We show a connection between the semidefinite relaxation of unique
  games and their behavior under parallel repetition.
  Specifically, denoting by $\val(G)$ the value of a two-prover unique
  game $G$, and by $\sval(G)$ the value of a natural semidefinite
  program to approximate $\val(G)$, we prove that for every
  $\ell\in\N$, if $\sval(G) \geq 1-\delta$, then
  \begin{math}
    \val(G^{\ell}) \geq 1 - \sqrt{s\ell\delta\,} \mper
  \end{math}
  Here, $G^{\ell}$ denotes the $\ell$-fold parallel repetition
  of $G$, and $s=O(\log(k/\delta))$, where $k$ denotes the alphabet
  size of the game. For the special case where $G$ is an XOR game
  (i.e., $k=2$), we obtain the same bound but with $s$ as an absolute
  constant.
  %
  Our bounds on $s$ are optimal up to a factor of $O(\log(\nicefrac 1
  \delta))$.
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  For games with a significant gap between the
  quantities $\val(G)$ and $\sval(G)$, our result implies that
  $\val(G^{\ell})$ may be much larger than $\val(G)^{\ell}$, giving a
  counterexample to the strong parallel repetition conjecture. In a
  recent breakthrough, Raz (FOCS '08) has shown such an example using the
  max-cut game on odd cycles. Our results are based on a
  generalization of his techniques.