# What are the Implications of Bell's Theorem?

Consider the following game show: two friends Tom and Jerry (X and Y) are selected from an audience to compete for a grand prize, a brand new Ferrari.

The game description:

1. The two contestants are space-like separated.

2. Each contestant will be asked one of three questions {A, B, C} and the questions which are asked of the two contestants need not be the same. Each of these questions has two possible answers, which we will quantify as {+1, -1}.

3. Repeat 2 a large number of times.

4. When the two contestants are asked the same question, they must always give the same answer.

5. When the two contestants are asked different questions, they must agree $\frac 14$ of the time.

Now, how can Tom and Jerry win this game?

It's simple: They create a strategy, whereby they pre decide on a set of answers to give to the three questions {A, B, C}. This will guarantee that they always give the same answer when they are asked the same question. For example, they may decide on {+1, +1, -1} to {A, B, C}. Let us denote these answers as $v_i(\alpha) = \pm1$ for $\mathcal i$ = X, Y, $\alpha$ = A, B, C.

This will not allow Tom and Jerry to satisfy 4.

$\mathscr Theorem:$

There do not exist random variables $v_i(\alpha), \mathcal i$ = X, Y, $\alpha = A, B, C$ such that:

$$1. v_i(\alpha) = \pm1$$ $$2. v_X(\alpha) = v_Y(\alpha)\forall \alpha$$ $$3. Pr(v_X(\alpha) = v_Y(\beta)) = \frac 14 \forall \alpha, \beta, \alpha \neq \beta$$

$\mathscr Proof:$

Assume for contradiction that there do exist random variables $v_i(\alpha), \mathcal i$ = X, Y, $\alpha = A, B, C$ such that (1-3 hold).

Since $v_i(\alpha)$ can only take on the two values $\pm1$, we must have $Pr(v_X(A) = v_X(B)) + Pr(v_X(A) = v_X(C)) + Pr(v_X(B) = v_X(C)) \geq 1$

By condition 2, we then have $Pr(v_X(A) = v_Y(B)) + Pr(v_X(A) = v_Y(C)) + Pr(v_X(B) = v_Y(C)) \geq 1$

Now, by condition 3, we have $\frac 14 + \frac 14 + \frac 14 \geq 1$ a contradiction.

But, if you look at the predictions of quantum mechanics, it is possible to satisfy (1-3). Experiments have validated quantum mechanics, thus the correlations achieved cannot be due to a pre existing strategy. Then we must wonder how could Tom and Jerry always assure that property (2) holds, if they can not simply pre decide on a set of answers to give to the three questions {A, B, C}. It must be that when Tom and Jerry are asked a question, they communicate with each other what question is being asked, and agree to an answer, for if not, one would have $Pr(v_X(\alpha) = v_Y(\alpha)) = \frac 12$

$\mathscr Bell's\; Theorem\; Implication:$ Quantum mechanical correlations are not due to pre existing properties $\Rightarrow$ there must exist an exchange of information between entangled subsystems about what properties are being measured on them, and what values of these properties are to be taken on. Combining this with rule (1) of the game implies that this must take place faster than light, recent experiments overwhelmingly suggest instantaneously.

My question is why is this salient point so muddled in the literature? Bell's theorem is often stated as follows.

$\mathscr No\; theory\; of\; local\; hidden\; variables\; can\; produce\; the\; predictions\; of\; quantum\; mechanics$

That's missing the point. The hidden variables (pre existing properties) are a derived consequence of no exchanging of information about what properties are being measured and what values to take on. Bell simply showed that pre existing properties fail as an explanation. Consequently, we must have the implication above.

Credit to Tim Maudlin for the game description.

• You seem to be opining that Bell implies FTL communication. Most physicists take locality/causality extremely seriously, and would argue that Bell implies no hidden variables. Elevating a theory of functions to $\mathbb{R}$ to a theory of functions to PDFs is a lot less unsettling than extending a nice local theory to one in which causality doesn't hold. – user10851 May 27 '14 at 22:47
• No locally realistic theory can reproduce QM. To retain locality you must reject realism (or embrace superdeterminism). In other words, one rejects your implicit (and rather imprecise) assumption that measurement outcomes correspond to underlying "properties" of the universe. Instead, measurements are viewed simply as a class of physical processes which produce some clicks on some detector with some frequency that can be predicted by QM. As Chris White states, this is the view adopted by most physicists. It is a purely philosophical choice, obviously. – Mark Mitchison May 27 '14 at 23:09
• @user7348 You wrongly assume that I'm espousing the anti-realist view. I was just explaining it. Your post (and your comment) makes the implicit philosophical assumption that measurement outcomes are correlated with an underlying state space of "properties" or hidden variables. In the anti-realist viewpoint, one does not need to "account" for the perfect correlations: they are simply a consequence of quantum mechanics, which gives a prescription for prediction via non-commuting observables (explicitly not properties). If this viewpoint leaves you unsatisfied then join the club buddy... – Mark Mitchison May 28 '14 at 10:39
• @user7348 Sorry but I don't know how to explain it to you in more simple terms. Bell's theorem absolutely does not enforce anti-realism. Indeed, your concept of FTL information transfer is a realist (non-local) one, that is compatible with Bell's theorem. It might help you to try to explicitly construct a model for the "information transfer". You will see that you are implicitly ascribing "properties" or "beables". – Mark Mitchison May 28 '14 at 18:46
• For example, in the CHSH game (the game you described), we assume the existence of correlations between Tom & Jerry's input (A,B,C) and output (+1,-1). A problem of non-locality only arises if you assume that these correlations are a property belonging to Tom & Jerry. In the anti-realist viewpoint this correlation between input (i.e. measurement basis) and output (i.e. observable outcome) exists inside the observer's head only, it is not a property of the physical system itself. (The observer being, for example, some hapless individual watching the game show on their home television.) – Mark Mitchison May 28 '14 at 18:49

The conclusion that the violation of Bell's inequalities imply faster than light communication can only be reached in counterfactual reasoning. In our quantum universe counterfactuals ("would X have been measured, Y would have been observed") don't correspond to elements of reality. Accepting this notion removes any weird conclusions (such as faster-than-light communication) one would otherwise be forced to draw.

So an alternative to your conclusion runs as follows:

$\mathscr Bell's\; Theorem\; Implication:$ Quantum mechanical correlations can not be ascribed solely to pre-existing jointly measurable properties $\Rightarrow$ the deeply-ingrained classical notion that correlations between measurements must derive from a pre-existing correlation between variables which - although not measured - can be allocated a value, is faulty.

I am not sure about actual numbers, but I'd guess that a vast majority of quantum physicists subscribe to this conclusion.

A much more eloquent way of arguing against 'realism of counterfactuals' can be found in David Mermins book Boojums All The Way Through. A transcript of a key chapter can be found here: Quantum Baseball.

• I gave +1. But, I still don't think you get it. – user7348 Oct 31 '14 at 16:26
• @user7348 - ok, we won't convince each other. So let's shut up and calculate. – Johannes Nov 1 '14 at 3:53
• @ Johannes. :) Well taken. I'd however recommend you read Bell's papers and articles on the meaning of his theorem. For example: " It is important to note that to the limited degree to which determinism plays a role in the EPR argument, it is not assumed but inferred. What is held sacred is the principle of local causality' -- or no action at a distance'. Continued... – user7348 Nov 1 '14 at 3:58
• ...Continued Of course, mere correlation between distant events does not by itself imply action at a distance, but only correlation between the signals reaching the two places. These signals, in the idealized example of Bohm, must be sufficient to determine whether the particles go up or down. For any residual undeterminism could only spoil the perfect correlation." -- John Bell – user7348 Nov 1 '14 at 3:59

The problem you highlight is a result of taking at face value the idea that since you see a single outcome to each measurement every physical quantity relevant to understanding the evolution of a physical system must represent some outcome of this kind. Quantum mechanics violates this assumption. The physical quantities that describe the evolution of a quantum system are Hermitian operators that evolve entirely locally:

http://xxx.lanl.gov/abs/quant-ph/9906007

http://arxiv.org/abs/1109.6223

These operators describe physical reality as being a more complex structure than the universe as described by classical physics that, in some approximations, resembles multiple non-interacting versions of the world as described by classical physics.

For each measurement on a pair of electrons in the singlet state, say, there will be two versions of the measuring apparatus after the measurement. One of the versions of the measuring apparatus will record spin up, the other will record spin down. When a joint measurement is done on records of each result they then become correlated. Tom and Jerry don't have to agree on answers ahead of time because the correlation is only established after some set of observables carrying the relevant information have interacted with one another.

• Sorry, but I have no sympathy for the many worlds interpretation of quantum mechanics. – user7348 May 28 '14 at 15:45
• Do you have an argument against it? – alanf May 28 '14 at 15:51
• No, and I don't have an argument against Santa Claus. – user7348 May 28 '14 at 16:35
• You don't have an argument against Santa Claus? I do. His existence doesn't solve any problems that can't be solved by simpler ideas, i.e. - parents putting presents under Christmas trees. In addition, SC raises many problems like how he would manage to deliver presents to all the children in a single night. The MWI by contrast solves problems like the alleged non-locality of quantum physics. More generally it takes QM seriously as a description of how the world works making it easier to come up with testable models rather than having ad hoc exceptions. – alanf May 29 '14 at 9:46
• +1 hard to agree but consistant – user46925 Nov 30 '15 at 5:35

Here is a tongue-in-cheek implication. If special relativity holds true and spatial relations were to disappear then time would disappear with them. Conversely, if special relativity holds true and time were to disappear, all spatial relations would go with it. In entangled subsystems, time has disappeared. It follows that in entangled subsystems spatial relations have disappeared as well. The only physically realizable system of which we are aware without spatial relations is prior to the big bang. Therefore, using the principle of Occam's Razor, it seems our simplest explanation is that the big bang has not happened yet.

According to the Relativity Theory, time as we measure it, is not the same for every reference system. The relationship between speed and time dictates, that if something moves at the velocity c, then time will stop flowing forward for that system.

Interestingly enough, if something (anything) was able move faster that the velocity of light, then how do you think we would experience it? Not as a fluid motion from A to B, but rather as an instantaneous 'teleportation'.

Assuming time travel is impossible: If I could place a reflector on the moon and send an FTL signal there and back again. I would not receive the signal BEFORE I send it, as that is impossible, but if the signal moves faster than light, then it also moves faster than time itself, and so I would receive the signal faster than any clock could possibly measure.

In other words: If a signal exists that can move faster than lights, then it could allow two entangled systems to exchange data instantaneous.

Allowing for the possibility of FTL communication would really mess with our current understanding of physics, but it would resolve certain problems Quantum Mechanics is struggling with.

I remember seeing an print of the electron shells of an atom. No detector can trace an electron's movement, so instead it was a bunch of dots where the electrons had been encountered over many reading. I just wanted to reiterate: If FTL motion really was possible, then it would also account for our difficulties tracking the absolute position and momentum of a particle - how could you ever trace a particle that teleported around?

• Not much of this makes sense. – Kyle Kanos Nov 29 '15 at 23:34