  For many protocols, quantum strategies have advantages compared with their
classical counter-partners, and these advantages have attracted many interests
and applications. One of the famous examples is the Clauser-Horne-Shimony-Holt
(CHSH) game, which recasts Bell's theorem~\cite{2} into the framework of a
game. In the CHSH game, two space-like separated players, Alice and Bob are
each assigned a classical bit $a$ and $b$ respectively. Then they return bits
$x$ and $y$ according to some pre-agreed strategies. They will win the game
when $x\oplus y= a\cdot b$. In the game, if the players use the classical
strategies, the optimal success probability $w(\text{CHSH})=0.75$.However, if
they add some quantum resources, the success probability will increase and up
to maximal value $cos^2(\pi/8)$, which is know as the Tsirelson's bound.
Moreover, Popescu and Rohrlich noted that the perfect success probability $1$
can also be achieved in a more general theory without violating the
no-signaling assumption
