Today, I learned something new: matryoshka nesting dolls aren’t quantum mechanical objects. Surprisingly enough, this was not something I should have been certain of. After reading a paper about how quantum mechanics within quantum mechanics can lead to contradictions, I will never look at matryoshka dolls the same way again.

The paper is not just theoretical in nature but seems to overcomplicate what should be a fairly straightforward argument. That probably means I’ve misunderstood it pretty thoroughly. So, let the mistakes begin.

The innermost doll is solid

Imagine that I have a single particle that has a quantum mechanical property called spin. We don’t care what spin is, just that it has an orientation in space. I can measure the orientation of the spin. But quantum mechanics doesn’t let me measure it in a general way; I can't ask "hey, Frank-the-quantum-particle, which direction is your spin pointing?" Instead, I can only ask "hey, Frank, is your spin pointing up or down?" To which he will always reply with an “up” or a “down.”

This is true even if Frank’s spin was sideways (and has no up-ness or down-ness at all). In this case, he will pick up or down at random. We cannot predict the result of the first measurement at all, but all future measurements on Frank will give the same result: once up, always up. On the other hand, if Frank’s spin was down before the measurement, the answer will always be down.

Let’s extend the experiment. My spin-generating system and I are placed in a sealed box. Outside the box, Bob will try to predict the outcome of the measurement and then measure the entire box to obtain the outcome. No information about my measurements can leak out to anyone, but information about the spin state as it was before my measurement is available.

If the prepared spin state is up or down, then Bob and I always agree on the outcome of the experiment. But if the spin is sideways, we don’t. Bob argues that the spin is in a superposition state of up and down, while I, having already made the measurement, say that it is down.

This isn’t much of a paradox: Bob simply doesn’t know that I’ve made the measurement. As soon as he is informed, we agree on the result. Note that Bob doesn’t initially make a measurement; he makes a prediction, and it is the prediction that he gets wrong.

If Bob and I disagree about the direction of up and make our measurements, we may obtain different results. I measure the spin and record an up. Bob measures the entire box and reports that the spin was down. But we can resolve this by sharing what direction we believe up to be in. In all cases, the disagreement revolves around the lack of shared knowledge.

Cruel living doll experiments

Now that we have our first layer of quantumness, let’s make it even more complicated.

I am—despite running low on oxygen—still in the box. The spin state is now prepared by Alice (who is also in a box), who uses a coin flip to select either spin sideways (tails) or spin down (heads). She sends me the object with its spin, as well as what she measured.

Outside the boxes are two observers, Bob who will measure the state of my box and Bert who measures the state of Alice's box. In this case everyone knows about everyone else, and everyone understands quantum mechanics. The game is to try to predict the outcome of a measurement on the two boxes by Bob and Bert—and what they conclude about Alice's coin flip.

Alice throws tails. That means that I will only measure spin down with a 50-percent probability. If I measure the spin to be up, then I know that Alice threw tails. And, Bob, having made his measurement, will also state that Alice threw tails. Bert agrees. Not only do they all agree, but Bob, Alice, Bert, and I can, based on our own knowledge, predict that we will all agree.

Let’s run the experiment again. Alice throws tails again. But I measure spin down. I know the probabilities, so my box is in a superposition state of Alice-threw-heads and Alice-threw-tails. When Bob makes his measurement, he gets one of those results at random. Bert, on the other hand, will always obtain tails because Alice and her box know that the coin toss was tails. Bob, based on his measurement and knowledge of the rest of us, cannot correctly predict our predictions. And, not only that, Bob and Bert don’t agree on heads or tails.

Of course, once Bob knows that Bert measured tails, he can understand the disagreement, but by then it is too late, and both Alice and I have died of asphyxiation. Bob and Bert later get five years for the accidental death of two graduate students.

You can’t all be quantum-mechanical dolls

What is the significance of the disagreement between Bob and Bert? It tells us, seemingly, that a quantum system with knowledge of quantum mechanics cannot be self-consistent. That, in turn, means that one of the following three statements about reality has to be true:

Quantum mechanics does not apply at all scales. For quantum systems, the statement “Bob knows that I know the spin is up, so Bob knows the spin is up” is false. (The worst) “If the spin is up with probability of one, then the spin is not down” is false.

To put it more simply, statement one says that somewhere, as we scale from microscopic to the everyday, quantum mechanics is no longer true. The second statement says that quantum mechanics does not allow logical inference. And statement three says that measurement outcomes can have more than one value.

From there we fall into the depths of philosophy. We can, and have, argued for decades about what a measurement really means in quantum mechanics. This paper highlights the fact that we still haven’t—and may not ever—resolved that problem.

Nature Communications, 2018, DOI: 10.1038/s41467-018-05739-8 (About DOIs)