Algebraic nature of quantum nonlocality

Cite this problem as Problem 34.


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

P(a_1,...,a_n|x_1,...,x_N)=\mbox{tr}(\rho E^{1,x_1}_{a_1}\otimes...\otimes E^{N,x_N}_{a_N}),

s.t. \rho\geq 0, \mbox{tr}(\rho)=1

E^{j,x}_{a}\geq 0, \sum_{a=1}^KE^{j,x}_{a}=\mathbb{I}

This is the multipartite analog of the set Q' of quantum correlations defined in Problem 33. The question is whether the closure of Q' in each nonlocality scenario is a semi-algebraic set, or, in other words, whether, for any N,M,K, there is a finite set of polynomials \{F_i\}_{i=1}^n such that P(a_1,...,a_N|x_1,...,x_N)\in \mbox{closure}(Q') iff F_i(P(a_1,...,a_N|x_1,...,x_N))\geq 0 for i=1,...,n.


For M=K=2, the answer is positive. Indeed, as shown by Tsirelson [1] (for N=2) and independently by Masanes [2] (for all N), in this Bell scenario the extreme points of the quantum set can be realized by conducting projective measurements over an N-qubit pure state. Since the dimension of the real space where P(a_1,...,a_N|x_1,...,x_N) lives is 3^N-1, by Caratheodory’s theorem [3], each point in Q' must be a convex combination of at most 3^N of these extreme points.

Hence in this scenario the set Q' can be seen as the projection of a large set of variables* subject to a finite number of polynomial constraints** to the P(a_1,...,a_N|x_1,...,x_N) space. By the Tarski-Seidenberg theorem [4], this implies that the probabilities P(a_1,...,a_N|x_1,...,x_N) themselves are characterized by a finite set of polynomial inequalities.

*Namely, P(a_1,...,a_N|x_1,...,x_N), the matrix entries of the projectors E^{j,x}_a and the state \rho for each of the 3^N extreme points, together with their weights.

**Namely, the polynomials which constrain the matrices E^{j,x}_a, \rho 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 1; and the constraint that applying the Born rule over the weighted ensemble of extreme points we recover the probabilities P(a_1,...,a_N|x_1,...,x_N).


[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.