User:BenRG/Bell's theorem as a game show

Bell's inequality as a game show

The game is a twisted variation of The Newlywed Game. There are two cooperating contestants, Alice and Bob, playing against the house. They are allowed to meet to agree on a strategy, then taken to separate rooms where they can't communicate. Each player is then shown a randomly selected number, 1, 2, or 3 (the "question") and must answer either "yes" or "no". The goal of the players is to give the same answer when shown the same number and different answers when shown different numbers. The payoff matrix is:

	Same number	Different numbers
Same answer	0	−2
Different answers	−∞	1

That is, the players win $1 when they give different answers to different questions, lose $2 when they give the same answer to different questions, and lose everything they've won and then some if they give different answers to the same question.

The game is repeated many times. What's the best long-term strategy, assuming you have to play? In order to avoid the severe penalty, each player must know with certainty what answer the other player will give for each number. This means they must pre-agree on answers for each number. There are only 2³ = 8 possible answer sets, so this leaves only 8 possible strategies. Two of those, namely YYY and NNN, are bad ideas because they'll always lose when the numbers are unequal. The other six are all equivalent, because all of the rules of the game are symmetric under exchange of "yes" and "no" and under permutation of the numbers 1, 2, and 3. In all six of these strategies, the players win for 4 of the 6 possible combinations of unequal numbers. Since the numbers are chosen randomly, they win 4/6 = 2/3 of the time when the numbers are unequal, which means that they just break even overall.

Stop here and convince yourself that there's no way they can do better than this.

Actually, they can. In the strategy phase, Alice and Bob make a pair of electrons with opposing spins (a Bell pair) and each take one. Bob rotates his electron by 180° in a pre-agreed plane. When they see the number, they use it to choose among three different spin axes separated by 120° from each other in that plane, and give an answer depending on the spin measured on that axis (see the diagram and ignore the black boxes). If the numbers are the same, they both measure along the same axis and get the same answer. If the numbers are different, they measure along axes separated by 120° and the quantum prediction is that the answers will be different sin² (120°/2) = 3/4 of the time. This is better than 2/3, so they make a profit.

Discussion

Since the players can make a profit in the real world, there must be something wrong with the argument that they can't.

The assumption that sounds the most dubious is that the players don't communicate. Maybe their electron-measurement devices are communication devices. The question-and-answer phases for the two players can be spacelike separated, so that the communication would have to be faster than light. Bohmian mechanics is a version of quantum mechanics in which there's explicit faster-than-light communication through the "quantum potential". However, most physicists don't believe in it or anything like it. Part of the problem is that it doesn't extend well to quantum field theory (the basis of the ridiculously successful Standard Model), but there's a deeper problem that is implicitly understood by most physicists but not, I think, by most of the interested public: in Bohmian mechanics there's no reason for special relativity or quantum mechanics to appear to be true. Fundamentally, in this model, they're completely wrong. The fact that we can't build a faster-than-light radio and win 100% of the time instead of only 75% is a very special, essentially unique, property of the combination of quantum mechanics and special relativity. Almost every Bohmian world would allow us to build the radio, as well as computers that could efficiently solve NP-complete problems, which would revolutionize the world in other ways. Special relativity and quantum mechanics must be right because they're right, not by accident.

Another assumption is that the question is chosen at random. The questioner must be using a device of some sort to choose the question, and that device is subject to the same laws as the players' electrons, and maybe it somehow colludes with the electrons to give the players better-than-expected odds. It's mathematically possible to do this while still passing all statistical tests for randomness, but it has the same problem as the previous attempt—it seems to require some sort of conscious conspiracy by the universe to skew our experimental results. If the device is the questioner's brain, it also seems to conflict with free will.

A crazy-sounding idea is that the outcome of the measurements isn't actually determined until later. The answers have to be sent back to a common location to be compared, and they can't be sent faster than light, so there's time for the particles to agree on an outcome before the discrepancy can be detected. The trouble with this is that Alice and Bob are going to remember giving a particular answer, so at least one of them must exist in two copies, with different memories, until the ultimate determination of the outcome. This is more or less the "solution" offered by the relative-state interpretation (a.k.a. many-worlds interpretation) of quantum mechanics, in which the copies persist forever and lead independent lives.

A variation on that idea without the human clones is that the electrons communicate by a signal that travels along their past worldlines—that is, along the V-shaped curve in spacetime traced out by the generated and then separated Bell pair. This sits a bit more comfortably with special relativity since the signal travels slower than light (albeit back in time). It still seems to suffer from the why-can't-we-build-a-radio problem—in fact, we ought to be able to communicate with our own past, leading to grandfather paradoxes, which can't happen in Bohmian mechanics. And it's not clear why the particles would do this or what it even means—it makes no sense (millions of science fiction stories notwithstanding) to talk about "changing history" because change is already incorporated into the idea of history. If it can change then there would have to be two different kinds of time, the history kind and the changing-history kind, and you'd think we would have noticed that by now.

Another possibility is that Alice and Bob can't really win this game, i.e., the quantum prediction is wrong and we haven't noticed yet. But quantum mechanics works far too well to be so grossly wrong about such a simple experiment. Also, experiments resembling Alice and Bob's quantum strategy have been done and seem to be consistent with the quantum prediction. But it's hard to reliably generate Bell pairs and very hard to measure their spins along axes that randomly change more rapidly than the light travel time between the detectors, so it might not be impossible to hack up a model consistent with the experimental data and Bell's inequality. But if it's not impossible, it at least looks like special pleading.

Non-assumptions

It's worth listing some assumptions this argument doesn't make. There's nothing about "hidden variables" (nor about quantum mechanics, for that matter). There's no assumption that the objects available to Alice and Bob are in any definite state (whatever that means) at any given time, or that they behave predictably. You can give Alice and Bob a source of true randomness if you like, and of course you can give them free will; the only requirement is that the number shown to Bob is not available to Alice until after she's given her Final Answer, and vice versa. Some Bell-type arguments do make additional assumptions and are weaker as a result.