  We study the behavior of the entangled value of two-player one-round
projection games under parallel repetition. We show that for any projection
game $G$ of entangled value 1-eps < 1, the value of the $k$-fold repetition of
G goes to zero as O((1-eps^c)^k), for some universal constant c\geq 1.
Previously parallel repetition with an exponential decay in $k$ was only known
for the case of XOR and unique games. To prove the theorem we extend an
analytical framework recently introduced by Dinur and Steurer for the study of
the classical value of projection games under parallel repetition. Our proof,
as theirs, relies on the introduction of a simple relaxation of the entangled
value that is perfectly multiplicative. The main technical component of the
proof consists in showing that the relaxed value remains tightly connected to
the entangled value, thereby establishing the parallel repetition theorem. More
generally, we obtain results on the behavior of the entangled value under
products of arbitrary (not necessarily identical) projection games. Relating
our relaxed value to the entangled value is done by giving an algorithm for
converting a relaxed variant of quantum strategies that we call "vector quantum
strategy" to a quantum strategy. The algorithm is considerably simpler in case
the bipartite distribution of questions in the game has good expansion
properties. When this is not the case, rounding relies on a quantum analogue of
Holenstein's correlated sampling lemma which may be of independent interest.
Our "quantum correlated sampling lemma" generalizes results of van Dam and
Hayden on universal embezzlement.
