The set of Kirkwood-Dirac positive states is almost always minimal

TL;DR

This paper shows that for randomly chosen observables, the set of classical states in Kirkwood-Dirac distributions is minimal and simple, with a small number of vertices, revealing fundamental properties of quantum-classical boundaries.

Contribution

It proves that the set of classical states in KD distributions is almost always a minimal simple polytope for random observables.

Findings

The classical state set forms a simple polytope of size 2d in dimension 2d-1.

Almost all KD distributions have a resource theory with a small set of free states.

The polytope has 2d known vertices, indicating minimal complexity.

Abstract

A central problem in quantum information is determining quantum-classical boundaries. A useful notion of classicality is provided by the quasiprobability formulation of quantum theory. In this framework, a state is called classical if it is represented by a quasiprobability distribution that is positive, and thus a probability distribution. In recent years, the Kirkwood-Dirac (KD) distributions have gained much interest due to their numerous applications in modern quantum-information research. A particular advantage of the KD distributions is that they can be defined with respect to arbitrary observables. Here, we show that if two observables are picked at random, the set of classical states of the resulting KD distribution is a simple polytope of minimal size. When the Hilbert space is of dimension $d$, this polytope is of dimension $2d-1$ and has $2d$ known vertices. Our result implies, $\textit{e.g.}$, that almost all KD distributions have resource theories in which the free states form a small and simple set.