Today I learned something new: Matryoshka nesting dolls are not quantum mechanical objects. Surprisingly, this was not something I should have been sure of. After reading a paper on how quantum mechanics can lead to contradictions within quantum mechanics, I will never look at matryoshka dolls the same way again.
The article is not only theoretical in nature, but seems to overcomplicate a fairly simple argument. That probably means I’ve misunderstood it pretty thoroughly. So, let the mistakes begin.
The inner doll is solid
Suppose I have a single particle with 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 spider. But quantum mechanics won’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, all I can do is ask, “Hey, Frank, is your spider pointing up or down?” on which he will always answer 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, it picks up or down randomly. We cannot predict the result of the first measurement at all, but all future measurements on Frank give the same result: once up, always up. On the other hand, if Frank’s spider 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 get the outcome. No one can leak information about my measurements, but information about the spin status as it was for my measurement is available.
If the prepared spin status is up or down, Bob and I always agree on the outcome of the experiment. But if the spin is sideways, we don’t. Bob states that the spider is in a superposition state of up and down, while after I take the measurement I say it is down.
This isn’t really a paradox: Bob just doesn’t know I took the measurement. As soon as he is informed, we will agree on the result. Note that Bob doesn’t measure initially; he makes a prediction, and it is the prediction that he is wrong.
If Bob and I disagree on the upward direction and take our measurements, we can get different results. I measure the spin and record an up. Bob measures the entire box and reports that the spider was down. But we can solve this by sharing which direction we think we are going. In all cases, the disagreement revolves around the lack of shared knowledge.
Cruel live doll experiments
Now that we have our first quantum layer, let’s make it even more complicated.
I’m – even though I’m almost out of oxygen – still in the box. The spin state is now being prepared by Alice (who is also in a box), who uses a coin to select 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 condition of my box and Bert who will measure the condition of Alice’s box. In this case, everyone knows everyone and everyone understands quantum mechanics. The game is to try to predict the outcome of a reading on the two boxes by Bob and Bert – and what they conclude about Alice’s coin toss.
Alice throws tails. That means I will only measure spin-down with a 50 percent chance. If I measure the spider to be up, then I know that Alice threw tails. And Bob, having taken his measurement, will also say that Alice threw tails. Bert agrees. Not only do they all agree, but Bob, Alice, Bert and I can predict, based on our own knowledge, 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 superimposition of Alice-threw-heads and Alice-threw-tails. When Bob takes his measurement, he gets one of those results at random. Bert, on the other hand, will always get coins because Alice and her chest know that the toss was coins. Based on his measurements and knowledge of the rest of us, Bob cannot correctly predict our predictions. And not only that, Bob and Bert disagree on heads or tails.
Sure, once Bob knows that Bert has measured tails, he can understand the disagreement, but by then it’s too late and both Alice and I have died of suffocation. Bob and Bert are later given five years for the accidental death of two graduate students.
You can’t all be quantum mechanical puppets
What is the meaning of the disagreement between Bob and Bert? It apparently tells us 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 must be true:
- Quantum mechanics does not apply to all scales.
- For quantum systems, the statement “Bob knows I know the spin is up, so Bob knows the spin is up” is incorrect.
- (Worst) “If the spin is up with a probability of one, then the spin is not down” is false.
To put it more simply, the theorem says that somewhere, when we scale from the microscopic to the mundane, quantum mechanics is no longer true. The second statement says that quantum mechanics does not allow any logical inference. And statement three says that measurement results can have more than one value.
From there we fall into the depths of philosophy. We can and have debated for decades what a measurement really means in quantum mechanics. This article highlights the fact that we still haven’t — and may never have — solved that problem.
nature communication2018, DOI: 10.1038/s41467-018-05739-8 (About DOIs)