One of the universal postulates of quantum mechanics is that the information nearly a physical arrangement may exclusively live obtained through a measuring of an observable. The observable must live mathematically represented yesteryear a Hermitian linear operator \(\hat L\) as well as the final result of the measuring has to live 1 of the eigenvalues of the operator. The probability that a exceptional eigenvalue \(\lambda\) is produced yesteryear the measuring is calculable yesteryear Born's rule. After the measurement, the nation vector is projected to travel 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 graphic symbol of the predictions is unavoidable. The modification of the nation yesteryear whatsoever measuring is unavoidable, too. This is genuinely a straightforward paragraph that summarizes most of the full general laws of quantum mechanics. But some people just give away this straightforward axiom nearly the right way to acquire the information nearly whatsoever physical arrangement impossibly difficult. So fifty-fifty to a greater extent than than xc years after the nascence of quantum mechanics, they are coining lots of stupid names of "measurements" that completely contradict the constabulary inward a higher house – or, equivalently, that contradict the dubiety principle. They are clearly convinced that they're amazing as well as they may completely ignore as well as circumvent operators as well as eigenvalues as well as stair out properties of physical objects inward the direct old-fashioned way, i.e. classically.
I receive got discussed the nonsensical "weak measurements" several times. It's supposed to live a way to stair out the arrangement without modifying it. It is just what is not possible inward quantum mechanics.
Influenza A virus subtype H5N1 similar but non quite equivalent buzzword (to the "weak measurement") is the so-called "protective measurement". It was coined to a greater extent than than 25 years agone inward the next newspaper yesteryear Yakir Aharonov as well as Lev Vaidman:
Measurement of the Schrodinger moving ridge of a unmarried particleAs the championship says, these ii "thinkers" claimed that it was possible to stair out the moving ridge part inward a unmarried repetition of the experiment. In particular, it was possible to direct stair out the expectation value of an operator, too.
Now, every undergraduate educatee who deserves an Influenza A virus subtype H5N1 inward a quantum mechanics course of written report must know that this claim is totally incorrect inside a infinitesimal as well as he must live able to supply a consummate plenty proof alongside the localized error inside an hour. Too bad, at that spot are no people who would deserve an Influenza A virus subtype H5N1 from quantum mechanics amidst the "interpreters of quantum mechanics". You may banking concern 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 yesteryear Vaidman or yesteryear Aharonov or both), you lot may utilization the listing of followups every bit a fast database of clueless anti-quantum zealots. The followups to this totally incorrect newspaper are nevertheless beingness written.
OK. So these ii Gentlemen claimed that if a physical arrangement is institute inward the nation \(\ket\psi\), you lot may pick out whatsoever observable \(A\) as well as utilization a special apparatus that volition spit out the value\[
\langle Influenza A virus subtype H5N1 \rangle = \bra\psi Influenza A virus subtype H5N1 \ket\psi.
\] Holy crap. Just recall nearly the simplest illustration to run into how self-evidently incorrect it is. Consider the spin of an electron, a two-dimensional Hilbert space, as well as pick out \(A = j_z\). Can you lot stair out the expectation value \(\langle j_z\rangle\) inward a unmarried repetition of an experiment? It would live similar doing something to a flight croak that tells you lot that you lot volition acquire six alongside the probability of 1/6 – as well as that number could travel to 0 or 1 if you lot did the same physical care for later. That would live quite a weird gadget to quantify your hopes.
Quantum mechanics allows you lot to notice whatsoever observable which is, inward the spin's case, a Hermitian \(2\times 2\) matrix that acts on the two-dimensional Hilbert space. But whatsoever \(2\times 2\) Hermitian matrix may live written as\[
Influenza A virus subtype H5N1 = a + \vec b \cdot \vec \sigma
\] where \(\sigma\) are the 3 Pauli matrices, as well as \(\{a,b_1,b_2,b_3\}\subset \RR\). If you lot stair out \(A\), it is clearly equivalent to measuring the spin along the \(\hat b\) axis, inward the administration of \(\vec b\). You may acquire ii results, "up" or "down" along that axis, as well as the corresponding eigenvalues of \(\hat b\cdot \vec \sigma\) are \(\{+1,-1\}\). The possible outcomes of the measuring of \(A\) – the most full general measuring you lot tin brand on the electron's spin – are but \(a\pm |\vec b|\) where the sign follows from the spin's beingness "up" or "down" along the \(\hat b\) axis.
Done.
Any nontrivial measuring done on the electron's spin but has ii possible outcomes. It is a "qubit", so whatsoever measuring is some variation of "up" as well as "down".
But Aharonov as well as Vaidman boldly wrote that they may utilization quantum mechanics to invalidate the basic axioms of quantum mechanics as well as "measure" \(\langle j_z\rangle\). Oh, really? So let's travel through page 3 of their 1993 paper.
They review a protocol described yesteryear John von Neumann. How did von Neumann stair out things? He measured observables using the proper, "strong" measuring (i.e. inward understanding alongside the firstly paragraph of this weblog post). All measurements that are allowed inward Nature are "strong" inward the terminology of the anti-quantum zealots.
One tin beak nearly the "direct" measuring of an observable. That's basically what the Copenhagen schoolhouse assumed most of the time. You notice things yesteryear your eyes as well as you lot know how the perceptions from your eyes (or other organs) may live interpreted. So you're non interested inward how they just work. But to brand things to a greater extent than amusing, John von Neumann was the guy who inserted – somewhat redundantly – the apparatus into the description, inward the chain betwixt your eyes as well as the measured system. (Much of Everett's fame arose from Everett's plagiarism of von Neumann's comments nearly the entangled apparatuses – Everett also added lots of wrong claims every bit his added value.) So you lot receive got the measured arrangement as well as pair it to the apparatus (which has some canonical \(q,p\) describing the seat of a pointer) yesteryear the interaction Hamiltonian\[
H_{\rm int}(t) = g(t)p A
\] where \(A\) is the observable of the measured arrangement that you lot desire to measure, \(g(t)\) is a coupling constant that integrates to \(\int g(t) dt = 1\) (this coupling constant is exclusively nonzero during some brusk catamenia of fourth dimension \(t\) around the measuring time), as well as \(p\) is the dual variable to \(q\), the seat of the pointer on the apparatus. OK, the apparatus is affected yesteryear the observable \(A\) of the measured system. The larger \(A\) is, the to a greater extent than \(p\) acts on the pointer, as well as every bit the result, the pointer's place \(q\) is equal to \(A\) or some multiple of it after the interaction. You may imagine that the pointer starts inward a Gaussian package around \(\langle q\rangle = 0\) at the beginning, as well as this package is just shifted to a dissimilar location.
So nosotros just transferred the information nearly \(A\) to the place of the pointer of the apparatus, \(q\). The width of the pointer's package survives every bit a contribution to the error margin of this measuring procedure. We stair out \(q\) every bit 1 of the allowed eigenvalues of \(A\). So far so good.
How hit they claim to stair out \(\langle A\rangle\) instead? They hit the same thing but instead of a speedy plenty interaction betwixt the measured arrangement as well as the apparatus, they allow the interaction to live ho-hum as well as adiabatic. For the adiabatic alter as well as a modest perturbation of the Hamiltonian, the eigenvalues may live computed yesteryear the first-order perturbation theory every bit \[
\delta E = \langle H_{\rm int} \rangle = \frac{\langle Influenza A virus subtype H5N1 \rangle p}{T}
\] where \(T\) is the long plenty duration of the interaction betwixt the measured arrangement as well as the apparatus. At whatsoever rate, \(\delta E\), the perturbation of the loose energy eigenvalue, is proportional to the expectation value of the interaction Hamiltonian i.e. to the expectation value of \(A\), as well as that's how you lot volition run into \(\langle A\rangle \) every bit the place of the pointer of the apparatus.
Instead of measuring the discrete eigenvalue "up" or "down" of the electron's spin, for example, you lot may stair out the continuous quantity \(\langle Influenza A virus subtype H5N1 \rangle \) i.e. an aspect of the moving ridge function. It's wonderful. The exclusively occupation is that this determination is a final result of a short mistake.
In the displayed equation inward a higher house which is borrowed from first-order perturbation theory for loose energy eigenstates, the expectation value \(\langle H_{\rm int}\rangle \) must live 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 inward an eigenstate of the unperturbed Hamiltonian. But Aharonov as well as Vaidman wanted the expectation value to live interpreted every bit the expectation value inward the actual pre-measurement nation of the physical system. So their usage of the first-order perturbation theory exclusively industrial plant if as well as when the pre-measurement nation is an eigenstate of \(H_{\rm sys}\), the Hamiltonian of the measured arrangement (without the interactions alongside the apparatus).
And they must genuinely know what the corresponding eigenvalue \(E_0\) of \(H_{\rm sys}\) is.
But they nevertheless desire to pair the measured arrangement to the apparatus as well as give away something about the measured system, namely some aspect (expectation value) of the operator \(A\). It agency that at that spot are several allowed eigenvalues of \(A\), similar "up" as well as "down" inward our illustration of the spin, as well as both of them or all of them must live assumed to live the eigenstates of \(H_{\rm sys}\) alongside the same eigenvalue.
However, when it's so, nosotros can't just utilization the first-order perturbation theory rule\[
\delta E_{n,1} = \bra{\psi_{0,n}} H_{\rm int} \ket{\psi_{0,n}}.
\] This dominion exclusively industrial plant for non-degenerate loose energy eigenstates. But because the whole indicate of the measuring is that at that spot are at to the lowest degree ii possible as well as allowed eigenvalues of \(A\), these loose energy \(H_{\rm sys}\) eigenstates are degenerate. The relevant infinite is at to the lowest degree two-dimensional. For this reason, every bit every introductory undergraduate quantum mechanics course of written report teaches (in the department nearly perturbation theory), nosotros receive got to utilization the degenerate perturbation theory.
Degenerate perturbation theory is a method that starts alongside the isolation of the proper eigenstates inside this degenerate infinite – eigenstates that stay eigenstates when you lot plow on the perturbation. Needless to say, this search for the "proper, well-behaved footing of eigenstates" guarantees that your measuring volition genuinely yield 1 of the eigenvalues of \(A\) again. If the apparatus is exclusively affected yesteryear the measured arrangement through its observable \(A\), as well as so the pointer may exclusively human activeness every bit a part of \(A\) i.e. stair out \(A\) as well as selection 1 of the eigenvalues encoded to the pointer inward some way. None of the details of the interaction Hamiltonian tin alter this fact. You just can't stair out \(\psi(x)\) or whatsoever functional of it yesteryear a unmarried repetition of an experiment. It's an axiom of quantum mechanics as well as it's totally internally consistent. If you lot could stair out \(\psi(x)\) inward a unmarried repetition of a situation, quantum mechanics would live self-contradictory.
But it's not. You may pair your arrangement to whatsoever apparatuses of whatsoever pattern inward whatsoever way. What is ultimately possible is to notice an observable, a linear operator, as well as all properly calculated, well-defined (linear or nonlinear) functions or functionals of linear operators on whatsoever Hilbert infinite are nevertheless linear operators on the Hilbert space! So this whole endeavour to examine to pattern the measuring of some "completely novel things" that are done yesteryear "new as well as clever ways" is every bit futile every bit the endeavour to gear upwardly a perpetuum mobile. It is just clear from the firstly infinitesimal what is the laid of all possible things that you lot may measure. It's the laid of all linear Hermitian operators on the Hilbert space! Period.
If the Hilbert infinite is \(N\)-dimensional, the infinite of linear Hermitian operators is existent \(N^2\)-dimensional. It's that easy. There's a exceptional continuous infinite of things that may live measured. Most of the pairs of observables reject to commute alongside each other. Some observables are (linear or nonlinear) functions or functionals of others but they're nevertheless linear operators as well as yesteryear combining many observables into complicated functions, you lot won't acquire whatsoever qualitatively novel clever things to measure. The laid of things that may live measured is clear from the beginning. You shouldn't overlook this woods because of trees.
For to a greater extent than than 25 years, the anti-quantum crackpots kept on writing papers occasionally referring to the "protective measurements" as well as similar things. They just didn't assist that this concept flatly cntradicted basic axioms of quantum mechanics. They couldn't figure out that the stair from perturbation theory that was claimed to live usable was genuinely non usable because the degenerate perturbation theory should receive got been used instead.
You may search e.g. through Google Scholar for 2018 papers mentioning the "protective measurements". In the listing of authors, you lot volition give away "who is who" inward the soil of anti-quantum zealots. Pusey, Leifer, Rovelli, Pussy, Cunt, Gao, as well as tons of others. Over 25 years, over 1,500 papers mentioned the "protective measurement[s]". You may verify that non a unmarried 1 manages to figure out that the whole concept is completely incorrect – that the emperor has no clothe – despite the fact that they receive got been doing this materials for to a greater extent than than 25 years.
These people are staggering morons. They are children who were left behind. This whole "field" is a plain of college dropouts who just couldn't acquire this basic undergraduate materials correctly. But they were able to persuade other morons that they're wonderfully clever, anyway.
P.S.: If the moving ridge part or expectation values could live measured inward this way, as well as so the "collapse of the moving ridge function" would travel a physical number that objectively changes the predictions for hereafter ("protective") measurements, as well as it would thing when it happens. The collapse next the firstly measuring of an entangled composite arrangement would indeed impact the results of some "protective measurements" of the moving ridge part instantaneously. That would live a conflict alongside special relativity. The prohibition of the "protective measurement" as well as other weird methods to stair out the moving ridge part every bit if it were an observable is indeed a necessary status for the consistency of quantum mechanics alongside locality (and thus special relativity).