Cite this problem as **Problem 34**.

**Problem**

Given a nonlocality scenario, with parties, settings and outcomes, consider the set of probability distributions of the form

,

s.t.

This is the multipartite analog of the set of quantum correlations defined in Problem 33. The question is whether the closure of in each nonlocality scenario is a semi-algebraic set, or, in other words, whether, for any , there is a finite set of polynomials such that iff for .

**Background**

For , the answer is positive. Indeed, as shown by Tsirelson [1] (for ) and independently by Masanes [2] (for all ), in this Bell scenario the extreme points of the quantum set can be realized by conducting projective measurements over an -qubit pure state. Since the dimension of the real space where lives is , by Caratheodory’s theorem [3], each point in must be a convex combination of at most of these extreme points.

Hence in this scenario the set can be seen as the projection of a large set of variables* subject to a finite number of polynomial constraints** to the space. By the Tarski-Seidenberg theorem [4], this implies that the probabilities themselves are characterized by a finite set of polynomial inequalities.

*Namely, , the matrix entries of the projectors and the state for each of the extreme points, together with their weights.

**Namely, the polynomials which constrain the matrices of each extreme point to be, respectively, projectors and normalized quantum states; the weights of the different points to be non-negative and add up to ; and the constraint that applying the Born rule over the weighted ensemble of extreme points we recover the probabilities .

**References**

[1] B.S. Cirel’son, Letters in Mathematical Physics **4**, 93-100 (1980).

[2] Ll. Masanes, arXiv:quant-ph/0512100.

[3] Eggleston, H. G. (1958). Convexity. Cambridge University Press.

[4] J. Bochnak, M. Coste and M.-F. Roy, Real Algebraic Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 36, Springer Berlin Heidelberg, 1998.