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In a recent work, Moshkovitz [FOCS '14] presented a transformation ontwo-player games called "fortification", and gave an elementary proof of an(exponential decay) parallel repetition theorem for fortified two-playerprojection games. In this paper, we give an analytic reformulation ofMoshkovitz's fortification framework, which was originally cast incombinatorial terms. This reformulation allows us to expand the scope of thefortification method to new settings.  First, we show any game (not just projection games) can be fortified, andgive a simple proof of parallel repetition for general fortified games. Then,we prove parallel repetition and fortification theorems for games with playerssharing quantum entanglement, as well as games with more than two players. Thisgives a new gap amplification method for general games in the quantum andmultiplayer settings, which has recently received much interest.  An important component of our work is a variant of the fortificationtransformation, called "ordered fortification", that preserves the entangledvalue of a game. The original fortification of Moshkovitz does not in generalpreserve the entangled value of a game, and this was a barrier to extending thefortification framework to the quantum setting.

Parallel repetition via fortification: analytic view and the quantum case