Concavity of the set of quantum probabilities for any given dimension

Abstract

Let us consider the set of all 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 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 the generalized measurement associated with positive operator-valued measures allowed may contain points that cannot be achieved with only projective measurements.

Locations

• Physical Review A - View
• arXiv (Cornell University) - View - PDF
• DataCite API - View