Is there an information exchange in this game? (Bell's inequality)

Question

This question concerns quantum mechanics experiment. But I believe it belongs here, on MathOverflow.

So, we have two players. They play a simple game and either both win or both loose, so they cooperate.

A fair coin is thrown in front of each player. (Each player sees only "his" coin). Having seen the result of the throw, each player tells a number: either 0 or 1. The outcome of the game is determined by the following rules: if both coins are "heads", players win if they named different numbers. Otherwise players win if they named the same number. Otherwise players loose.

Of course, if players can communicate (send some information to each other), they can easily win each game. But each player can't see the other coin and the game master made his best not to allow any information exchange.

In this case it's easy for players to achieve 75% win rate. Just always tell 1. (Players can discuss the strategy before the game starts).

The question is, can players achieve higher win rate?

It's "obvious" they can't. I can prove that if they use strategy like "if outcome the the coin throw is this, I tell number 0 with probability X, etc..." they can't beat 75% result. In other circumstances I would be satisfied, because no other strategy comes to my mind. But things are more complicated.

As I mentioned the question relates to quantum mechanics and physics.

One of the best ways to ensure that players can't exchange information is to put them quite far away from each other and require that they name the number fast enough. Relativity theory says that if "speed of light" * "given time" < "distance between players" they have absolutely no way to send each other information about the coin throw result. Sending information faster than light would make time-machine possible.

But players can "cheat" using quantum mechanics. The strategy looks like this: they prepare a pair of entangled particles and each player takes one of these particles. After the coin is thrown, a player makes an experiment with his particle: he positions the measurement device in either of two ways (depending the coin throw result) he measures the state of the particle: the outcome of measurement is either 0 or 1 and the player names the outcome. Quantum mechanics predicts win rate of about 85% in this case.

My description of the strategy is not at all complete. You can easily find full description if you want. My point is that the strategy exists, and it was experimentally confirmed (!).

So, there are three statements:

1. 75% is a maximum win rate if there is no information exchange between players
2. there IS no information exchange between players in experiments
3. win rate is above 75% in experiments

These statements contradict each other. At least one of them must be wrong. I am pretty sure the second and third statements are correct. I think the first statement is wrong. Could you please confirm this and explain why exactly is it wrong?

Answer 1

The protocol you describe satisfies 2 and 3 but not 1, so when that protocol is adopted, 1 is wrong.

The correct form of statement 1 is that 75% is a maximum win rate if each player must choose a strategy that is contingent on the realization of some classical random variable (so that in particular there exists a joint distribution for the set of all random variables that are available for at least one player to use).

In a world governed by classical physics, this correct form of statement 1 is equivalent to the statement you've given about information exchange. In that world, all observables can be modeled as classical random variables.

But for some sets of quantum observables, that's no longer true, so in a world governed by quantum mechanics the two forms of Statement 1 are not equivalent. In your example, if we take four observables --- the outcome of Player $i$'s measurement with the device in position $j$, where $i$ and $j$ each have two possible values --- there is no joint probability distribution for the outcomes (i.e. they cannot be modeled as classical random variables satisfying Kolmogorov's axioms).

Edited to add: For a general formalism that specializes both to the case of classical game theory with mixed strategies (where your version of statement 1 holds) and to the quantum case, see here.