The Hudson theorem in LCA groups and infinite quantum spin systems

TL;DR

This paper extends the Hudson theorem to a broad class of LCA groups and infinite quantum spin systems, characterizing functions with nonnegative Wigner distributions and exploring their properties in various group settings.

Contribution

It provides a complete characterization of functions with nonnegative Wigner distributions on general LCA groups, including infinite quantum spin systems, based on measure-preserving properties.

Findings

Functions with nonnegative Wigner distributions are subcharacters of second degree under certain conditions.

If the measure-preserving condition fails, the Wigner distribution is always negative.

The results apply to infinite sums, products, n-adic systems, and solenoid groups.

Abstract

The celebrated Hudson theorem states that the Gaussian functions in $\mathbb{R}^d$ 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 $\mathbb{Z}_d^n$, for $d$ odd - corresponding to a system of $n$ 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 $2$-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 $x\mapsto 2x$ 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 $n$-adic systems, where $n\geq 2$ is an arbitrary integer (not necessarily a prime), as well as solenoid groups.