We give a converging semidefinite programming hierarchy of outer approximations for the set of quantum correlations of fixed dimension and derive analytical bounds on the convergence speed of the hierarchy. In particular, we give a semidefinite program of size exp(O(T^12(log^2(AT)+log(Q)log(AT))/eps^2)) to compute additive eps-approximations on the values of two-player free games with TxT-dimensional quantum assistance, where A and Q denote the numbers of answers and questions of the game, respectively. For fixed dimension T, this scales polynomially in Q and quasi-polynomially in A, thereby improving on previously known approximation algorithms for which worst-case run-time guarantees are at best exponential in Q and A. For the proof, we make a connection to the quantum separability problem and employ improved multipartite quantum de Finetti theorems that we prove via quantum entropy inequalities. Joint work with Hyejung H. Jee, Carlo Sparaciari, and Omar Fawzi.
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