Strong Parallel Repetition for a Monogamy-of-Entanglement Game
We consider a game in which two players collaborate to prepare a quantum system and are then asked to independently guess the outcome of a measurement in a random basis on that system. Intuitively, by the monogamy of entanglement, the probability that both players simultaneously succeed in guessing the outcome correctly is bounded. We are interested in the question of how the success probability scales when this guessing game is repeated in parallel. We show a perfect parallel repetition theorem for this game, that is, we show that any strategy that maximizes the probability to win every round individually is also optimal for the parallel repetition of the game. In particular, our result implies that the optimal guessing probability can be achieved without the use of entanglement. We explore several applications of this result. First, we show that it implies security for standard BB84 quantum key distribution when one party uses fully untrusted measurement devices. Second, we show that our result can be used to prove security of a one-round position-verification scheme. Finally, our techniques can be used to generalize a well-known uncertainty relation for the guessing probability to quantum side information.