The violation of Bell's inequality

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.

But I think it is interesting not only for laymen. It describes a practical application of a device which violates Bell's inequality. It allows to win in a game which, in a world where no faster than light information transfer exists, would be fair.

Rules of a simple game:
"Find cards with equal color"

I invite you to play with me the following game:

I put three cards of my choice on the table so that you cannot see their color.

The value doesn't matter. All what matters is the color — red or black.

You choose two of the three cards.

Your aim is to find two cards with equal color — or two red cards, or two black cards.

Once you have made your choice, I open the two cards of your choice.

If the two cards have different color — that means, one card is black, and the other card is red — you have lost this round.

In this case, you have to pay me 100 $.

Instead, if the two cards have equal color — that means, above red, or above black — I have lost this round.

In this case, I have to pay you 200 $.

Is this game fair for you?

You have three possibilitities to choose two out of three cards. If I choose at least one black as well as at least one red card, I win in two of these three cases. But I have no better possibility: I cannot choose cards in such a way that I win in more than two of three rounds. You chance to win is, in this case, 13. Given the payments, the game seems fair to you.

I have yet another, rather stupid, choice: All cards of the same color. In this case, you always win. Of course, I'm not that stupid. But, in theory, I have this possibility. We can prove, therefore, only an inequality:

Your chance to win is at least 13.

Is it possible to cheat?

Unfortunately, it is not extremely difficult to replace a card at the moment of opening it. You need some advice how to do it, and some time for exercise, that's all. Thus, it is possible for me to cheat, as long as I open the cards. Thus, you will probably insist on a modification of the rules, and want to open the cards yourself.

Unfortunately, I'm as well afraid that you want to cheat, and don't want to accept your proposal.

What can we do? It would be nice to have a variant of the game where it doesn't matter if somebody cheats when opening the cards. And this, indeed, possible.

A Modification To Avoid Cheating

We need a variant of the game so that, at the moment of opening one card, you don't know enough to cheat successfully. For this purpose, we need a more complicate version of the game. We have to distribute the game into two rooms, which are isolated from each other. And, instead of a single person on each side, we need now teams. I will be in the first room, my friend in the second room. As well, you have also a friend in the second room.

We have already six cards in this variant of the game, three in each of the two rooms. But the cards in above rooms have to be the same, and have to be ordered in the same way. At least this is what my team has to promise.

But don't be afraid, you don't have to believe me — you will have a possibility to verify my claim.

Of course, my team has the possibility to communicate before the start of the game. Thus, I can tell my friend about the cards I will present you in the first room. As well, he can tell me about his plans. But this is only before the start of the game.

At the beginning, your team (that means, you in one room, your friend in the other room) doesn't see the color of the cards. All you see are the three cards, with hidden color, in your room. Your friend sees the other three cards, also with hidden color, in the other room. The doors will be closed, so that communication is no longer possible between the two rooms.

Now, your team has to choose. But now you have to choose only one of the three cards in each of the two rooms. My team opens the cards — without knowing what happens in the other room, especially which card your team has chosen in the other room.

Only after above cards have been opened, we meet each other to compare the results.

For the outcome, we have to distinguish two different situations (each with two possible results):

Verification of the promise:
The same choice in above rooms

The first situation is that your team chooses the same card in above rooms. This possibility allows you to test our promise: The cards have to be identical.

If the two cards really appear to be identical, this round does not count. (I think, it would be fair if you pay us 1 $ for the inconvenience.)

But if, instead, the cards appear to be different, you have found that we have broken our promise.

This is considered to be a very serious violation of the rules of the game. So we have to pay you a serious fine of, say, 1,000,000 $. Moreover, we have to go to jail for some time.

The original game:
Different choices in above rooms

The other situation — you choose different cards in the two rooms — is similar to the first game.

If the two cards have different color — that means, one card is black, and the other card is red — you have lost this round.

In this case, you have to pay us 100 $.

Instead, if the two cards have equal color — that means, above red, or above black — we have lost this round.

In this case, we have to pay you 200 $..

Is this variant of the game fair?

Of course, there are possibilities for cheating. Let's assume, that changing the color of a card at the time of opening is not a problem.

Are there other possibilities? The question can be subdivided into two parts: First, does my team have a possibility to break our promise? If not, is the game fair if we follow our promise?

To answer the first question, let's consider my situation. I'm sitting in room 1, and you ask me to open the third card. I don't know what your friend asks in the other room. Thus, I have to expect that, with some sufficiently large probability, your friend also wants to see the third card in the other room. Given the high penalty for giving a different answer, it would be stupid for us if we do not establish, before, what to answer in this case, and, then, to give the same answer. My friend is in the same situation. Thus, our best choice is to give the prepared answers, thus, to hold the promise.

If the promise is not broken, we are in the same situation as in the first game — but without the possibility for me to change the color while opening it. The color I will show you is always the color we have prepared before the start of the game. Thus, our modification has, indeed, reached it's aim — to close this particular possibility for cheating.

Precautions against hypnosis

If you are aware of the danger of hypnosis, you can prepare yourself. Instead of making some arbitrary choice at that moment, you follow some well-defined algorithm, say:

Even if I would be able to manipulate one or two of the random number algorithms, it would not help me to win.

Precautions against a hidden information channel

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 tells about its choice at approximately the same time (Earth time or Mars time doesn't matter here), and my team has only a few seconds to open the card. Now, one of the most fundamental laws of nature — Einstein causality — forbids communication between the two rooms.

Now, it seems, all loopholes for cheating are closed. 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. The formal proof is also not very difficult.

If you doubt, I highly recommend to think yourself, if it is possible to cheat for my team. It's sometimes simpler to think about something yourself, than to follow an explanation, moreover, to follow an abstract mathematical proof. And this seems to be such a case. Your everyday common sense is sufficient here: There are no loopholes, no ways to cheat for my team.

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) * (-200)$)  +  ((3/4) * 100$) = 25 $

The device is based on quantum mechanics. The main idea is from Bell, that's why let's name it the Bell 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 +1 if -1
left card 120 red black
middle card 0 red black
right card -120 red black

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

Your friends question My friends measurement angle if -1 if +1
left card 120 red black
middle card 0 red black
right card -120 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 120,0,-120), 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 pay the fine.

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, because this angle is 120 degree, you find the same color and win only with probability 1/4, instead of 1/3.


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. An application in our game would require that they work without failure. Indeed, if the detector fails, I would have to say something like "Sorry, I have lost the cards of this round, so, we cannot count this round."

This is, of course, something you cannot accept. It would allow us to prepare the cards as 1: red, 2: black, 3: invalid. For this preparation, you can never win, because there are no cards with the same color.

Now, the detectors are not players, they don't know that the scientists who use them are testing Bell's inequalities, and, moreover, they are not interested to cheat. They simply fail to detect some photons. It, therefore, does not seem very likely that with better detectors the observed violations of Bell's inequality disappear. Thus, most scientists think that detector efficiency is only a technical problem. But, maybe, if you read this file, the loophole has been already closed by a new, better experiment. Anyway, a loophole is a loophole is a loophole. We cannot say — yet — that the existing experiments have observed violations of Bell's inequality.

Despite this, below we will not care about this loophole, and assume that a Bell device can be constructed and really works.


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 a quantum measurement. There are two possible directions of this information transfer: Or, my friend's measurement of his particle P2 influences immediately the state of my particle P1. Or, my measurement of my particle P1 influences immediately the state of his particle P2.

The consequence of above explanations is rigorous: Quantum theory is nonlocal, that means, it violates Einstein causality. Moreover, every theory which recovers, in some limit, the predictions of quantum theory for Bell's device, should be nonlocal too.

This is the explanation favoured by the author of these pages.

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. It's 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 observations. This explanation is the simplest one, it is in agreement with common sense.

Other explanations?

Are there explanations not based on some hidden information transfer in a preferred frame? If yes, how do they explain that Bell's device allows to win in this game?

The mainstream position (shut up and calculate)

To answer this question is not easy. On one hand, our explanation is not accepted by the mainstream of modern physics. Instead, following the mainstream, a preferred frame is not necessary to explain the violation of Bell's inequality.

But what is, then, the mainstream explanation for these violations? There is none. At least I have been unable to find such an explanation. For the average physicist, the violation of Bell's inequality is simply part of the general "quantum strangeness". The strangeness of quantum theory is something which has to be accepted as a fact of life, but should not be discussed: "Shut up and calculate". To consider these foundational problems in more detail is not recommended: "Younger physics were strongly discouraged from pursuing such questions. Those who persisted generally had difficult careers, and much of the careful thinking about quantum foundations was relegated to departments of philosophy" (G. Baccialuppi, A. Valentini, Quantum theory at the crossroads, Cambridge University Press, 2006, quant-ph/0609184)

In other words, the mainstream rejects the obvious common sense solution — a preferred frame — but otherwise simply ignores the question. But there is, of course, more to say: Last but not least, there are scientists considering foundational questions of quantum theory, and most of them don't accept the necessity of a preferred frame. What is their explanation of the violation of Bell's inequality?

Consistent histories

Unfortunately, this question cannot be answered. The problem is that there is no agreement about the meaning of "explanation". Consider, for example, the following statement about the "consistent histories" interpretation by Griffiths, one of its proponents: "The principle of unicity does not hold: there is not a unique exhaustive description of a physical system or a physical process. Instead, reality is such that it can be described in various alternative, incompatible ways, using descriptions which cannot be combined or compared." (Griffiths, R. B., Consistent Quantum Theory. Cambridge: Cambridge University Press, p. 365, 2002, as quoted by Wallace, arxiv:0712.0149, p.20, 2007)

This is not only in itself in clear and obvious conflict with common sense. It is also quite obvious that this position leads to a different notion of explanation: For common sense, the best explanation of a physical process is a unique exhaustive description — that means, it is what, according to the consistent histories interpretation, simply does not exist.

Does that mean that consistent histories does not have an explanation for the violation of Bell's inequality? In some sense, yes: There is nothing consistent histories can present that I would accept as a sufficient explanation — because the thing I would accept as an explanation does not exist according to consistent histories.

On the other hand, for a proponent of consistent histories, there is no open question. What serves as an explanation in the consistent histories interpretation — namely, a "consistent history" — can be given. You don't accept this as an explanation? That's your personal classical prejudice.

Many worlds

Instead, many worlds claims to be a realistic interpretation, but, nonetheless, claims to be Einstein-causal as well. Unfortunately, it is impossible to verify this. The reason is that the notion of probability is notoriously difficult in many worlds. It is quite easy to argue that there is no reasonable, consistent way to introduce probabilities into many worlds. Many worlders, of course, disagree, and they have some proposals how to introduce them.

But essentially the same arguments against a reasonable notion of probability in many worlds can be used as well to argue that the proof of Bell's inequality is impossible in many worlds. Of course — the proof of Bell's inequality uses probabilities, and if it is impossible to introduce them, the proof fails..

Now, if we cannot introduce probabilities in a reasonable way into many worlds, many worlds is dead — it would not be an interpretation of standard quantum theory, but something much weaker. If, on the other hand, it has standard Born probabilities, then the Bell inequalities will be violated as well. And it remains to see where, in this case, the proof of Bell's inequalities fails. The most probable candidate is Einstein-locality: Last but not least, the wave function ψ(q), which defines reality in many worlds, is a function on the configuration space Q, and, therefore, inherently non-local.

Subjective interpretations

One way to handle this problem is to consider almost everything in quantum theory as subjective. If, in particular, our probability assignments do not describe some objective properties of the world around us, but only our subjective expectations, it becomes impossible to prove Bell's inequalities: Subjective beliefs can, of course, violate them.

But this is, as well, not an explanation of how the device we use to cheat in the game works. It is, at best, an "explanation" of the type "explanation in the classical sense is not available". And, in particular, I do not see that it survives the test of my FTL phone argument: If probabilities are only subjective, can we prove that a working FTL phone violates Einstein causality? What we observe, as facts, are only some frequencies, which show some correlations between inputs and outputs of the phone.