We study the quantum moment problem: Given a conditional probability distribution together with some polynomial constraints, does there exist a quantum state rho and a collection of measurement operators such that (i) the probability of obtaining a particular outcome when a particular measurement is performed on rho is specified by said conditional probability distribution, and (ii) the measurement operators satisfy the constraints. For example, the constraints might specify that some measurement operators must commute. A special case of the quantum moment problem is to compute the value of one-round multi-prover games with entangled provers, or as it is known in physics, the maximum violation of Bell inequalities in quantum mechanics.
We show that if an instance of the quantum moment problem is unsatisfiable, then there exists a certificate of a particular form proving this. Our proof is based on a recent result in algebraic geometry, the noncommutative Positivstellensatz of Helton and McCullough [Trans. Amer. Math. Soc., 356(9):3721, 2004].
Finally, we discuss open problems and conjectures.
Joint work with A. Doherty, B. Toner and Y. Liang.