The tendency of semidefinite programs to compose perfectly under product has been exploited many times in complexity theory: for example, by Lovász to determine the Shannon capacity of the pentagon; to show a direct sum theorem for non-deterministic communication complexity and direct product theorems for discrepancy; and in interactive proof systems to show parallel repetition theorems for restricted classes of games. Despite all these examples of product theorems—some going back nearly thirty years—it was only recently that Mittal and Szegedy began to develop a general theory to explain when and why semidefinite programs behave perfectly under product. This theory captured many examples in the literature, but there were also some notable exceptions which it could not explain—namely, an early parallel repetition result of Feige and Lovász, and a direct product theorem for the discrepancy method of communication complexity by Lee, Shraibman, and Špalek. We extend the theory of Mittal and Szegedy to explain these cases as well. Indeed, to the best of our knowledge, our theory captures all examples of semidefinite product theorems in the literature.
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