Nonlocal games and XOR games

In previous lectures, we discussed a few fairly direct connections between quantum information theoretic notions and semidefinite programs. For instance, the semidefinite program associated with an optimization over measurements is quite simple and direct, and if you are familiar with the Choi representation of channels the same can be said about our semidefinite program for optimizing over channels. Similarly, the semidefinite programs for the trace norm and fidelity are straightforward, once you know about the lemma from the previous lecture characterizing 2-by-2 positive semidefinite block operators. In this lecture and the next, we will study an example of a semidefinite programming formulation of a quantum information theoretic notion where the relevance of semidefinite programming is not at all evident from the start—sometimes some work is required (and perhaps some luck as well) before a connection between a particular notion and semidefinite programming becomes apparent.