Bell's Theorem

This is a popular introduction - without formulas - into Bell's inequality. IMO, the introduction given in the Physics FAQ does not show the seriousness of the problem.

This introduction is not only for laymen - it describes an application of the violation of Bell's inequality which is not widely known.

Rules of a Simple Game

I invite you to play with me the following game: There are three (red or black) cards on the table so that the color is not visible to you. Obviously at least one of the following three statements should be false:

Now, you have to guess which, and test this by opening two of these three cards. If you find a false one, you win two dollar, else you loose one dollar.

Is the game fair? If you simply use a "probability strategy" with probability 1/3 for each statement, your chance to find a false statement is at least one-third (at least because there may be also three false statements). Thus, we have:

((1/3) * 2.00$) + ((2/3) * (-1.00)$) = 0.00 $

That means, the game is fair.

A Modification To Avoid Cheating

A nice property of this game is that I have no possibility to manipulate the game before I know your question. One of the statements must be false. But, unfortunately, there may be some other possibilities for manipulation. One cheating strategy is to change the color of the other cards after you have choosen the first card. Indeed, having this possibility I can win every game.

To avoid this, let's consider a variant of the game so that this type of manipulation is not possible. The idea is that the cards have to be opened separately, without information which other card was choosen.

For this purpose, we consider two rooms without any possibility for communication. In every room is a member of my team and your team. My team claims that we have fixed the colors of three cards. You and your friend have the right to ask one of the following three questions:

"What is the color of the left card?"
"What is the color of the middle card?"
"What is the color of the right card?"

You can choose the same or a different question for above rooms. My team gives an answer "red" or "black" in each room.

Now, if your friend asks the same question as you, our answer should always be the same. If not, we have to pay you 100,000.00$ fine and go to jail for cheating. If they coinside, the game does not count. (You pay 0.03$ for our expenses.)

This rule gives you sufficient warranty that we really have fixed our anwers before we know your question and give consistent answers. Indeed, assume you ask the left card. I anwer "red". I have no information about the question of your friend. My friend has to return the same answer "red" if your friend asks him about the left card. Because I cannot send him the information about my answer, the answer "red" should have been fixed before. My friend does not know if you ask me about the left card or not. Thus, he should be afraid that you ask about the left card, and should answer "red" if your friend asks for the left card.

If your friend asks a different question, the rules are the same as before. You win now 2.09$ if left and middle (resp. middle and right) card have a different color or if left and right card have the same color. Else we win 1.00$. In the average you win now 0.03$.

If you use a strategy to ask by choice, independent of the choice of the other, you ask the same question (and loose 0.03$) in 1/3 of the cases, else you win 0.03$. That means, in the average you win 0.01$ in each game.

To cheat, we need some communication. How to avoid this? For this purpose, let's use Einstein causality. We use two rooms far away from each other, one on Earth, the other on the Mars. Your team asks the question at approximately the same time (Earth time or Mars time doesn't matter here), and my team has only one second to answer. Now, one of the most fundamental laws of nature - Einstein causality - forbids communication between the two rooms.

Now we have in principle no strategy to cheat. Really?

Bell's Inequality

The claim that the game we have described now is really fair for your team is one of the variants of Bell's Inequality. Thus, our previous argumentation that the game is fair is already a proof of Bell's inequality. Of course, it is not a formal proof, we have not derived the whole stuff from the axioms of probability and so on. But, I hope, the argumentation was clear enough to show you that the game is fair. If you doubt, I highly recommend to think yourself if it is possible to cheat for me, that means, to search for loopholes. believe we should be as formal as necessary to understand the problem. And in this informal sense we have already proven Bell's inequality.

The Violation of Bell's Inequality

And now comes the most interesting thing: There is a device which allows me to win the game. Using this device, I win the game with probability 3/4 instead of 2/3, that means in average I win in every game

((1/4) * (-2)$)  +  ((3/4) * 1$) = 0.25 $

The device is based on quantum mechanics. The main idea is from Einstein, Podolsky and Rosen, that's why let's name it the EPR device, and it really seems to work even in the relativistic domain.

We have to create a pair of particles P1 and P2 of spin 1/2 so that the sum of the spins S1 + S2 of the two particles is zero. I take particle P1, and my friend takes particle P2, with the aim to provide a measurement of the spin later, far away from each other.

In the moment you ask me about the card, with the justification of "looking into the cards", I make a measurement of the spin of the particle. The point is that the angle of the polarizer I use for the measurement depends on your question:

Your question   |    My measurement angle: | if +    | if -
----------------+--------------------------+---------+--------------
left card       |       60		   | red     | black
middle card     |        0		   | red     | black
right card      |      -60		   | red     | black

My friend is using the same strategy, with the obvious small modification of giving the reverse answer:

Your question   |    My measurement angle: | if -    | if +
----------------+--------------------------+---------+--------------
left card       |       60		   | red     | black
middle card     |        0		   | red     | black
right card      |      -60		   | red     | black

How it works? According to quantum mechanics, after your question to me and my measurement my particle P1 is in a state with spin +1/2 or -1/2 of direction x (x is 60,0,-60), and caused by the conservation law S1 + S2 = 0 the particle P2 of my friend is in the reverse state (spin -1/2 resp. +1/2 in the same direction).

If asked about the same card, my friend measures the same direction and obtains the "correct" answer - correct means the same as I have given, so that we don't have to go into jail.

If asked about another direction, quantum mechanics predict the probability of each result. The probability of getting the same answer for the measurement of the spin in another direction depends on the angle between the axes x and is cos2(x/2). Thus,

Thus, for each of the three statements, the probability that it is "verified" if it is tested is 3/4. Thus, independent on your strategy, my team wins with probability 3/4, and receives an average income of 0.25$ per game.

Nice?

Real Experiments

Thus, we have a prediction of quantum theory that such a device may be created and allows to win in our game. But quantum theory is only a theory, it may be wrong. Only experiment can decide if Bell's inequality may be violated in reality.

Unfortunately, there are some hard technical problems to solve. Not all of them have been solved. Thus, we do not have such a device yet which allows to cheat in our game in reality.

Nonetheless, there have been made real experiments starting with the famous experiment of A. Aspect and others. These experiments verify the interesting predictions of quantum theory in the relativistic domain. They really show a violation of Bell's inequality in this domain.

Unfortunately, the devices which are available now are not ideal. They often fail to measure the spin of the particle. Our thought experiment requires that they work without failure. Thus, with the best available particle detectors we cannot really "cheat" in our game.

This seems to be only a technical problem. But it opens a loophole for other explanations. If the failure of the detector is only by chance - the simples assumption - then Bell's inequality is violated in reality. But there may be some strange correlation between the failures of the detectors and the measurements - strange, because it violates quantum theory and Occam's razor. Thus, there is a loophole in the experiments, but it seems to be only a technical problem. May be, if you read this file, the loophole has been already closed by a new, better experiment.

Explanations

Once my team manages to win in this game, there should be some weak place in our argumentation or in the assumptions we have made.

There is one simple explanation: There is some hidden information transfer related with the measurement. If my friend makes his measurement of his particle P2, this influences immediately the state of my particle P1. This influence is what is called in quantum theory the "collapse of the wave function".

Such a hidden information transfer violates Einstein causality. Does that mean that we have to give up causality? No. We have a much simpler solution - simply to go back to Lorentz ether theory. Its absolute time also violates relativistic symmetry and is also hidden from observation. Lorentz ether theory makes the same predictions as special relativity, thus, is in agreement with special-relativistic experiments. I have generalized it to gravity so that it makes the same predictions as general relativity, thus, it is also in agreement with general-relativistic experiments. This explanation is the simplest one, it is in agreement with common sense.

There are other explanations. But these other explanations are in contradiction with common sense. Of course, common sense is not the holy of holies, but an explanation which is in agreement with common sense is simpler, we prefer it because of Occam's razor.