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For the classes of synchronous, binary constraint systems, and XOR nonlocalgames, we show that near-optimal finite-dimensional quantum strategies witharbitrary states are approximate representations of their affiliated nonlocalgame algebra. We also show that finite-dimensional approximate representationsof these nonlocal game algebras are close to near-optimal strategies where theplayers employ a maximally entangled state. As a corollary, we show thatnear-optimal quantum strategies are close to a near-optimal quantum strategythat uses a maximally entangled state.

Rounding near-optimal quantum strategies for nonlocal games to strategies using maximally entangled states