The Parable of the Muffins

Let’s try to make a profound statement about reality by thinking hard about baked goods.

I promise this is going somewhere.

A certain bakery has a special deal on muffins. They sell mystery boxes for those who like to live dangerously: mix-and-match sets of three muffins apiece. Each day, Alice, Bob and Charlie buy a mystery box together, and each day, Alice, Bob and Charlie take one muffin apiece back to their respective laboratories for analysis. They each have two testing devices — say, a device that can test whether a muffin is positive for dairy, and another device that can test whether it is positive for tree nuts. We’ll call these $X$ tests and $Y$ tests for short. Each day, Alice chooses either to do an $X$ test or a $Y$ test. Bob likewise chooses, independently of Alice, and so does Charlie. Importantly, each muffin can only be tested once. Maybe the test destroys the muffin, or maybe it takes so long to do one that they eat their muffins immediately afterward. Whatever the rationale, one test per muffin — that’s a rule of the parable.

We can write what they choose to do in a compact way. For example, if all three of them choose to do the $X$ test on their respective muffins, we’ll write $A_X B_X C_X$. If Bob and Charlie choose to do the $Y$ test but Alice goes instead with the $X$ test, we’ll write $A_X B_Y C_Y$. And so on. We can also write the results compactly, using $+1$ to stand for a positive result and $-1$ to stand for a negative one. (We could also record the outcomes with zeros and ones, or with trues and falses, greens and blues, etc. Using $+1$ and $-1$ is just a notation that will turn out to be helpful in a moment.) So, for example, if Alice chooses $Y$, Bob chooses $X$ and Charlie goes with $Y$, the results might be $(+1, -1, -1)$. Or they might be $(+1, +1, +1)$, or perhaps $(-1, +1, -1)$.

Over many days of muffin investigation, comparing their notes, they find a dependable pattern. Whenever two of them choose to do the $Y$ test, then the product of their results is always $+1$. The specific outcome varies randomly from day to day, but there’s never only one $-1$, and they never get all three results being $-1$. From this pattern, they can draw a couple conclusions. First, once two of them obtain their results, the result of the third is predictable. Let’s say their choices are $A_Y B_X C_Y$, as in the previous example, and both Alice and Bob get the result $+1$. Then we can predict that Charlie will get $+1$, because that’s the only way the product of the three numbers can be $+1$. Or, suppose their choices are $A_Y B_Y C_X$, and both Bob and Charlie get a $-1$. The two of them report their results and wait for Alice. Knowing Bob and Charlie’s results, we can predict that Alice will report a $+1$ outcome, because that’s the only way the product of the three outcomes is $+1$. A minus times a minus makes a plus, and so a third minus would spoil the plus.

If we had used a different notation for the outcomes, like “green” and “blue” instead of $+1$ and $-1$, then we could express this pattern by saying that whenever two of them choose to do the $Y$ test, an even number of the results will be blue.

Now, we deduce something else from the pattern. We can make a prediction about what happens under very different conditions. What about the days when all three choose to measure $X$?

We’ve used capital letters to write their choices. Let’s use lowercase letters to denote the properties of the muffins being measured. An $A_X$ measurement — that is, Alice doing the $X$ test — uncovers the value of $a_X$. Likewise, Bob doing the $Y$ test reveals the $b_Y$ muffin property, and so on. The pattern so far is that
$$a_X b_Y c_Y = +1,$$
and
$$a_Y b_X c_Y = +1,$$
and
$$a_Y b_Y c_X = +1.$$
Here comes the neat trick. We multiply all three of these facts together.
$$(a_X b_Y c_Y)(a_Y b_X c_Y)(a_Y b_Y c_X) = +1.$$
We rearrange the variables, putting like with like:
$$a_X b_X c_X a_Y^2 b_Y^2 c_Y^2 = +1.$$
On any given day, we don’t know the value of $a_Y$ or $b_Y$ or $c_Y$ before the test, and if Alice chooses $X$ instead of $Y$, then we’ll never learn $a_Y$ for that day, and likewise for Bob and Charlie. But the values do have to be waiting there, don’t they? Each has to be either $+1$ or $-1$, even if we never learn which: That’s just a fact determined at the bakery.

The square of $-1$ is the same as the square of $+1$, so on every day,
$$a_Y^2 = b_Y^2 = c_Y^2 = +1.$$
Therefore, we can drop all the $Y$ factors from our previous equation!
$$a_X b_X c_X = +1.$$
And now we have a prediction: on those days when all three independently choose to do the $X$ test, the product of their answers will be
$$A_X B_X C_X = +1.$$
This follows from the dependability of the pattern found on the two-$Y$-test days and the basic assumption that the tests are testing properties intrinsic to the muffins, properties waiting to be found. The tests don’t have to be for dairy and tree nuts — any properties will do. The objects being tested don’t have to be muffins, either. Anything that can come in sets of three is suitable. From the dependable pattern in the two-$Y$-test events, we can draw a conclusion about the triple-$X$-test events. The product of the $X$ results is going to be $+1$.

How general this result is! From the one pattern, we deduce the other, whether we find that first pattern in muffins, cookies, apples, sand grains…

Electrons? Photons?

Ay, there’s the rub.

It is possible to make triplets, not of baked goods but of subatomic particles, that fit the two-$Y$-test pattern. After doing the $Y$ test on the first two particles, for example, the result of an $X$ test is predictable, using the same rule we described above.

But the prediction about what happens when each particle gets the $X$ test does not hold.

The rules of quantum mechanics imply that there is a way an experimenter can prepare triplets of particles such that they should predict that the product of an $X$ result and two $Y$ results will always be $+1$, while the product of three $X$ results will always be $-1$.

By rights, this ought to be impossible. But the imagination of nature is more subtle than our own!

Doing the calculation to get the quantum-mechanical answer is not all that hard. It’s a spot of matrix algebra that’s doable by hand and doesn’t require knowing much more than what an “eigenvector” is. (The readings below have more details.) The much more difficult part is knowing what to make of it!

The predictions of quantum mechanics — which have been checked in the lab, directly and indirectly, and found to work superbly — clash with the basic and seemingly bulletproof calculation we have done here. How can that calculation fail to apply? Where is the gap that means one pattern doesn’t have to imply the other? If an argument whose fundamental premise is that measurements record a property of the thing being measured is inconsistent with quantum physics, our spectacularly successful guide to living in reality, then what does that say about reality?

Well, ask two rabbis and you’ll get three opinions. It’s not even clear what making progress on that kind of “what does it all mean?!” question even looks like. If people disagree on the meaning but all use the same math to make the same predictions, on what grounds can we even prefer one proposal about the meaning over another? Gut reaction? Preserving intuitions from classical physics? Remember, classical physics is what turned out to be wrong, so we had to invent quantum mechanics to fix it!

Perhaps one way to keep the question from going stale and vacuous is to find a new way of expressing quantum theory itself. After all, there’s no grand guarantee that the formulation of the theory which is good for finding the spectrum of atomic hydrogen (and all the other topics we traditionally assign to undergraduates each semester) is equally good for all questions. And one feature of the standard presentation is that the really weird stuff, such as what we’ve confronted today, tends to be buried several chapters in. When the books lay out “axioms” or “postulates” for quantum mechanics, blatant defiance of intuition about intrinsic properties isn’t among them. Instead, it is deduced as a consequence, deep into the algebra. But perhaps that is a historical accident, due to the way we got here and not to nature itself!