Speaking of Bell tests, I’ve noticed that science writers often struggle to find a good metaphor that illustrates just what’s so weird about the correlation between entangled particles. Bell tests are complex, and because they squat in the middle of quantum weirdness, they’re hard to explain in any classical terms.
I thought I had the beginnings of a good metaphor, at least the classical part. But the quantum part is definitely a challenge. (All the more so because I’m still not entirely clear on the deep details of Bell’s theorem myself.)
Worse, I think my metaphor fails the ping-pong ball test.
I saw “the ping-pong ball test” referred to in a paper I read (one, not entirely coincidentally, about Bell’s theorem). The idea is that any analogy to quantum behavior in which the “particles” could be replaced by ping pong balls without damaging the analogy… is probably missing the point.
Because quantum “particles” aren’t anything like ping pong balls.
That’s the whole problem with the quantum world, really. It’s nothing like ping pong balls. Or any kind of balls. Or anything we know from experience.
It’s a bunch of mathematics that works really well for all sorts of predictions and work, but which drives everyone crazy because what the hell does it mean?
(Well, don’t ask me. I don’t know anymore than anyone does.)
Anyway, I’ve sought a good analogy for Bell’s Inequality tests for a long time. Most I’ve seen are either too detailed or too vague. Finding a Goldilocks version is a real challenge. What I’ve toyed with so far is this:
When I wrote about quantum spin I explicitly stayed away from “magic” boxes and discussed Stern-Gerlach devices, but a classical analogy seems to require them. My magic box contains either one glove or one shoe, depending on whether one opens the top (glove) or side (shoe).
Two boxes can be entangled such that, opening both from the top always results in matching right- and left-handed gloves. Likewise, opening both from the side always results in matching shoes. Note that which box contains which glove or shoe is random.
However, opening one box from the top and the other from the side results in a random glove or shoe with no correlation to each other.
The analogy stars Alex and Blair, who each receive one of an entangled pair of boxes. They are free to choose how to open their boxes but they cannot communicate their choice. Neither knows how the other opened their box.
If both open their box the same way there are four possible outcomes:
- Alex opens Top: Left Glove + Blair opens Top: Right Glove
- Alex opens Top: Right Glove + Blair opens Top: Left Glove
- Alex opens Side: Left Shoe + Blair opens Side: Right Shoe
- Alex opens Side: Right Shoe + Blair opens Side: Left Shoe
We say that when Alex and Blair open their boxes the same way, the results are always anti-correlated. The important aspect is that they are fully correlated. It just so happens that, in this case, the correlation is anti-wise.
If they open their box not the same way there are eight possible outcomes:
- Alex opens Top: Left Glove + Blair opens Side: Left Shoe
- Alex opens Top: Left Glove + Blair opens Side: Right Shoe
- Alex opens Top: Right Glove + Blair opens Side: Left Shoe
- Alex opens Top: Right Glove + Blair opens Side: Right Shoe
- Alex opens Side: Left Shoe + Blair opens Top: Left Glove
- Alex opens Side: Left Shoe + Blair opens Top: Right Glove
- Alex opens Side: Right Shoe + Blair opens Top: Left Glove
- Alex opens Side: Right Shoe + Blair opens Top: Right Glove
Here we say that when Alex and Blair don’t open their boxes the same way, the results are not correlated at all. In the first case there is 100% correlation, in this case there is 0% correlation.
This, in a nutshell, is a Bell’s test, but as described so far it can be fully explained classically. Clearly the boxes can be prepared in such a way to account for any result Alex and Blair get. We need to add a weird wrinkle to bring out the quantum nature of entangled pairs.
Unfortunately, this analogy doesn’t lend itself to what’s needed.
What’s needed is a magic dial on the box that sets the opening from 100% glove (and 0% shoe) to 100% shoe (and 0% glove) or any setting between (say 90% glove and 10% shoe).
There’s a 100% of getting something, so the two probabilities have to sum to 100%. But anything from 100/0 through 50/50 to 0/100 is allowed within that constraint.
If the dial is set, say, to 75% glove (and 25% shoe), then when Alex or Blair open their box, there is a 75% chance of getting a glove (and a 25% chance of a shoe).
When Alex and Blair open their boxes the constraints involving matching pairs applies. If both get shoes, those shoes must match (one left, one right). Likewise if both get gloves. Who gets a left or right, as always, is random, but matching pairs are always (anti-)correlated.
Classically, one might assume a linear probability distribution. But quantum probability follows a cosine-squared distribution. At the 100/0, 50/50, and 0/100, settings, classical and quantum behavior happens to match, but at other settings they don’t.
What the analysis of results seems to prove is that entangled states are not separable. The wavefunction describes a single system that acts in a singular manner no matter how far apart its components are.
But the glove and shoe box isn’t any more helpful in illustrating this than any other analogy I’ve seen. It really all does boil down to experimental results and some rather heavy math.
Stay gloved, my friends! Go forth and spread beauty and light.