Realism is the principle that the world exists independent of any observer. If a tree falls in a forest with no one around to hear it, it still makes a sound. Locality is the principle that an occurrence can only directly affect its immediate surroundings. Its effects can eventually propagate far and wide, but this propagation has a speed limit, specifically the speed of light. Local realism is the logical conjunction of locality and realism.

Locality and realism are both very appealing and intuitive properties for a physical theory to have. The problem is that local realism has been empirically disproven, over and over again. This is pretty disconcerting, but the first rule of science is ideas are tested by experiment. No way around it. This post explains what the Bell test experiments are, and why the results of those experiments are incompatible with local realism. This explanation does not rely on an understanding of quantum mechanics, or actually on basically any physics at all. (It does use a bit of math though.)

## Setup

A device of some kind — let’s call it a particle generator — periodically emits a pair of particles that quickly zoom off in opposite directions. For the argument we want to make, it doesn’t matter what the particle generator is, or what exactly the particles are. Just two particles, zooming away from each other, that we have reason to believe may be related to each other in some way. The particles head towards two distant locations, say A and B. At each of A and B there is another device — let’s call these detectors. When the particle gets to the detector, it goes through it and interacts with it in some way, and the detector records something about how it was affected by the particle. This process is often called a measurement. Again, the details don’t matter for the argument we need to make. The detector records something that has something to do with the particle; that’s all we need to know. (Note that in discussions of quantum mechanics, the word “measure” is often used in a confusing way. In what follows, “measure” means simply that the particle passed through the detector and the detector recorded something.)

While the particles are en route, the detector at A is pseudorandomly set to one of two configurations, called a and a’. The one at B is simultaneously and independently pseudorandomly set to b or b’. These detector settings happen in time to affect their own measurement, but too late and too far away to affect the other detector’s measurement of the corresponding particle (assuming locality). There are two possibilities for the recorded measurement, called + and -.

In the experiment, the particle generator generates many pairs of particles, and we look at the relative frequencies of A and B getting different combinations of + and -, conditioned on the settings of a or a’, b or b’. In 1964, John Bell proved Bell’s Theorem, which states that if local realism is true, some inequalities have to hold between these relative frequencies, and that in some situations, quantum mechanics predicts that these inequalities will be violated. For the case against local realism, we don’t care about the predictions of quantum mechanics. We only care that under local realism, the inequalities must hold, and that empirically, they do not.

## The CHSH Inequality

### Statement

Many of the Bell test experiments demonstrate the violation of a variant of Bell’s original inequality called the CHSH inequality (named after the initials of its developers). The inequality says that, *assuming local realism is correct*, there is a relationship that must hold on the frequencies with which the two detectors record the same measurement as each other, both + or both -, between the four combinations of detector settings, (a, b), (a, b’), (a’, b), and (a’, b’). We define to be, looking only at the measurements where the detectors were in the states a and b, the probability that the measurements are the same, minus the probability of them being different. So if, whenever the detectors were set to a and b, they always recorded either both + or both -, . If (a, b) always leads to +- or -+, . CHSH says:

Defining as the probability of mismatch (+- or -+) given settings a and b, the probability of a match is , and so . (The difference between probabilities and proportions of results is discussed below in Statistics.) CHSH can therefore be rewritten as

,

which is equivalent to saying that

.

### Assumption of Local Realism

We’re going to prove that, assuming local realism, the CHSH inequality must hold. The assumptions of locality and realism come into play separately. In the proof, we consider a pair of particles heading towards the two detectors. Realism comes into play when we consider the probabilities of a detector recording + for a particle, under both possible settings of the detector. Realism asserts that the particle exists, and either measurement could theoretically be performed on it, so it makes sense to talk about both detector settings for the same particle, even though only one is actually applied. This point of view specifically is called counterfactual definiteness: experiments that were not performed can still be reasoned about.

The assumption of locality comes into play when we look at the simultaneous measurement of one pair of particles by the two detectors. Because of the timing and distance between the detectors, locality says that there is no way one measurement can affect the other one. This means that the simultaneous recordings of + or – by the two detectors must be independent of each other. Below, we use the fact that when two outcomes are independent, the probability of both happening is the product of the probabilities of the two outcomes.

### Proof

We first prove that the CHSH inequality holds for a specific pair of particles, and then later for the full space of particle pairs that might be emitted by the particle generator. Imagine a pair of particles moving towards the two detectors. For the particle going towards A, there must be some probability of + if the detector is set to a, and some probability of + if the detector is set to a’. We call these probabilities p and p’. And let the probabilities of b and b’ recording + for the particle going towards B be q and q’, respectively.

Even though the two particles may be related to each other in some way, once they are spatially separated from each other, measurements on the two particles must be independent. is the probability of +-, , plus the probability of -+, . Similarly, , and so on. This means that we can rewrite as:

It turns out that this statement is true whenever . This is a purely mathematical statement, and it is proven in the appendix. Next, we consider that the particle generator may emit different pairs of particles at different times. The value of for the total distribution of particle pairs is the expected value, over the space of particle pairs, of for each pair of particles. Since CHSH is true for each pair of particles, the inequality must also hold for the total distribution of particle pairs. This follows from the fact that if X is always greater than or equal to Y, the expected value of X must be greater than or equal to the expected value of Y.

We have proven that . As discussed above, this is equivalent to the original CHSH inequality: .

### Symmetrical Results

Since the detector settings were named arbitrarily, there was nothing special about (a’, b’) that enabled us to prove that . So, it’s also true that , or either of the other combinations. Also, + and – were named arbitrarily. Interchanging them at one detector only has the effect of interchanging matches with mismatches, which negates all the E’s. This means we also know that , or equivalently, .

These symmetrically equivalent versions are often combined. For example, the paper Rowe et al. uses the statement . The implications of this, for different signs inside the absolute values, are all symmetrically equivalent versions of the CHSH inequality proved above.

## Empirical Results

There have been many experiments performed of the form described above. These experiments have consistently found that the inequality being tested (such as CHSH) is empirically violated, demonstrating that local realism is impossible. The rest of this section discusses a couple of caveats about these experiments.

### Statistics

Above, we defined in terms of the probabilities of the different measurement results. Although CHSH is sometimes stated in terms of the counts of the different measurement results, all that can be proven is the inequality on the probabilities. The corresponding inequality on the counts is only guaranteed to be true on average, in the long-run.

When the experiments are run and the counts violate the CHSH inequality, it’s always still theoretically possible that the underlying probabilities satisfy the inequality, and the difference between the two was simply caused by luck. However, generally, sufficiently many particle pairs are measured that the probability of that happening is extremely unlikely. That low probability is reported in the papers as the p-value, which in some cases is as low as . When something is that unlikely, we consider it to definitely not have happened, without reservation. If you’re still looking for a way out, I suggest looking for more loopholes.

### Loopholes

What if not all of the pairs of particles were detected, and the ones that were detected were biased in a way that effected the relative probabilities of the measurement outcomes? What if one detector was set early enough to affect the other measurement before it happened? What if information from a common source affected the particle generation and the detector settings? These what-ifs, and a few others, are the Bell test loopholes. I won’t go into detail about all the possible loopholes here, but recent experiments have increasingly claimed to close all the loopholes.

I think it’s hard to say with absolute certainty that all possible loopholes are closed, but it seems to me that at this point the evidence is overwhelming that CHSH has been repeatedly violated in a way that is inconsistent with local realism.

## Conclusion

If local realism is wrong, what are we left with? One answer is to look to the interpretations of quantum mechanics for guidance. The most common one, the Copenhagen interpretation, sacrifices both locality and realism. The violation of locality occurs in the assumption of wavefunction collapse, which happens instantaneously over large distances. As for realism, the Copenhagen interpretation treats “measurement” as a first-class concept, but asserts that the property being measured didn’t have a value until the measurement happened. This is certainly strange, but perhaps not as overtly anti-realist as, say, the “consciousness causes collapse” interpretation.

In any case, the rejection of local realism doesn’t require one to accept the Copenhagen interpretation of quantum mechanics. There are many other interpretations, although none of them have gained widespread acceptance. Nevertheless, there is a lot of philosophical space between the observer-dependent Copenhagen interpretation and the sense of local realism that is ruled out by the argument above. Perhaps a middle ground can be found: an interpretation that feels philosophically coherent (no pun intended) without adhering strictly to the requirements of local realism.

## Appendix

We’ll prove that for all ,

Note that all six terms on the left are always nonnegative. We look at cases. Case 1: and . Then LHS terms and are RHS terms and respectively. Case 2: and . Then LHS terms and are and respectively.

Case 3 is that and . For this we use three of the LHS terms: , , and . First,

.

We also know that and . Adding these three inequalities gives:

Case 4 is and . The proof is the same as in case 3, but with all instances of p and q interchanged (and p’ and q’).