To continue with the series of “public service” posts, I will write the presentation of Bell’s theorem that I would like to have read when I was learning it. My reaction at the time was, I believe, similar to most students’: what the fuck am I reading? And my attempts to search the literature to understand what was going on only made my bewilderement worse, as the papers disagree about what are the assumptions in Bell’s theorem, what are the names of the assumptions, what is the conclusion we should take from Bell’s theorem, and even what Bell’s theorem even is! Given this widespread confusion, it is no wonder that so many crackpots obsess about it!

This is the first of a series of three posts about several versions of Bell’s theorem. I’m starting with what I believe is by consensus the simplest version: the one proved by Clauser, Horne, Shimony, and Holt in 1969, based on Bell’s original version from 1964.

The theorem is about explaining the statistics observed by two experimenters, Alice and Bob, that are making measurements on some physical system in a space-like separated way. The details of their experiment are not important for the theorem (of course, they are important for actually doing the experiment). What is important is that each experimenter has two possible settings, named 0 and 1, and for each setting the measurement has two possible outcomes, again named 0 and 1.

Of course it is not actually possible to have only two settings in a real experiment: usually the measurement depends on a continuous parameter, like the angle with which you set a wave plate, or the phase of the laser with which you hit an ion, and you are only able to set this continuous parameter with finite precision. But this is not a problem, as we only need to define in advance that “this angle corresponds to setting 0” and “this angle corresponds to setting 1”. If the angles are not a good approximation to the ideal settings you are just going to get bad statistics.

Analogously, it is also not actually possible to have only two outcomes for each measurement, most commonly because you lost a photon and no detector clicked, but also because you can have multiple detections, or you might be doing a measurement on a continuous variable, like position. Again, the important thing is that you define in advance which outcomes correspond to the 0 outcome, and which outcomes correspond to the 1 outcome. Indeed, this is exactly what was done in the recent loophole-free Bell tests: they defined the no-detection outcome to correspond to the outcome 1.

Having their settings and outcomes defined like this, our experimenters measure some conditional probabilities $p(ab|xy)$, where $a,b$ are Alice and Bob’s outcomes, and $x,y$ are their settings. Now they want to explain these correlations. How did they come about? Well, they obtained them by measuring some physical system $\lambda$ (that can be a quantum state, or something more exotic like a Bohmian corpuscle) that they did not have complete control over, so it is reasonable to write the probabilities as arising from an averaging over different values of $\lambda$. So they decompose the probabilities as

\[ p(ab|xy) = \sum_\lambda p(\lambda|xy)p(ab|xy\lambda) \]

Note that this is not an assumption, just a mathematical identity. If you are an experimental superhero and can really make your source emit the same quantum state in every single round of the experiment you just get a trivial decomposition with a single $\lambda$ (incidentally, by Caratheodory’s theorem one needs only 13 different $\lambda$s to write this decomposition, so the use of integrals over $\lambda$ in some proofs of Bell’s theorem is rather overkill).

The first assumption that we use in the proof is that the physical system $\lambda$ is not correlated with the settings $x$ and $y$, that is $p(\lambda|xy) = p(\lambda)$. I think this assumption is necessary to even do science, because if it were not possible to probe a physical system independently of its state, we couldn’t hope to be able to learn what its actual state is. It would be like trying to find a correlation between smoking and cancer when your sample of patients is chosen by a tobacco company. This assumption is variously called “freedom of choice”, “no superdeterminism”, or “no conspiracy”. I think “freedom of choice” is a really bad name, as in actual experiments nobody *chooses* the settings: instead they are determined by a quantum random number generator or by the bit string of “Doctor Who”. As for “no superdeterminism”, I think the name is rather confusing, as the assumption has nothing to do with determinism — it is possible to respect it in a deterministic theory, and it is possible to violate it in a indeterministic theory. Instead I’ll go with “no conspiracy”:

- No conspiracy: $p(\lambda|xy) = p(\lambda)$.

With this assumption the decomposition of the probabilities simplifies to

\[ p(ab|xy) = \sum_\lambda p(\lambda)p(ab|xy\lambda) \]

The second assumption that we’ll use is that the outcomes $a$ and $b$ are deterministic functions of the settings $x$ and $y$ and the physical system $\lambda$. This assumption is motivated by the age-old idea that the indeterminism we see in quantum mechanics is only a result of our ignorance about the physical system we are measuring, and that as soon as we have a complete specification of it — given by $\lambda$ — the probabilities would disappear from consideration and a deterministic theory would be recovered. This assumption is often called “realism”. I find this name incredibly stupid. Are the authors that use them really saying that they cannot conceive of an objective reality that is not deterministic? And that such a complex concept such as realism reduces to merely determinism? And furthermore they are blissfully ignoring the existece of collapse models, which are realistic but fundamentally indeterministic. As far as I know the name realism was coined by Bernard d’Espagnat in a Scientific American article from 1979, and since them it caught on. Maybe people liked it because Einstein, Podolsky and Rosen defended that a deterministic quantity is for sure real (but they did not claim that indeterministic quantities are not real), I don’t know. But I refuse to use it, I’ll go with the very straightforward and neutral name “determinism”.

- Determinism: $p(ab|xy\lambda) \in \{0,1\}$.

An immediate consequence of this assumption is that $p(ab|xy\lambda) = p(a|xy\lambda)p(b|xy\lambda)$ and therefore that the decomposition of $p(ab|xy)$ becomes

\[ p(ab|xy) = \sum_\lambda p(\lambda)p(a|xy\lambda)p(b|xy\lambda) \]

The last assumption we’ll need is that the probabilities that Alice sees do not depend on which setting Bob used for his measurement, i.e., that $p(a|xy\lambda) = p(a|x\lambda)$. The motivation for it is that since the measurements are made in a space-like separated way, a signal would have to travel from Bob’s lab to Alice’s faster than light in order to influence her result. Relativity does not like it, but does not outright forbid it either, if you are ok with having a preferred reference frame (I’m not). Even before the discovery of relativity Newton already found such action at a distance rather distasteful:

It is inconceivable that inanimate Matter should, without the Mediation of something else, which is not material, operate upon, and affect other matter without mutual Contact… That Gravity should be innate, inherent and essential to Matter, so that one body may act upon another at a distance thro’ a Vacuum, without the Mediation of any thing else, by and through which their Action and Force may be conveyed from one to another, is to me so great an Absurdity that I believe no Man who has in philosophical Matters a competent Faculty of thinking can ever fall into it.

Without using such eloquence, my own worry is that giving up on this would put into question how can we ever isolate a system in order to do measurements on it whose result does not depend on the state of the rest of universe.

This assumption was called in the literature “locality”, “no signalling”, and “no action at a distance”. My only beef with “locality” is that this word is overused, so nobody really knows what it means; “no signalling”, on the other hand is just bad, as the best example we have of a theory that violates this assumption — Bohmian mechanics — does not actually let us signal with it. I’ll go again for the more neutral word and stick with “no action at a distance”.

- No action at a distance: $p(a|xy\lambda) = p(a|x\lambda)$ and $p(b|xy\lambda) = p(b|y\lambda)$.

With this assumption we have the final decomposition of the conditional probabilities as

\[ p(ab|xy) = \sum_\lambda p(\lambda)p(a|x\lambda)p(b|y\lambda) \]

This is what we need to prove a Bell inequality. Consider the sum of probabilities

\begin{multline*}

p_\text{succ} = \frac14\Big(p(00|00) + p(11|00) + p(00|01) + p(11|01) \\ p(00|10) + p(11|10) + p(01|11) + p(10|11)\Big)

\end{multline*}

This can be interpreted as the probability of success in a game where Alice and Bob receive inputs $x$ and $y$ from a referee, and must return equal outputs if the inputs are 00, 01, or 10, and must return different outputs if the inputs are 11.

We want to prove an upper bound to $p_\text{succ}$ from the decomposition of the conditional probabilities derived above. First we rewrite it as

\[ p_\text{succ} = \sum_{abxy} M^{ab}_{xy} p(ab|xy) = \sum_{abxy} \sum_\lambda M^{ab}_{xy} p(\lambda)p(a|x\lambda)p(b|y\lambda) \]

where $M^{ab}_{xy} = \frac14\delta_{a\oplus b,xy}$ are the coefficients defined by the above sum of probabilities. Note now that

\[ p_\text{succ} \le \max_\lambda \sum_{abxy} M^{ab}_{xy} p(a|x\lambda)p(b|y\lambda) \]

as the convex combination over $\lambda$ can only reduce the value of $p_\text{succ}$. And since the functions $p(a|x\lambda)$ and $p(b|y\lambda)$ are assumed to be deterministic, there can only be a finite number of them (in fact 4 different functions for Alice and 4 for Bob), so we can do the maximization over $\lambda$ simply by trying all 16 possibilities. Doing that, we see that

\[p_\text{succ} \le \frac34\]

for theories that obey *no conspiracy*, *determinism*, and *no action at a distance*. This is the famous CHSH inequality.

On the other hand, according to quantum mechanics it is possible to obtain

\[p_\text{succ} = \frac{2 + \sqrt2}{4}\]

and a violation of the bound $3/4$ was observed experimentally, so at least one of the three assumptions behind the theorem must be false. Which one?