By Michael Nielsen, December 2018

We have always had a great deal of difficulty understanding the world view that quantum mechanics represents. At least I do, because I’m an old enough man that I haven’t got to the point that this stuff is obvious to me. Okay, I still get nervous with it…. You know how it always is, every new idea, it takes a generation or two until it becomes obvious that there’s no real problem. I cannot define the real problem, therefore I suspect there’s no real problem, but I’m not sure there’s no real problem. – Richard Feynman

In popular articles about quantum computing it’s common to describe qubits as having the ability to “be in both $|0\rangle$ and $|1\rangle$ states at once”, and to say things like “quantum computers get their power because they can simultaneously be in exponentially many quantum states!”

I must confess, I don’t understand what such articles are talking about.

What seems to be implied – it’s rarely spelled out, although some accounts come close – is that quantum computers work by preparing a superposition $\frac{1}{\sqrt 2^n} \sum_x |x\rangle|f(x)\rangle$, with $x$ varying over possible solutions to the problem – maybe it’s tours in a travelling salesman problem. And $f(x)$ is some associated quantity of interest, such as the distance through the tour. Then, somehow, voila!, you get to read out the desired answer $f(x)$ from the quantum computer.

The only trouble is that this is provably impossible to do in general, or even just in typical cases.

What I think is going on is this: when people remark that the state $0.6|0\rangle+0.8|1\rangle$ is simultaneously $0$ and $1$, they’re trying to explain the quantum state in terms of classical concepts they’re already familiar with. That sounds sort of okay at first, and fills a vacuum of meaning for people unfamiliar with quantum mechanics. But the more you think about it, the worse things get. Saying $0.6|0\rangle+0.8|1\rangle$ is simultaneously $0$ and $1$ makes about as much sense as Lewis Carroll’s nonsense poem Jabberwocky:

’Twas brillig, and the slithy toves
Did gyre and gimble in the wabe:
All mimsy were the borogoves,
And the mome raths outgrabe.

I call the implied way of thinking the “word salad interpretation of quantum mechanics”. The main (sole?) virtue of the word salad interpretation is that it does fill a vacuum of meaning. Because it is a genuinely good question: what does the quantum state mean?

For me, it’s also a deeply uncomfortable question. I genuinely don’t know the answer, despite having spent tens of thousands of hours thinking about quantum mechanics. And I cannot, with conviction, tell you what the quantum state means. It’s frankly a pretty strange situation.

Now, there are some people who will very confidently tell you that they “know” the correct way to think about the quantum state. Trouble is, different people will tell you different things! That includes deeply knowledgeable experts on quantum mechanics. Individually, each can sound pretty convincing. But when you get them together in a room, the result is sometimes some pretty unpleasant conflagrations. I’ve seen physicists shout at one another over the issue, on more than one occasion.

I’m not alone in my discomfort with the question. A lot of physicists respond to this discomfort with a sort of reserved agnosticism. A pretty common approach is what the physicist David Mermin dubbed the “shut-up-and-calculate interpretation of quantum mechanics”.

In the shut-up-and-calculation interpretation, you think of the quantum state as a calculational device. At most you have a sort of vague meaning in mind, perhaps thinking of the quantum state as being a bit like a probability distribution over states, but satisfying slightly different mathematical rules (different for reasons that are never made quite clear). You become fluent in those mathematical rules, and use them to solve lots of different problems. Gradually, you build up a library of higher-order tricks and intuitions, understanding emergent rules hidden inside the rules of quantum mechanics – ideas like quantum teleportation, or the no-cloning theorem, for instance. It’s a very instrumental way of making meaning of the quantum state.

As a practical matter, and for students starting out, I’m pretty sympathetic to adopting the shut-up-and-calculate interpretation, at least most of the time. It builds up many handy skills, as well as intuition about how quantum mechanics work. That’s extremely useful background when investigating interpretational issues.

Why does the meaning of the quantum state matter? Sure, maybe people would feel better if they had a way of interpreting the quantum state beyond it being a calculational device. But maybe that’s just an irrelevant human prejudice. Nature doesn’t need to conform to our prejudices! But I think there’s a genuine problem here, beyond our prejudices about what our theories should look like. Quantum mechanics isn’t a final theory. We don’t have a convincing understanding of the measurement process in quantum mechanics. Nor do we have a convinving quantum theory of gravity. And maybe those problems are connected to having a better understanding what the quantum state means. In which case having a better understanding of the quantum state may help in solving those other problems.

I attributed the term “shut-up-and-calculate” to David Mermin. Mermin is one of the deepest thinkers about interpretational issues, and he certainly didn’t intend the term as a compliment! But despite that, I’m somewhat sympathetic to shut-up-and-calculate not just as a practical strategy, but also as a strategy for (eventually) better understanding quantum states.

In particular, the situation reminds me of the study of human consciousness. Many scientists and philsophers spend a great deal of time pondering consciousness, writing about the “hard problem of consciousness” and so on. In the meantime, there’s an army of scientists doing very plain nuts-and-bolts experiments, trying to understand all the myriad details of action potentials, neural circuits, and so on. I suspect the latter group will ultimately make far more contribution to our understanding of consciousness than the former. Sometimes, when you solve enough tiny problems the big problems just melt away. And I wonder if the same will be true of the meaning of the quantum state, that we’ll understand it by gradually building up our detailed knowledge of quantum mechanics, and eventually understand things like the interpretation of the quantum state almost en passant. If that’s the case, then the current lack of a universally-agreed upon interpretation is a nuisance, and regrettable, but no more.

My own current preference is thus for the this-is-an-open-problem interpretation of quantum mechanics: I think we don’t yet have enough evidence to know, and won’t for decades. I know some readers will dislike this: they’d much prefer if I shouted with conviction that the right way to interpet the quantum state is etc But I don’t know, and I don’t think anyone else does either. I do have opinions about how to get to such an interpretation, but will omit them in the interests of brevity. The main thing I want you to take away from this essay is that determined agnosticism is a possible approach, and is also consistent with a deep interest in actually solving the problem.

Will all that said, there are people who’ve thought long and hard about the meaning of the quantum state, and who do have definite opinions about the right way to think about it. As a starting point, I recommend reading Hugh Everett and David Deutsch on the many-worlds interpretation of quantum mechanics; Chris Fuchs on the idea that the quantum state is a state of knowledge; David Bohm on the idea that it’s a sort of pilot wave, guiding particles in the system. And, although it’s not exactly an interpretation of the quantum state, I like Richard Feynman’s paper recasting quantum mechanics in terms of (sometimes negative!) probability distributions, rather than quantum states. Those are just a few ideas, to give you a sample of some of the (very different) ideas out there. Many more points of view have been put forward! Be aware that many of these people disagree (or disagreed, while alive) strongly with one another. Don’t necessarily expect to solve the problem yourself – although maybe you will make some contribution. And do come back to just plain working with the theory, boots on the ground. No matter how you think about the quantum state, quantum mechanics is a beautiful theory, and remarkably fun to work with.