[I just noticed this sitting in the “Drafts” section of my blog. (You can tell it’s old, because I mention something that I’m “going to be teaching in the fall.”) I don’t know why I didn’t post it at the time. Did I notice something wrong with the physics, and put it aside until I fixed it? I don’t think so — it looks right to me.

This is a subject lots of people have written about, and what I say here is just a summary of some standard stuff, but it’s an incredibly cool subject, so if you don’t already know all this, check it out.]

Einstein’s big problem with quantum physics was that it involved “spooky action at a distance.” His most famous quote on quantum physics is about randomness — “God does not play dice” — but in fact he seems to have been much more bothered by nonlocality than randomness.

The sort of nonlocality that bothered Einstein, and that bothers many other people, is kind of hard to explain. In particular, one key ingredient is a thing called Bell’s Inequality, which is somewhat technical. There are some very good explanations out there, especially a couple by N.D. Mermin, e.g., one in the *American Journal of Physics *(paywall) and one that originally appeared in *Physics Today*. But recently I came across a nice way of thinking about it in an unexpected place: a puzzle blog. Even though this way of formulating the result seems to be well known in certain circles (and is apparently close to what Bell wrote in his original paper), I’d never encountered it before.

If you want to understand this stuff, you can read any or all of the above. But I’m going to try to summarize the main idea here too, mostly because I’m going to be teaching this topic in the fall, and I can use the practice. So here goes.

Let me start with a quote from the puzzle blog:

Suppose three friends A, B and C take a test with 100 yes-no questions. If you compare the answers given by A and B, 98 of the 100 are the same. Likewise, if you compare the answers given by B and C, again 98 of the 100 are the same. What is the minimum number of questions that A and C have answered in the same way?

The answer, of course, is 96. Now let’s connect this with quantum physics.

In quantum physics, particles like electrons have a property called “spin.” The main thing you need to know about electron spin is that when you measure it you always get one of two values, which are usually called “spin-up” and “spin-down.”

Spin-up and spin-down have to be specified relative to some chosen axis: you can pick any direction you want, and measure the spin of an electron along an axis oriented in that direction, and you’ll get either spin-up or spin-down. (If the axis is horizontal, then the terminology is kind of stupid: it’d make much more sense to call it “spin-left” and “spin-right”. But we still say “up” and “down” even in this case.)

It’s relatively easy (or so they tell me) to produce pairs of particles that are “entangled,” meaning that their spins are related to each other. In particular, it’s possible to produce a pair of particles that have the following properties:

- There’s no way to predict in advance what the result will be of any spin measurement on either of the particles.
- If you measure the spin of one particle, the spin of the other particle, measured about the same axis, is guaranteed to be the same.

(Actually, it’s technically easier to create pairs where the spins are guaranteed to be opposite, but it’s possible to flip one around after the fact, and anyway it’s easier to explain this way.)

So far, there’s nothing “spooky” going on. In fact, these electrons are just like my socks, which — take my word for it — are not at all spooky. You don’t know what color my socks are, but knowing what a fastidious dresser I am, you can bet that they match. So there’s no way to predict the result of one sock-color measurement, but once you’ve measured one you can predict the other with certainty.

(If you actually tried to do this experiment, you’d have a problem: I’m not wearing socks at the moment. The sock analogy comes from an essay by John Bell, by the way, although I think in his case he imagined someone whose socks always failed to match.)

Einstein believed that the electrons in this system really were just like my socks: my socks have a definite color, even if you don’t know it, and the electrons (according to Einstein) have a definite spin-value even before we’ve measured it. If we repeat this experiment many times, each pair of electrons will leave the source with a plan in mind about whether to say “up” or “down” when the spin is measured. That plan may be chosen randomly, but the two particles have the same plan, so of course they give the same answer. The name for this point of view is “local realism,” by the way.

Bell’s inequality says that local realism is impossible. To see why, we have to bring in the fact that the spins can be measured about different axes — that is, that we can rotate our measuring apparatus before the electron hits it. Rule 2 above says what happens when spins of both particles are measured with respect to the same axis: the results agree 100% of the time. It doesn’t matter what axis we choose, as long as it’s the same for both measurements. But what if we rotate one measurement apparatus relative to the other? There’s another rule for that:

3. If the two measurement axes are rotated with respect to each other by an angle x, then the results of the measurements will agree cos^{2}(x/2) of the time.

In particular, say one axis is tilted 16.3 degrees away from the other. Then that number works out to 0.98. That is, the results will agree 98% of the time.

Now suppose we have a machine that can mass-produce pairs of entangled particles. We can imagine doing an experiment in which we measure the spin of either particle about a vertical axis. (It doesn’t matter which particle: we know they’ll come out the same.) The series of outcomes of those measurements (up, down, down, up, up, down, down, down, …) is like student B’s test.

Now suppose that instead we measure the first particle’s spin about an axis that’s tilted 16.3 degrees to the left. The resulting sequence is like student A’s test: it’ll agree with the first list 98% of the time.

Finally, suppose that instead we measure the second particle’s spin about an axis that’s tilted 16.3 degrees to the right. The results here are (you won’t be surprised to hear) like student C’s test. Once again, there’s a 98% match with the first list.

Here’s the thing: there’s nothing stopping us from measuring both the second and third lists simultaneously (particle 1 with an axis tilted to the left, and particle 2 with an axis tilted to the right). If we do, we find that the results match only 92% of the time. (That’s rule 3, but with x=32.6 degrees, which is how far apart these two axes are.) That should be impossible: if both sets of tilted-axis results are 98% correlated with the (hypothetical) vertical-axis results, then they must be at least 96% correlated with each other.

In summary, there is no way to consistently assign values to all three sets of possible measurements (vertical axis, tilted to the left, tilted to the right) that agrees with the probabilities found in quantum mechanics. In effect, that means that the particle doesn’t “decide” whether it’s going to tell you it’s spin-up or spin-down until you decide which axis you’re going to measure. And yet the pairs of particles manage to decide the same way, even if they’re very far apart.

By the way, Bell’s inequality was originally a theoretical result. Rules 1-3 were known to be the predictions made by quantum mechanics, but they hadn’t been tested experimentally at the time. So originally all you could conclude was that either quantum physics was wrong or local realism was wrong. But later, the experiment was done, and the quantum physics predictions were confirmed to be correct. So local realism is wrong, and electrons are not like my socks.