Paper Review: Bell’s Theorem: The Price of Locality

Quantum phenomena–and the theories built to account for them–can be strange. One of the most fundamental and (to some), spooky, things about quantum mechanics is action at a distance.

What exactly is action at a distance in quantum mechanics, and what are its implications? This is one of the central questions of Tim Maudlin‘s book Quantum Non-Locality & Relativity. The first chapter of his book is Bell’s Theorem: The Price of Locality, in which he explains some of the physical phenomena that for which any adequate physical theory must account. He also briefly discusses the associated Bell’s theorems and their consequences.

***The original book can be found here.***

I’ve mentioned Bell inequalities/Bell’s theorem(s) on this blog before, though I’ve never gone into them in detail. Maudlin does an excellent job in this chapter of explaining both some of the actual physical phenomena (such as photons emitted by calcium vapour) and the mathematical structure of the phenomena. For this post I will focus on his explanation of the latter since I think it captures the interesting part of the physics, though I recommend everyone read his book for more of the physical details.

The general set-up is as follows: we have two particles, say, photons, emitted from a common source. We can verify with experiments that some of the properties of the two photons are correlated. In particular, the correlations can have the following structure.

There are three different experiments we can run on each photon once they are far apart–we’ll call them A, B, and C (in Maudlin’s example these experiments are the orientations of polarization filters, but I will ignore such details from now on–be assured that the abstract correlations I describe have physical analogues). We can think of the experiments as questions we can ask the photons. Each photon can give either a “yes” or “no” answer when it is asked a question. For example, we might ask photon 1 the question A, and it might reply “yes”. We might ask photon 2 question C and it might reply “no”.

Each photon is asked and answers only one question once they are sufficiently far apart. There are certain empirical facts about how the photons answer the questions:

  1. If both photons are asked the same question (for example, A), they need to give the same answer (either “yes” and “yes”, or “no” and “no”).
  2. If the questions the photons is asked are next to each other (A and B, or B and C), then their answers need to agree 3/4 of the time.
  3. If one photon is asked A and the other C, then their answers need to agree 1/4 of the time.

These are the correlations we empirically observe when we run this experiment with photons again and again and again.

Now, so far on the surface this doesn’t seem odd. The photons are emitted from a common source, and so it seems reasonable that their properties could be correlated. However, things are more puzzling than they initially seem.

To draw out why these correlations are interesting (disturbing, even, or profound), he asks us to consider a game in which you are one of the photons. You have a partner playing the game with you. Here is how the game works: you both start out in the same room, and you are allowed to talk and devise a strategy before leaving the room. You walk to different rooms, in which you cannot communicate with each other. Once you arrive at the new rooms you will each be asked of the three questions A, B, or C, at random (you can’t predict which one you will be asked). For example, you might both be asked A, or maybe your friend is asked C and you are asked B. You play the game many times, for an arbitrary number of rounds. You have to reply in a way that mimics the statistics exhibited by the photons. That is, your answers must obey the following conditions:

  1. If both of you are asked the same question (for example, A), you need to give the same answer (either “yes” and “yes”, or “no” and “no”).
  2. If the questions you both are asked are next to each other (A and B, or B and C), then your answers need to agree 3/4 of the time.
  3. If one of you is asked A and the other is asked C, then your answers need to agree 1/4 of the time.

You and your friend can use any strategy you want, and you can change your strategy from one round to the next.

This is the set up of the game. Now, take a minute (with a pen and paper, if you like), and try to think of what strategy you and your friend would use. You may assume your friend is fully coöperative. And remember, you can vary your strategy from round to round.

Have you thought about it? What did you find? Try to summarize any insights before we continue our analysis together.

Welcome back. Let’s think through the possible strategies together.

One of the first things to which Maudlin calls our attention is that the only kind of strategies we can consider our deterministic strategies. What do I mean by this? I mean strategies that don’t involve things like coin-flipping our dice-rolling once you are outside of the first room. This is because if you want to employ such a strategy, you might as well flip the coins already in the first room–it makes no difference, you can just remember the result. You could use a flip of a coin to vary which strategy you use to employ round to round, but the strategy itself won’t involve flipping a coin. If this is a little fuzzy, I think this becomes even clearer with the next insight.

If we consider the first condition–that you and your friend’s answers on the same question always have to agree–then we realize that you and your friend have to choose in advance the same strategy. This is because you do not know in advance with questions either of you will get asked. Thus, since you might both get asked A, or B, or C, you will need to ensure that the strategies you both use are the same and deterministic. For example, if you both use the strategy [if A then “yes”, if B then “no”, if C then “yes] (which I will from now on write as <Y,N,Y>), then this will ensure that condition 1 is met. If you do not both use the same such strategy, there is a chance you might mis-coördinate on the same question, and you would lose the game.

As Maudlin points out, this means that you can only use 8 possible strategies, which we can break into four different categories:

(a) <Y,Y,Y> (b) <N,N,N> (ONE)
(b) <N,Y,Y> (c) <Y,N,N> (TWO)
(e) <Y,N,Y> (f) <N,Y,N> (THREE)
(g) <Y,Y,N> (h) <N,N,Y> (FOUR)

These are the only 8 deterministic strategies. For example, you and your partner could both choose (f), and then you would both answer “no” to A, “yes” to B, and “no” to C, then ensuring that you satisfy condition 1.

Maudlin calls to our attention that since we only care about whether or not your answer agrees with your friends, that we don’t really care whether you both answer “no” or “yes” on one question, as long as it is the same answer. Thus, for our purposes, (a) and (b), for example, are equivalent. Thus we can consider just the four categories of strategies, (ONE), (TWO), (THREE), and (FOUR).

This has greatly simplified things; we only have four viable strategies, where you and your partner always play the same strategy. Now, the only question is, how often should you use each strategy?

This matters because we need to not fulfill only condition 1, but also 2 and 3. Just to recall these conditions, I repeat them here:

  1. If both of you are asked the same question (for example, A), you need to give the same answer (either “yes” and “yes”, or “no” and “no”).
  2. If the questions you both are asked are next to each other (A and B, or B and C), then your answers need to agree 3/4 of the time.
  3. If one of you is asked A and the other is asked C, then your answers need to agree 1/4 of the time.

Good. In order to try to satisfy 2 and 4, you need to choose the proportion of the time you will use each of the four strategies. For example, you might choose to follow (ONE) half the time, (TWO) one quarter of the time, (THREE) one quarter of the time, and (FOUR) never. This particular combination would not let you satisfy 2 and 3, but perhaps another such proportioning would.

Let’s make this a little more precise. There are four numbers you need to choose:

  • The proportion you will play (ONE)–call this \alpha
  • The proportion you will play (TWO)–call this \beta
  • The proportion you will play (Three)–call this \gamma
  • The proportion you will play (FOUR)–call this \delta

Since these are proportions and these are our only options, we have the condition that \alpha+\beta+\gamma+\delta = 1.

We can figure out conditions that these proportions need to satisfy if we are to meet conditions 2 and 3. Condition 2 says if the questions you both are asked are next to each other (A and B, or B and C), then your answers need to agree 3/4 of the time. We can also rewrite this as saying that they need to disagree 1/4 of the time. All of the strategies except for (ONE) allow for the possibility of disagreement on adjacent questions. Thus, we see that

\gamma + \delta = 0.25
\gamma + \beta = 0.25

since in strategy (THREE) (which corresponds to \gamma) adjacent ones will always agree, in strategy (FOUR) (which corresponds to \delta) only A and B will disagree, and in strategy (TWO) (which corresponds to \beta) only A and B will disagree. So the first equality captures the proportion of rounds in which you and your friend disagree when asked B and C, and the second one when asked A and B.

Okay, so if the proportions satisfy those two conditions then we fulfill condition 2. Now all that is left in condition 3, which says that if you and your friend are asked A and C then you need to agree 3/4 of the time. The only strategies in which you disagree on A and C are (TWO) and (FOUR), and so we have

\beta + \delta = 0.75

as our final condition. So in total our assignment our numbers to \alpha, \beta, \gamma, and \delta must satisfy the following three conditions:

\gamma + \delta = 0.25
\gamma + \beta = 0.25
\beta + \delta = 0.75

If we do this then we have a proportion of strategies that can win the game, and thus replicate what photons do.

However, let us see what these conditions imply. If we add the first two, we get

\gamma + \delta + \gamma + \beta = 0.25 + 0.25 = 0.5

But we could also rearrange it like thus:

\gamma + \delta + \gamma + \beta = 2*\gamma + (\beta+\delta)

if we substitute in our last constraint for (\beta+\delta) then we get

\gamma + \delta + \gamma + \beta = 2*\gamma + 0.75

But then, if we put these two results together we get that

0.5 = 2*\gamma + 0.75

But this means that 2*\gamma = -0.25, and thus \gamma = -0.125. But these are proportions! \gamma cannot be 0! You can’t play strategy (THREE) negative 12.5% of the time. That doesn’t make any sense. Thus, there is no proportions that you can choose to satisfy the conditions. You cannot do what photons do.

There is no overall strategy that you and you friend can use to get the correct statistics. This holds even of the photons–all of the reasoning would be exactly the same. The situations share the same structure. So what gives?

In the game, you and your friend were not allowed to talk to each other after you left the first room. Physicists call this kind of constraint locality–very loosely speaking, only local things can affect each other. Because of this locality, you could not share information about which question either of you had been asked–if you could (say, with walkie-talkies) then you could easily coordinate your answers so that they satisfy the three constraints.

But how do particles do this? This is the puzzle. They don’t have walkie-talkies. Furthermore, it is MUCH worse than that. Maybe, you might think, they do send signals, we just can’t detect them. Sure, that would work. However, we have done experiments in which the particles exhibit these statistics, and the particles were far enough apart that if they did send signals then those signals would have had to have gone faster than the speed of light. As far as we can tell, this communication is instantaneous.

This is Wild with a capital W indeed. Special Relativity — one of our other best physical theories — says that the speed of light in a vacuum is absolute physical limit. Nothing can move faster than light. Yet, the particles seem to share information faster that this speed. This is troubling.

John Bell (whom I’ve discussed before) proved a more general version of this using the famous eponymous inequality, but the upshot is the same. Locality, it seems, is being violated.

This result is very subtle, and there is a large literature concerning it. I am very curious to find out more about how we understand Bell’s result, and what it means for quantum mechanics and special relativity. I look forward to reading the rest of Maudlin’s book; if the first chapter is any indication, it will be quite the adventure.

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