  We present two parallel repetition theorems for the entangled value of
multi-player, one-round free games (games where the inputs come from a product
distribution). Our first theorem shows that for a $k$-player free game $G$ with
entangled value $\mathrm{val}^*(G) = 1 - \epsilon$, the $n$-fold repetition of
$G$ has entangled value $\mathrm{val}^*(G^{\otimes n})$ at most $(1 -
\epsilon^{3/2})^{\Omega(n/sk^4)}$, where $s$ is the answer length of any
player. In contrast, the best known parallel repetition theorem for the
classical value of two-player free games is $\mathrm{val}(G^{\otimes n}) \leq
(1 - \epsilon^2)^{\Omega(n/s)}$, due to Barak, et al. (RANDOM 2009). This
suggests the possibility of a separation between the behavior of entangled and
classical free games under parallel repetition.
  Our second theorem handles the broader class of free games $G$ where the
players can output (possibly entangled) quantum states. For such games, the
repeated entangled value is upper bounded by $(1 -
\epsilon^2)^{\Omega(n/sk^2)}$. We also show that the dependence of the exponent
on $k$ is necessary: we exhibit a $k$-player free game $G$ and $n \geq 1$ such
that $\mathrm{val}^*(G^{\otimes n}) \geq \mathrm{val}^*(G)^{n/k}$.
  Our analysis exploits the novel connection between communication protocols
and quantum parallel repetition, first explored by Chailloux and Scarpa (ICALP
2014). We demonstrate that better communication protocols yield better parallel
repetition theorems: our first theorem crucially uses a quantum search protocol
by Aaronson and Ambainis, which gives a quadratic speed-up for distributed
search problems. Finally, our results apply to a broader class of games than
were previously considered before; in particular, we obtain the first parallel
repetition theorem for entangled games involving more than two players, and for
games involving quantum outputs.
