In this paper we study bipartite quantum correlations using techniques fromtracial noncommutative polynomial optimization. We construct a hierarchy ofsemidefinite programming lower bounds on the minimal entanglement dimension ofa bipartite correlation. This hierarchy converges to a new parameter: theminimal average entanglement dimension, which measures the amount ofentanglement needed to reproduce a quantum correlation when access to sharedrandomness is free. For synchronous correlations, we show a correspondencebetween the minimal entanglement dimension and the completely positivesemidefinite rank of an associated matrix. We then study optimization over theset of synchronous correlations by investigating quantum graph parameters. Weunify existing bounds on the quantum chromatic number and the quantum stabilitynumber by placing them in the framework of tracial optimization. In particular,we show that the projective packing number, the projective rank, and thetracial rank arise naturally when considering tracial analogues of the Lasserrehierarchy for the stability and chromatic number of a graph. We also introducesemidefinite programming hierarchies converging to the commuting quantumchromatic number and commuting quantum stability number.
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