Concavity of the quantum body for any given dimension

TL;DR

This paper investigates the shape of the set of joint probabilities from local measurements on quantum systems of fixed dimension, revealing it is non-convex and highlighting differences between measurement types.

Contribution

It demonstrates the non-convexity of the quantum body for fixed dimension and shows POVMs can achieve points unattainable by projective measurements.

Findings

Quantum body is non-convex for fixed dimension d.

Certain points in the convex hull are not attainable by d-dimensional quantum systems.

POVMs can reach points that projective measurements cannot.

Abstract

Let us consider the set of all joint probabilities generated by local binary measurements on two separated quantum systems of a given local dimension d. We address the question of whether the shape of this quantum body is convex or not. We construct a point in the space of joint probabilities, which is on the convex hull of the local polytope, but still cannot be attained by measuring d-dimensional quantum systems, if the number of measurement settings is large enough. From this it follows that this body is not convex. We also show that for finite d the quantum body with POVM allowed may contain points that can not be achieved with only projective measurements.