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Nature prohibits "protective measurements"

Hundreds of anti-quantum zealots cannot find a trivial mistake even after 25 years

One of the universal postulates of quantum mechanics is that the information about a physical system may only be obtained through a measurement of an observable. The observable must be mathematically represented by a Hermitian linear operator \(\hat L\) and the result of the measurement has to be one of the eigenvalues of the operator. The probability that a particular eigenvalue \(\lambda\) is produced by the measurement is calculable by Born's rule. After the measurement, the state vector is projected to become an eigenstate of the measured operator i.e. \[

\hat L \ket{\psi}_{\rm after} = \lambda_{\rm outcome} \ket{\psi}_{\rm after}.

\] For this reason, the probabilistic character of the predictions is unavoidable. The modification of the state by any measurement is unavoidable, too. This is really a straightforward paragraph that summarizes most of the general laws of quantum mechanics. But some people just find this straightforward axiom about the right way to get the information about any physical system impossibly difficult. So even more than 90 years after the birth of quantum mechanics, they are coining lots of stupid names of "measurements" that completely contradict the law above – or, equivalently, that contradict the uncertainty principle. They are clearly convinced that they're amazing and they may completely ignore and circumvent operators and eigenvalues and measure properties of physical objects in the direct old-fashioned way, i.e. classically.

I have discussed the nonsensical "weak measurements" several times. It's supposed to be a way to measure the system without modifying it. It is exactly what is not possible in quantum mechanics.



A similar but not quite equivalent buzzword (to the "weak measurement") is the so-called "protective measurement". It was coined more than 25 years ago in the following paper by Yakir Aharonov and Lev Vaidman:

Measurement of the Schrodinger wave of a single particle
As the title says, these two "thinkers" claimed that it was possible to measure the wave function in a single repetition of the experiment. In particular, it was possible to directly measure the expectation value of an operator, too.



Now, every undergraduate student who deserves an A in a quantum mechanics course must know that this claim is totally wrong within a minute and he must be able to provide a complete enough proof with the localized mistake within an hour. Too bad, there are no people who would deserve an A from quantum mechanics among the "interpreters of quantum mechanics". You may check that the preprint has over 200 citations now.

Although a huge fraction of the most well-known followups are self-citations (papers co-written either by Vaidman or by Aharonov or both), you may use the list of followups as a fast database of clueless anti-quantum zealots. The followups to this totally wrong paper are still being written.

OK. So these two Gentlemen claimed that if a physical system is found in the state \(\ket\psi\), you may choose any observable \(A\) and use a special apparatus that will spit out the value\[

\langle A \rangle = \bra\psi A \ket\psi.

\] Holy crap. Just think about the simplest example to see how self-evidently wrong it is. Consider the spin of an electron, a two-dimensional Hilbert space, and choose \(A = j_z\). Can you measure the expectation value \(\langle j_z\rangle\) in a single repetition of an experiment? It would be like doing something to a flying die that tells you that you will get 6 with the probability of 1/6 – and that number could go to 0 or 1 if you did the same procedure later. That would be quite a weird gadget to quantify your hopes.

Quantum mechanics allows you to observe any observable which is, in the spin's case, a Hermitian \(2\times 2\) matrix that acts on the two-dimensional Hilbert space. But any \(2\times 2\) Hermitian matrix may be written as\[

A = a + \vec b \cdot \vec \sigma

\] where \(\sigma\) are the three Pauli matrices, and \(\{a,b_1,b_2,b_3\}\subset \RR\). If you measure \(A\), it is clearly equivalent to measuring the spin along the \(\hat b\) axis, in the direction of \(\vec b\). You may get two results, "up" or "down" along that axis, and the corresponding eigenvalues of \(\hat b\cdot \vec \sigma\) are \(\{+1,-1\}\). The possible outcomes of the measurement of \(A\) – the most general measurement you can make on the electron's spin – are simply \(a\pm |\vec b|\) where the sign follows from the spin's being "up" or "down" along the \(\hat b\) axis.

Done.

Any nontrivial measurement done on the electron's spin simply has two possible outcomes. It is a "qubit", so any measurement is some variation of "up" and "down".

But Aharonov and Vaidman boldly wrote that they may use quantum mechanics to invalidate the basic axioms of quantum mechanics and "measure" \(\langle j_z\rangle\). Oh, really? So let's go through page 3 of their 1993 paper.

They review a protocol described by John von Neumann. How did von Neumann measure things? He measured observables using the proper, "strong" measurement (i.e. in agreement with the first paragraph of this blog post). All measurements that are allowed in Nature are "strong" in the terminology of the anti-quantum zealots.

One can talk about the "direct" measurement of an observable. That's basically what the Copenhagen school assumed most of the time. You observe things by your eyes and you know how the perceptions from your eyes (or other organs) may be interpreted. So you're not interested in how they exactly work. But to make things more amusing, John von Neumann was the guy who inserted – somewhat redundantly – the apparatus into the description, in the chain between your eyes and the measured system. (Much of Everett's fame arose from Everett's plagiarism of von Neumann's comments about the entangled apparatuses – Everett also added lots of wrong claims as his added value.) So you have the measured system and couple it to the apparatus (which has some canonical \(q,p\) describing the position of a pointer) by the interaction Hamiltonian\[

H_{\rm int}(t) = g(t)p A

\] where \(A\) is the observable of the measured system that you want to measure, \(g(t)\) is a coupling constant that integrates to \(\int g(t) dt = 1\) (this coupling constant is only nonzero during some short period of time \(t\) around the measurement time), and \(p\) is the dual variable to \(q\), the position of the pointer on the apparatus. OK, the apparatus is affected by the observable \(A\) of the measured system. The larger \(A\) is, the more \(p\) acts on the pointer, and as the result, the pointer's location \(q\) is equal to \(A\) or some multiple of it after the interaction. You may imagine that the pointer starts in a Gaussian packet around \(\langle q\rangle = 0\) at the beginning, and this packet is just shifted to a different location.

So we just transferred the information about \(A\) to the location of the pointer of the apparatus, \(q\). The width of the pointer's packet survives as a contribution to the error margin of this measurement procedure. We measure \(q\) as one of the allowed eigenvalues of \(A\). So far so good.

How do they claim to measure \(\langle A\rangle\) instead? They do the same thing but instead of a speedy enough interaction between the measured system and the apparatus, they allow the interaction to be slow and adiabatic. For the adiabatic change and a small perturbation of the Hamiltonian, the eigenvalues may be computed by the first-order perturbation theory as \[

\delta E = \langle H_{\rm int} \rangle = \frac{\langle A \rangle p}{T}

\] where \(T\) is the long enough duration of the interaction between the measured system and the apparatus. At any rate, \(\delta E\), the perturbation of the energy eigenvalue, is proportional to the expectation value of the interaction Hamiltonian i.e. to the expectation value of \(A\), and that's how you will see \(\langle A\rangle \) as the location of the pointer of the apparatus.

Instead of measuring the discrete eigenvalue "up" or "down" of the electron's spin, for example, you may measure the continuous quantity \(\langle A \rangle \) i.e. an aspect of the wave function. It's wonderful. The only problem is that this conclusion is a result of a trivial mistake.

In the displayed equation above which is borrowed from first-order perturbation theory for energy eigenstates, the expectation value \(\langle H_{\rm int}\rangle \) must be interpreted as\[

\bra{\psi_{0,n}} H_{\rm int} \ket{\psi_{0,n}}

\] i.e. the expectation value of the interaction Hamiltonian ("the perturbation") calculated in an eigenstate of the unperturbed Hamiltonian. But Aharonov and Vaidman wanted the expectation value to be interpreted as the expectation value in the actual pre-measurement state of the physical system. So their usage of the first-order perturbation theory only works if and when the pre-measurement state is an eigenstate of \(H_{\rm sys}\), the Hamiltonian of the measured system (without the interactions with the apparatus).

And they must really know what the corresponding eigenvalue \(E_0\) of \(H_{\rm sys}\) is.

But they still want to couple the measured system to the apparatus and find something about the measured system, namely some aspect (expectation value) of the operator \(A\). It means that there are several allowed eigenvalues of \(A\), like "up" and "down" in our example of the spin, and both of them or all of them must be assumed to be the eigenstates of \(H_{\rm sys}\) with the same eigenvalue.

However, when it's so, we can't just use the first-order perturbation theory rule\[

\delta E_{n,1} = \bra{\psi_{0,n}} H_{\rm int} \ket{\psi_{0,n}}.

\] This rule only works for non-degenerate energy eigenstates. But because the whole point of the measurement is that there are at least two possible and allowed eigenvalues of \(A\), these energy \(H_{\rm sys}\) eigenstates are degenerate. The relevant space is at least two-dimensional. For this reason, as every introductory undergraduate quantum mechanics course teaches (in the section about perturbation theory), we have to use the degenerate perturbation theory.

Degenerate perturbation theory is a method that starts with the isolation of the proper eigenstates within this degenerate space – eigenstates that remain eigenstates when you turn on the perturbation. Needless to say, this search for the "proper, well-behaved basis of eigenstates" guarantees that your measurement will actually yield one of the eigenvalues of \(A\) again. If the apparatus is only affected by the measured system through its observable \(A\), then the pointer may only act as a function of \(A\) i.e. measure \(A\) and pick one of the eigenvalues encoded to the pointer in some way. None of the details of the interaction Hamiltonian can change this fact. You just can't measure \(\psi(x)\) or any functional of it by a single repetition of an experiment. It's an axiom of quantum mechanics and it's totally internally consistent. If you could measure \(\psi(x)\) in a single repetition of a situation, quantum mechanics would be self-contradictory.

But it's not. You may couple your system to any apparatuses of any design in any way. What is ultimately possible is to observe an observable, a linear operator, and all properly calculated, well-defined (linear or nonlinear) functions or functionals of linear operators on any Hilbert space are still linear operators on the Hilbert space! So this whole effort to try to design the measurement of some "completely new things" that are done by "new and clever ways" is as futile as the effort to build a perpetuum mobile. It is just clear from the first minute what is the set of all possible things that you may measure. It's the set of all linear Hermitian operators on the Hilbert space! Period.

If the Hilbert space is \(N\)-dimensional, the space of linear Hermitian operators is real \(N^2\)-dimensional. It's that easy. There's a particular continuous space of things that may be measured. Most of the pairs of observables refuse to commute with each other. Some observables are (linear or nonlinear) functions or functionals of others but they're still linear operators and by combining many observables into complicated functions, you won't get any qualitatively new clever things to measure. The set of things that may be measured is clear from the beginning. You shouldn't overlook this forest because of trees.

For more than 25 years, the anti-quantum crackpots kept on writing papers occasionally referring to the "protective measurements" and similar things. They just didn't care that this concept flatly cntradicted basic axioms of quantum mechanics. They couldn't figure out that the step from perturbation theory that was claimed to be usable was actually not usable because the degenerate perturbation theory should have been used instead.

You may search e.g. through Google Scholar for 2018 papers mentioning the "protective measurements". In the list of authors, you will find "who is who" in the world of anti-quantum zealots. Pusey, Leifer, Rovelli, Pussy, Cunt, Gao, and tons of others. Over 25 years, over 1,500 papers mentioned the "protective measurement[s]". You may verify that not a single one manages to figure out that the whole concept is completely wrong – that the emperor has no clothes – despite the fact that they have been doing this stuff for more than 25 years.

These people are staggering morons. They are children who were left behind. This whole "field" is a field of college dropouts who just couldn't learn this basic undergraduate stuff correctly. But they were able to persuade other morons that they're wonderfully clever, anyway.

P.S.: If the wave function or expectation values could be measured in this way, then the "collapse of the wave function" would become a physical effect that objectively changes the predictions for future ("protective") measurements, and it would matter when it happens. The collapse following the first measurement of an entangled composite system would indeed affect the results of some "protective measurements" of the wave function instantaneously. That would be a conflict with special relativity. The prohibition of the "protective measurement" and other weird methods to measure the wave function as if it were an observable is indeed a necessary condition for the consistency of quantum mechanics with locality (and therefore special relativity).

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