Friday, December 14, 2018

The word salad interpretation of quantum mechanics

Michael Nielsen, an achieved quantum information researcher, just wrote a fun essay
What does the quantum state mean? (CognitiveMedium.com)
I largely agree with his view although he is way more "agnostic" about the meaning of the quantum state than I am.



The essay starts with a quote by Feynman saying that he still felt nervous about the lack of intuition why quantum mechanics was consistent – but suspected that 2 generations later, people would already feel comfortable with quantum mechanics, finding its consistency self-evident. It has surely worked for me.



Now, as I have done many times in the past, Nielsen disagrees with the assertion that "the superposition of two states means that the object is simultaneously in both of them". In the most notorious example, Schrödinger's cat is often said to be "both alive and dead". That's really what he calls the "word salad interpretation of quantum mechanics", a phrase that already exceeds the price you have to pay for his article.

In reality, Schrödinger's cat is still "dead OR alive", the word "OR" is still much more accurate than "AND", while quantum mechanics allows us to predict probabilities of "dead" and "alive" as well as the probabilities of different values of observables that don't commute with the dead/alive binary quantity. "OR" is more accurate than "AND" – but you must get rid of the idea that the very usage of the word "OR" means that there is an objective reality (and objective answer to the dead/alive question) at every moment.

Instead, "OR" is the logical addition of a sort, an operator that tends to add the truth values (0/1) or probabilities (probabilities are just the expectation values of truth values). When several options have nonzero probabilities, it means that one (the observer) is ignorant about the right answer.



In particular, Nielsen correctly says that this summary is often addressed to the laymen but it is also embraced "literally" by some professional physicists even though it's effectively or morally incorrect, and deeply incorrect. By the effective incorrectness, I mean the incorrectness of more specific conclusions that directly follow from such a way of thinking.

In particular, numerous people imagine that the state of the form\[

\frac{1}{\sqrt {2^n}} \sum_x |x\rangle|f(x)\rangle

\] where the sum over \(x\) goes over all candidate routes of the "traveling salesman" allows quantum mechanics to solve the "travelling salesman problem" (the homework to pick the shortest route through all the cities: see Kaggle.com for an ongoing Santa Claus contest of that type) really quickly.

In this way, the word-salad people imagine, the quantum state contains "much more classical information" than the classical state of the same object. And one can make a smaller number of operations – and for example, if \(f(x)\) is chosen to depend on the length of the candidate route, we may highlight the components of the wave function that correspond to the shortest route – and solve the travelling salesman problem in the exponentially accelerated way as if the quantum computer were a parallel computer.

But Nielsen knows quantum computation very well – which is why he can not only tell you that this is impossible but he also provides you with a rigorous enough paper that demonstrates the impossibility of this "quantum computer employed as a massively parallel computer" in detail.

Such technical papers on quantum computations may be interesting but I am confident that the mistakes of the word salad interpreters are much more elementary. They simply imagine that the wave function is a classical wave – so all the complex probability amplitudes (the coefficients of the different terms in the wave function) are some "objective classical data". There are exponentially many of them – for \(N\) qubits, you have \(2^N\) complex amplitudes – and they can be manipulated like an exponentially large number of classical data. And that's why you can try all the routes at once and do similar very effective things.

Except that you cannot. It is simply wrong to assume that the complex probability amplitudes are "objective classical data" or, equivalently, that the wave function is a classical wave. In particular, by their meaning, those probability amplitudes are much closer to the regular probabilities in probability distributions that already existed in classical physics when some uncertainty was present.

And probabilities encoding our ignorance about the state of a physical object or system are very different things from the "classical data that we know well". In particular, the evolution of the classical data may be almost arbitrary. If you describe planets by their coordinates \(x_j (t)\), then it's conceivable that the Universe evolves these functions of time through the most general, almost always nonlinear, differential equation – and the evolution could be even more general than one that may be called a differential equation.

On the other hand, the coefficients \(c_i\in\CC\) describing the state vector \(\ket\psi\) may only evolve through a rather general, but linear, equation – Schrödinger's equation where all the optional parameters of the evolution are encoded in the Hermitian linear operator \(H\), the Hamiltonian. So the wave function cannot be evolved in a "general way". In particular, you cannot have a gadget that evolves a wave function so that the amplitudes \(c_i\) are squared – so that they evolve into \(|c_i|^2\), for example. That's not linear, it's prohibited.

The linear evolution of the wave function forces you to evolve all the amplitudes in a correlated way. And the linearity prevents you from squaring or other nonlinear transformations – which would be equivalent to the quantum xeroxing and that's impossible.

On top of that, the complex probability amplitudes \(c_i\in\CC\) in the wave function \(\ket\psi\) cannot be separately measured, like \(x_j\) may be measured in classical physics e.g. by looking at the planets carefully. Instead, you may only measure the value of a Hermitian linear operator – and you get a random result whose probabilities are calculable from the amplitudes \(c_i\) by Born's rule. A simple subset of such measurements looks like you are not determining the value of the amplitude of \(c_i\) – instead, you are determining the value of the index \(i\) and you get a random result at probabilities dictated by \(c_i\).

Once again, there are aspects of quantum mechanics that are completely new – in particular, \(x\) and \(p\) much like the generic pairs of operators don't commute with each other which is the mathematical reason behind the uncertainty principle – but the meaning of Schrödinger's equation is much closer to the meaning of the Liouville equation on the phase space in the fancy, Hamiltonian classical mechanics. That equation is unavoidably linear. Also, you can't measure \(\rho(p,q)\) directly, by doing one measurement for each point parameterized by the coordinates \(p,q\). Instead, you can only try to measure \(p\) or \(q\) or their other functions.

Needless to say, the analogy between \(\rho(p,q)\) in classical physics and \(\ket\psi\) may be made even tighter: if you replace the pure state vector \(\ket\psi\) by the density matrix \(\rho\), you get the full-blown quantum counterpart of \(\rho(p,q)\). The moral meaning of the Schrödinger-like equation for the density matrix (it's called the von Neumann equation, not that I think such names are important or physical) is the same as the meaning of the Liouville equation obeyed by \(\rho(p,q;t)\) in classical physics. You may directly find a relationship between these two mathematical objects in the \(\hbar\to 0\) classical limit.

So the view that the wave function is a classical wave is wrong – equally wrong as the closely related meme that a quantum computer is an exponentially parallel quantum computer. Quantum mechanics simply isn't another classical theory. It seems clear that Nielsen agrees with these negative statements. And he sort of promotes the agnostic, shut-up-and-calculate interpretation as the optimal foundation for people who work on more well-defined "tiny problems" that are enough (in this case and other cases) to make the big problems "melt away".

I don't quite agree with his view that the "positive interpretation of the wave function" is something that good physicists can't agree about. The "negative" statements above sound negative but that's just an emotional classification of their meaning. They're "negative" because they tell us what the wave function isn't. They are a warning that is discouraging the listeners from some seriously incorrect conclusions.

But they're also "positive" information. We can learn something about Nature both from sentences that "sound" positive or negative – at the end, there is no universal well-defined way to separate statements to positive and negative ones. So these negative statements are meant to protect us from conceptual errors. But they may be viewed as a part of the positive description of the wave function, too. The wave function is an element of a multi-dimensional (usually infinite-dimensional) complex linear space that encodes the maximum knowledge of the observer about the physical system of interest. Its evolution is given by Schrödinger's equation, its testability is guaranteed by measurements that are labeled by Hermitian linear operators, whose outcomes are probabilistically determined by Born's rule, and whose extra effect is an adjustment of the wave function called the "collapse".

Because the wave function isn't a classical wave, the "collapse" isn't analogous to the collapse of any classical object that we may observe in real time. Instead, the quantum "collapse" is a complex generalization of the Bayes theorem that tells us how the pre-measurement (prior) probabilities should be replaced with the post-measurement (posterior) probabilities. That's it. More or less. One may talk about it for an extra while. And I have done so hundreds of times. But there's really nothing that will miraculously change the picture.

Let me say something equivalent about the positivity of the "negative statements". The discovery of quantum mechanics represented a really profound revolution in the foundations of physics which might be called a "meta-theory" and not just a "theory" by some people who only wanted to adjust details of the right theory of classical physics. In particular, quantum mechanics redefines what questions are meaningful and what should be done to make predictions or to experimentally test them. Like relativity but much more than relativity, quantum mechanics says that lots of questions that people may ask about Nature are physically meaningless. This is a negative assertion about some people's thinking but there's a lot of evidence supporting this negative assertion (and lots of the internal coherence in the whole logic). If some people aren't willing to accept this "meta-theory", it's too bad. But this refusal to accept to even consider it as a possible theory is these people's mistake, not a flaw of quantum mechanics.

It seems obvious to me that all the word salad and related misinterpreters of quantum mechanics want to find or promote some assertions that make the wave function look more (or completely) like a classical wave. But this viewpoint is wrong, as I said, and the statement "this general view is wrong" is really the final verdict about all their attempts – and therefore the final answer to all questions related to a would-be deeper interpretation of quantum mechanics that are being proposed by everyone out there!

If you adopt a more correct thinking, the quantum mechanical thinking, the problem about the meaning of the wave function becomes fully and completely settled. At least when we focus on all the questions and ideas that have actually been proposed and discussed as a possible "deepening" of the foundations of quantum mechanics, the "efforts to deepen" and the "introduction of fundamentally and demonstrably incorrect ideas" are absolutely inseparable!

It means that if you want to avoid being wrong, you also have to agree that "my" description of the wave function above is the complete and final answer. Maybe someone may find some extra questions or answers about the meaning of the wave function that have a chance not to be wrong but it hasn't happened yet in the first 93 years after the birth of quantum mechanics, despite the thousands of allegedly relevant pages that were filled with word salad.

In other words, some deeper questions or proposals that go beyond the orthodox or Copenhagen (or similar) interpretation of quantum mechanics may hypothetically have a merit or value in the future. But the subset of these questions, proposals, and efforts that have taken place up to December 2018 is worthless: all these efforts are at most a generalization of the word salad.

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