Polytopes of Absolutely Wigner Bounded Spin States

TL;DR

This paper explores the geometric structure of spin states with non-negative Wigner functions, characterizing absolutely Wigner-bounded states as polytopes and analyzing their properties in finite-dimensional quantum systems.

Contribution

It extends the characterization of absolutely Wigner positive states to finite-dimensional spins, defining them as polytopes with specific eigenvalue constraints and analyzing their geometric features.

Findings

Absolutely Wigner positive states form a polytope centered on the maximally mixed state.

A Hilbert-Schmidt ball provides a sufficient purity-based condition for states to be AWB.

Conjecture of a necessary condition for AWB states based on geometric analysis.

Abstract

Quasiprobability has become an increasingly popular notion for characterising non-classicality in quantum information, thermodynamics, and metrology. Two important distributions with non-positive quasiprobability are the Wigner function and the Glauber-Sudarshan function. Here we study properties of the spin Wigner function for finite-dimensional quantum systems and draw comparisons with its infinite-dimensional analog, focusing in particular on the relation to the Glauber-Sudarshan function and the existence of absolutely Wigner-bounded states. More precisely, we investigate unitary orbits of mixed spin states that are characterized by Wigner functions lower-bounded by a specified value. To this end, we extend a characterization of the set of absolutely Wigner positive states as a set of linear eigenvalue constraints, which together define a polytope centred on the maximally mixed state in the simplex of spin-$j$ states. The lower bound determines the relative size of such absolutely Wigner bounded (AWB) polytopes and we study their geometric characteristics. In each dimension a Hilbert-Schmidt ball representing a tight purity-based sufficient condition to be AWB is exactly determined, while another ball representing a necessary condition to be AWB is conjectured. Special attention is given to the case where the polytope separates orbits containing only positive Wigner functions from other orbits because of the use of Wigner negativity as a witness of non-classicality. Comparisons are made to absolute symmetric state separability and spin Glauber-Sudarshan positivity, with additional details given for low spin quantum numbers.