  The tendency of semidefinite programs to compose perfectly under product has
been exploited many times in complexity theory: for example, by Lovasz 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 Lovasz, and a direct product theorem
for the discrepancy method of communication complexity by Lee, Shraibman, and
Spalek.
  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.
