributions, respectively. We give then an application of our results concerning Weyl and localization pseudo-differential operators.
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.
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.
States in classical mechanics are probability measures, and their Fourier transforms are continuous functions of positive type. States in the phase-space formulation of quantum mechanics are Wigner distribution functions, and their (symplectic) Fourier transforms have been characterized by Kastler [Commun. Math. Phys. 1, 14 (1965)] and Loupias and Miracle-Sole [Commun. Math. Phys. 2, 31 (1966); Ann. Insto. H. Poincaré 6, 39 (1967)] as being continuous functions of ‘‘ℏ-positive type.’’ In this paper, (Schwartz) distributions of ℏ-positive type, are defined and studied. It is shown that if such distributions are bounded on a certain sequence of test functions, then they are symplectic Fourier transforms of Wigner distribution functions. These results, are applied to a variety of problems ranging from ones involving the quantum Liouville equation to a problem in signal analysis.