Positivstellensatz, SDPs and Entangled Games

Today we are going to talk about an infinite hierarchy of semidefinite programmings which attempts to approximate the entangled value of multi-prover games [1]. This work is based on a recent result called non-commutative Positivstellensatz [2] about the representation of positive polynomials. In a one-round two-prover cooperative game, a verifier asks questions to two provers, Alice and Bob, who cooperate with each other. A game G = G(π, V ) is specified by the set of questions S, T and answers A,B for Alice and Bob, a probability distribution π : S × T → [0, 1], and a predicate V : A×B×S × T → {0, 1}. The referee samples (s, t) ∈ S × T according to π, and sends question s to Alice and questions t to Bob. Alice replies with an answer a ∈ A, and Bob replies with an answer b ∈ B. The provers win if and only if V (a, b|s, t) = 1. The provers are allowed to agree on a strategy before the game starts, but not allowed to communicate with each other after receiving their questions. The classical value of this game, denoted by ωc(G), is the maximum probability with which the provers can win. In the entangled version of this game, we allow the provers to share arbitrary prior entanglement and perform arbitrary local quantum operations. We use ω∗(G) to denote the maximum probability with which any entangled provers can win. For example, CHSH game is a two-prover game in which A = B = S = T = {0, 1} and V (a, b|s, t) = 1 if a ⊕ b = s ∧ t, and 0 otherwise. For this game, any classical provers can win with probability at most ωc(G) = 3/4, but entangled provers can win with probability ω∗(G) = cos2(π/8) ≈ 0.85. Without loss of generality, we can assume the entangled prover’s strategy as follows. They share a pure state |ψ〉 ∈ Cd×d for some d ≥ 1. If Alice receives question s, then she performs a POVM {As} on her part of |ψ〉, (i.e. ∑ a A a s = I, A a s ≥ 0), and replies with answer a if the measurement outcome corresponds to As . Similarly, we define POVMs {Bb t} for Bob. So the probability that Alice answers a and Bob answers b is given by