Consider the following non-local game between Alice, Bob (who are spatially separated but share a maximally entangled state) and a referee:
- Referee samples two question bits "$x,y$" and sends them to Alice.
- Alice sends 1 answer bit "$a$" to Referee.
- Referee sends the answer bit "$a$" received from Alice as question to Bob.
- Bob sends back 1 answer bit $b$ to referee.
- Winning Condition: If $(a = x) \wedge (b = y)$ OR $(a = y) \wedge (b = x)$ they win and lose otherwise.
I want to optimize the winning probability of this game by a SDP or analytically upper bound it. It would also be nice if I could bound or optimize the average winning probability. It would be great if someone can provide me some hints upon how I could approach the problem.
Notice that this game is a variation of super-dense coding. Namely, in super-dense coding we send 1 qubit to recover two classical bits, here we quantumly correlate two classical bits in such a way that if we send one of them another one is recovered with high probability. We are also not violating no-signalling because this game is probabilistic and no-signalling tells us that no deterministic protocol exists.
My attempt so far:
Winning Strategy: Let Alice and Bob agree upon a winning strategy before playing the game. Let the winning strategy be denoted as choice of POVMs on Alice and Bob side as:
\begin{align} &\{A_{x,y}^{a} \otimes \mathbb{I}\} \\ &\{\mathbb{I}\otimes B_{a}^{b}\} \end{align}
The winning proability can then be written as
\begin{align}
\omega &= \sum_{x,y,a,b} p(x,y) \langle \Phi^{+} | A_{x,y}^{a} \otimes B_{a}^{b}|\Phi^{+}\rangle \delta_{x,a}\delta_{y,b}
+ \sum_{x,y,a,b} p(x,y) \langle\Phi^{+}| A_{x,y}^{a} \otimes B_{a}^{b}|\Phi^{+}\rangle \delta_{x,b}\delta_{y,a} \\
&= \sum_{x,y,a,b} p(x,y) \langle\Phi^{+}| A_{x,y}^{a} \otimes B_{a}^{b}|\Phi^{+}\rangle (\delta_{x,a}\delta_{y,b} + \delta_{x,b}\delta_{y,a})
\end{align}
where
\begin{align}
|\Phi^{+}\rangle &= \frac{1}{\sqrt{2}} \sum_{i} |i\rangle \otimes |i\rangle
\end{align}
and $p(x,y)$ is the probability of sampling questions $x,y$.
Now we can consider that the questions are sampled uniformly and independently, hence $p(x,y) = p(x)p(y) = \frac{1}{2^{2}}$.
Now, I would like some hints on how to analytically upper bound or use a SDP approach... Does the winning probability reach a maximum value 1? Does the shared quantum correlations help in this game? For analytical calculations does bounding the average winning probability over many rounds help? Does a multi-qubit version of this game yield better winning probability?
