Concavity of the set of quantum probabilities for any given dimension

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. ©2009 The American Physical Society