On a solution of the multidimensional truncated moment problem on vertices of the hypercube based on Bell inequalities

Motivated by applications in quantum information, we will consider the problem of establishing necessary and sufficient conditions so that linear functionals L : R n [ x 1 , … , x d ] → R L: R_n[x_1, … , x_d] → R with L ( 1 ) = 1 L(1) = 1 , where d ≤ n d ≤ n , have a representing measure on vertices of the hypercube in R d R^d , i.e., there exists a probability measure μ μ such that L ( p ) = ∫ R d p ( x ) d μ ( x ) ≔ ∫ ⋯ ∫ R d p ( x 1 , … , x d ) d μ ( x 1 , … , x d ) {equation*} L(p) = ∫ R^d p(x) \, dμ (x) ≔∫ ⋯ ∫ R^d p(x_1, … , x_d) dμ (x_1, … , x_d) {equation*} for p ∈ R n [ x 1 , … , x d ] p ∈ R_n[x_1, … , x_d] and s u p p μ ⊆ K d supp\, μ ⊆ K_d where K d ≔ { ( k 1 , … , k d ) ∈ R d : k j = ± 1 , for j = 1 , … , d } K_d≔\{(k_1,… ,k_d) ∈ R^d : \ k_j=± 1, for j=1, ,d\} . Taking advantage of the group structure of K d K_d and some tools from algebraic geometry and functional analysis, we will show that concrete necessary and sufficient conditions can be communicated in terms of certain consistency conditions and also a system of inequalities which are known in the quantum information theory literature as Bell inequalities. A variant of this problem (which is highly relevant to corresponding Bell nonlocality problems in quantum information theory), when L : A → R L: A → R , where A ⊆ R n [ x 1 , … , x d ] A ⊆ R_n[x_1, … , x_d] is a subspace, will also be considered. Finally, we will provide a formulation of the aforementioned concrete solution in a finite multiplicative Abelian group setting, which may be of independent interest.