Almost no experiments have classical Kirkwood-Dirac representations

A central problem in quantum information is determining quantum-classical boundaries.

In the quasiprobability 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 $d$-dimensional observables are picked at random, the set of classical (positive) states of the resulting KD distribution is a minimal polytope of dimension $2(d-1)$ with $2d$ explicitly known vertices.

This implies minimality of the sets of KD-real observables, of KD-positive measurement elements and of KD-positivity-preserving unitaries.

We show how these results have implications on robust observations of nonclassical phenomena, on classical simulations of quantum circuits, and on foundations of quantum theory.