Some unrelated news: I have recently appeared as a guest/expert in a video about NP-completeness on the popular Youtube channel Up and Atom. Take a look!
Context
Quantum mechanics has a reputation for being strange and counterintuitive. As a consequence, people have always wondered whether it is possible to replace it with a theory that would be more akin to the classical theories of physics but without sacrificing any of quantum mechanics’ predictions.
In 1964 John Bell published a paper in an obscure physics journal. He described a quantum mechanical experiment whose results, if observed, couldn’t possibly be accounted for by any classical theory of physics of the kind people were hoping to construct. At the heart of his argument was an inequality that has since come to be known as the Bell inequality. Bell argued that if the predictions of a theory obey this inequality, then they might be explained by a classical theory but if they don’t then they can’t be. The predictions of quantum mechanics in that experiment violate this inequality.
Almost sixty years later, the 2022 Nobel Prize in Physics was awarded to experimentalists who actually performed a version of the experiment and found that the predictions of quantum mechanics are corroborated. In the meantime Bell inequality has become quite well known. Almost everybody who becomes interested in quantum mechanics will eventually hear about it and it has been described in countless books, technical papers and YouTube videos.
So what is the experiment and why can’t its results be produced by classical physical systems? Frustratingly, even many people who encounter this result in technical books end up seeing it as little more than a lot of mathematics that “doesn’t add up” the way it was supposed to. Many don’t ever get any intuitive understanding of the result.
However, such an understanding is possible - and the secret to it is that Bell’s argument wasn’t really about physics at all. It’s about correlations and restrictions that exist on how multiple different things might be correlated with each other.
The same restriction that Bell pointed to for classical physical systems in fact applies to many other situations. And in many of those situations it is intuitive and arguably already a part of common sense!
We will illustrate his argument in one such context. We will then circle back to the experiment Bell proposed.
So forget all about quantum mechanics for a moment and consider a far removed situation. And that situation concerns political opinions.
A simple story – Bell inequality in politics
Imagine two people: a Democrat Alice and a Republican Bob. Alice and Bob passionately disagree on a number of controversial issues. When they talk, they notice they rarely agree on much of anything. When one of them thinks a particular policy should be adopted the other does not. However, Alice and Bob also share a friend Eve. The strange thing about Eve is that both Alice and Bob think she agrees with them on a lot of issues. When they each talk to her they experience her as being mostly in agreement with them.
Is this actually possible? Is it possible for Alice, Bob and Eve to have the right mix of opinions for this to all make sense or is Eve just being two-faced? Is it possible to be someone who agrees a lot with two people who otherwise disagree on almost everything?
Let’s consider a quantitative version of this question. Consider a situation where both Alice and Bob fill a long questionnaire of yes/no questions about various controversial political topics. Afterwards they compare their answers. They discover they gave different answers on 75% of the questions. One of them answered “yes” and the other answered “no” (and they gave identical answers on 25% of the questions). So far nothing surprising, Republicans and Democrats are known to disagree on a lot of issues.
But now Eve fills the same questionnaire and compares her answers with both Alice and Bob. She claims that she gave the same answer as Alice on 75% of the questions. And she also claims she gave the same answer as Bob on 75% of the questions!
Is this actually possible? Is it possible to produce three questionnaires which when compared produce these statistics?
It is not. And here is one way to see it. Consider every question where Alice and Bob gave different answers. One of them picked “yes” and the other picked “no”. Since Eve also has to choose one of these answers this means she will have to disagree on this question with either Alice or Bob. Hence if Eve keeps track of the number of questions she disagrees with Alice and the number of questions she disagrees on with Bob then the sum of these two numbers has to be at least the number of questions Alice and Bob disagree on.
But in our example, we have Alice and Bob disagreeing on 75% of the questions while the combined percentage of questions Eve disagrees with both Alice and Bob on is 50%. So this can’t work. There is, if you will, no “hidden opinions” theory that could account for the statistics she reported.
In general, if you denote by p(A,B) as the fraction of the time two people A and B disagree then we have the following constraint p(A,B) ≤ p(A,C) + p(C,B). This simple constraint is the Bell inequality. Or rather it is equivalent to all versions of it you will find in a typical textbook. (If you’re familiar with books the more common CHSH inequality can be obtained by applying this simple inequality twice.)
And indeed, this result makes intuitive sense. It is difficult to be or agree a lot with both a Democrat and a Republican simultaneously.
In a more abstract sense, we are saying that there is something approximately transitive about correlation. If A is highly correlated to B and B highly correlated to C then there are limits to how anti-correlated A and C can be.
So, this is what the Bell inequality looks like in the context of political opinions. It is the idea that only certain statistics about pairwise agreement between people are compatible with those people actually having definite opinions on the underlying issues. Or more concretely, if the reported statistics obey the inequality given above then we can find some collection of questionnaires that produce them and otherwise we cannot.
We will now see how we can find something like Alice, Bob and Eve in the context of quantum theory and why this makes it difficult for any classical theory to reproduce its predictions.
Quantum story
Quantum mechanics was developed to describe the behavior of very small particles like photons, electrons, protons and atoms. The experiments where violations of Bell inequality were observed used photons, but we don’t need to know much about them to be able to appreciate how Bell inequality violations arise.
The important thing to understand is that if you’re given a photon, you can perform a single measurement on it. We will limit ourselves to so called polarization measurements but again the physical details are not very important. The important facts about polarization measurements are the following.
There are a large number of different polarization measurements we can make, and each is associated with an angle between 0 and 90 degrees. When we pick an angle and perform the measurement associated with it we get one of two outcomes: +1 or -1. Quantum theory gives us only the probabilities of getting one outcome or the other. For example, for a given angle it might predict we get +1 with 80% probability and -1 with 20% probability. And the final important fact is that once we perform one of these measurements the photon is destroyed.
So given a photon you get to pick an angle (and perform the measurement associated with it), get back +1 or -1 and then the particle is gone. That’s all we have to work with.
Quantum theory allows two photons to be prepared in a so-called entangled state and it is there that we get to see Bell inequality violations. What does it mean for two particles to be entangled?
Everything we said about photons above is still true. We get to perform a single measurement on each photon and get back +1 or -1. However now, if we choose the same angle and perform the same measurement on both photons, we get back the same outcomes. If we pick an angle for both particles, say 20 degrees, perform the measurements and we get +1 for the first one, we will get +1 for the second one. And similarly for -1. This is so despite the fact that quantum mechanics might predict a 40% chance to get +1 and 60% chance to get -1 before we make the first measurement. And all this remains true even when particles are very far apart.
This fact about distant entangled particles led to the idea that any classical theory that would reproduce the predictions of quantum mechanics would have to have all measurement outcomes predetermined. Quantum theory gives only probabilities of measurement outcomes, but to explain these identical outcomes a classical theory would have to have measurement outcomes completely determined with the source of randomness being that when we prepare and entangled pair of photons, we are really preparing them in one of many possible states with fixed outcomes.
What Bell examined though was the question: what happens when we pick different angles for both particles?
Then it turns out that quantum theory predicts that the outcomes you get will still be correlated. We know that if we pick the same angles, we get the same answers 100% of the time but if we pick two angles that are very close together, we still get the same answer a very large fraction of the time. How often the measurements will agree is determined by the differences in their angles.
It now turns out we can find Alice, Bob and Eve if we pick the angles right. (For those familiar with the physics the angles are 0, 30 and 60 degrees.)
We can, if you will, pick a Republican angle (0 degrees), a Democrat angle (60 degrees) and a Two-Faced angle (30 degrees). Whenever we measure the Republican angle on one particle and a Democrat angle on the other, we get a different answer 75% of the time. But when we measure the Two-Faced angle on one of the particles and either the Republican or the Democrat angle on the other, we get the same answer 75% of the time. Exactly like in our story from the previous section.
Now let’s go back to the idea that measurement outcomes would be predetermined by some hidden states as the classical theories would require.
We can see that these hidden states correspond to the questions on the questionnaire, the chosen measurement angles correspond to Alice, Bob and Eve and the outcomes we obtain correspond to the answers Alice, Bob and Eve would give on that particular question. (With the exception that we only ever get to ask the question to two of the three people.)
But we saw before that there is no questionnaire and a set of answers that would produce these statistics before. And just like that led to the conclusion that there are no “hidden opinions” theory for those statistics in the case of political opinions, it leads to the conclusion that there are no so called local hidden variable theories that would reproduce the predictions of quantum mechanics.
It is not possible to be a Democrat and a Republican simultaneously but there are measurements you can perform on quantum particles that will agree with other measurements that otherwise disagree as much as Democrats and Republicans do.
This is what Bell showed and it is also what happens in experiments.
What does it all mean?
This is the idea behind Bell inequality and what its violations look like. But how are we to make sense of this? That remains a subject of surprisingly heated debate. Bell’s experiment is also at the heart of certain proposed technologies based on quantum mechanics such as certifiably random bits.
But perhaps what this phenomenon really shows is that there is still much to be understood about this mysterious world and that unfortunately many fascinating mysteries often unnecessarily lie buried under five pages of complicated algebra.
Thanks to Dorian Bandy, David Deutsch, Alycia Lee, Jake Orthwein and Tomaž Leonardis for reading drafts and suggesting improvements.