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Inverse problems for the zeros of the Wigner function

Luís Daniel Abreu, Ulysse Chabaud, Nuno Costa Dias, João Nuno Prata

TL;DR

This work studies how the zeros of a Wigner function constrain the underlying quantum state. By linking nodal sets to Bargmann/holomorphic structures and exploiting polyanalytic representations, the authors prove that a bounded nodal set forces f to be a generalized Gaussian times a polynomial, and they derive rigidity results for circles and ellipses in phase space. They show that a Wigner function vanishing on a circle anchored at the origin determines the state among low-order Hermite functions, and they establish a sign-uncertainty principle and several geometric constraints (lines, ellipses, radii) via Laguerre polynomial zeros. The results yield both sharp structural characterizations and foundational limits on the possible nodal configurations of Wigner distributions, with broader implications for phase-space analysis and Laguerre-zero theory.

Abstract

In this work we consider the inverse problem of determining the properties of a Wigner function from the set of its zeros (the nodal set). The previous state of the art of the problem is Hudson's theorem, which shows that an empty nodal set is associated only with generalized Gaussians. We extend this analysis to non-Gaussian functions. Our first main result states that, if the nodal set of the Wigner distribution of a function $f$ is bounded, then $f$ is equal to a generalized Gaussian times a polynomial. An immediate consequence of this result is that any open set is a uniqueness set for Wigner functions with bounded nodal set. Our second main result shows that the only Wigner function vanishing on a circle of radius $\sqrt{\hbar/2}$ and centered at the origin is the Wigner distribution of the first Hermite function. We prove similar results for the second and third Hermite functions. We also derive for Wigner functions a counterpart of the sign uncertainty principle of J. Bourgain, L. Clozel and J.-P. Kahane, which says that if the negative part of a Wigner function is contained in a ball, then the radius of the ball has a lower bound. Finally, we obtain various constraints on Wigner distributions whose bounded nodal sets contain circles, ellipses or line segments. As a by-product of our work we prove several non-trivial results about the zeros of Laguerre polynomials.

Inverse problems for the zeros of the Wigner function

TL;DR

This work studies how the zeros of a Wigner function constrain the underlying quantum state. By linking nodal sets to Bargmann/holomorphic structures and exploiting polyanalytic representations, the authors prove that a bounded nodal set forces f to be a generalized Gaussian times a polynomial, and they derive rigidity results for circles and ellipses in phase space. They show that a Wigner function vanishing on a circle anchored at the origin determines the state among low-order Hermite functions, and they establish a sign-uncertainty principle and several geometric constraints (lines, ellipses, radii) via Laguerre polynomial zeros. The results yield both sharp structural characterizations and foundational limits on the possible nodal configurations of Wigner distributions, with broader implications for phase-space analysis and Laguerre-zero theory.

Abstract

In this work we consider the inverse problem of determining the properties of a Wigner function from the set of its zeros (the nodal set). The previous state of the art of the problem is Hudson's theorem, which shows that an empty nodal set is associated only with generalized Gaussians. We extend this analysis to non-Gaussian functions. Our first main result states that, if the nodal set of the Wigner distribution of a function is bounded, then is equal to a generalized Gaussian times a polynomial. An immediate consequence of this result is that any open set is a uniqueness set for Wigner functions with bounded nodal set. Our second main result shows that the only Wigner function vanishing on a circle of radius and centered at the origin is the Wigner distribution of the first Hermite function. We prove similar results for the second and third Hermite functions. We also derive for Wigner functions a counterpart of the sign uncertainty principle of J. Bourgain, L. Clozel and J.-P. Kahane, which says that if the negative part of a Wigner function is contained in a ball, then the radius of the ball has a lower bound. Finally, we obtain various constraints on Wigner distributions whose bounded nodal sets contain circles, ellipses or line segments. As a by-product of our work we prove several non-trivial results about the zeros of Laguerre polynomials.
Paper Structure (17 sections, 23 theorems, 173 equations)

This paper contains 17 sections, 23 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Convolution
  4. Wigner functions and harmonic analysis in phase space
  5. Husimi function and Bargmann transform
  6. Proof of Theorem \ref{['TheoremSmootDecay']}
  7. Proof of Corollary \ref{['CorollaryUniquenessSet']}
  8. Proof of Theorem \ref{['TheoremEllipse']}
  9. A Proof of Conjecture \ref{['MainTheorem']} assuming the validity of Conjecture \ref{['PropositionLaguerre']}
  10. Proof of Theorem \ref{['TheoremHermite1']}
  11. Proof of Theorem \ref{['TheoremSignUPWignerfunctions']}
  12. Proof of Corollary \ref{['Corollary_SignUPWignerFunctions']}
  13. Proof of Theorem \ref{['TheoremLineSegment']}
  14. Proof of Theorem \ref{['TheoremArbitraryRadius']}
  15. Proof of Theorem \ref{['TheoremRationalRadius']}
  16. ...and 2 more sections

Key Result

Theorem A

Let $f \in L^2 (\mathbb{R}) \backslash \left\{0 \right\}$. Then $Wf$ is everywhere non-negative if and only if $f$ is a generalized Gaussian: where $a,b,c, \in \mathbb{C}$ and $\text{Re}(a) >0$.

Theorems & Definitions (40)

  • Theorem A: Hudson
  • Theorem B: Hudson-second version
  • Theorem 1: Extension of Hudson's Theorem
  • Example 1
  • Corollary 1
  • Theorem 2
  • Example 2
  • Definition 1
  • Remark 1
  • Theorem 3
  • ...and 30 more