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Frauchiger-Renner: QM is inconsistent with two Wigners and their friends

David Thornton has asked about a new paper that basically claims that quantum mechanics is inconsistent,

Quantum theory cannot consistently describe the use of itself (Nature).
Note that Renner has arguably done some non-rubbish work in the quantum information theory but as explained in an unbelievable video, he also employs a group of women who brag to be f*cking 16 hours a day, going from one pregnancy to another, and being paid as "physicists" – from some European taxpayers' money – for allowing their names to be used in some ludicrous papers about the "quantum foundations".

In April, I discussed one of these crackpot papers in which Renner and Frauchiger asserted that quantum mechanics required many worlds. They used a straightforward physical system of 2 qubits – and several bases of their 4-dimensional Hilbert space.

You're invited to click at the hyperlink in the previous paragraph and try to recall some of the equations. Why? Because the new paper in Nature uses exactly the same states and bases in the 4-dimensional Hilbert space. Just the conclusion is different this time: Instead of "proving" many worlds, they claim to "prove" a contradiction of quantum mechanics in the presence of people's thinking about the other people's thinking.

Needless to say, no such contradiction exists and it's a crackpot paper. What's going on? Let us repeat some mathematics. Alice A, renamed as \(\bar F\) (Wigner's f*ck toy in the bar) in the new paper, prepares two qubits in the state\[

\ket\psi = \frac{ \ket{h}_A\ket{0}_B+\ket{t}_A\ket{0}_B+\ket{t}_A\ket{1}_B }{\sqrt{3}}

\] Now Bob, renamed as \(F\) (Wigner's f*ck toy outside the bar) in the new paper, measures the second qubit, whether it's zero or one. Two more people, \(W\) and \(\bar W\), measure the labs \(L\) or \(\bar L\), respectively, which contain the people \(F\) and \(\bar F\), respectively. And they make various bases relevant for their measurements.

One of them makes the basis "ok/fail,0/1" relevant; the other makes "h/t,ok/fail" relevant. For the two qubits, ok/fail states are defined as the simple "sums" or "differences"\[

\ket{\rm fail,ok}_A &= \frac{\ket {h}_A\pm \ket {t}_A}{\sqrt{2}}\\
\ket{\rm fail,ok}_B &= \frac{\ket {0}_B\pm \ket {1}_B}{\sqrt{2}}

\] We know all the relationships between the bases to write three more forms for \(\ket\psi\) defined above:\[

\ket\psi &= \sqrt{\frac{1}{12}} \ket{\rm ok}_A \ket{\rm ok}_B
- \sqrt{\frac{1}{12}} \ket{\rm ok}_A \ket{\rm fail}_B+\\
&+\sqrt{\frac{1}{12}} \ket{\rm fail}_A \ket{\rm ok}_B
+ \sqrt{\frac{3}{4}} \ket{\rm fail}_A \ket{\rm fail}_B

\] and \[

\ket\psi &= \sqrt{\frac{2}{3}} \ket{\rm fail}_A \ket{0}_B
+ \sqrt{\frac{1}{3}} \ket{t}_A \ket{1}_B+\\
&=\sqrt{\frac{1}{3}} \ket{h}_A \ket{0}_B
+ \sqrt{\frac{2}{3}} \ket{t}_A \ket{\rm fail}_B .

\] Great. The mathematical content of the "paradox" remains exactly the same as in their older paper. But now all the transitions from the bases are reframed as "thinking of four people" who apply some "what do you care what other people think" methodology.

Mathematically, the "paradox" is claimed to arise because in the "ok/fail,ok/fail" basis, the result "ok/ok" has the probability \(P=1/12\) to occur assuming the same initial state \(\ket\psi\). On the other hand, in the bases privatized by the people \(W\) and \(\bar W\) and assuming that "ok,ok" was measured as the right state from the "ok/fail,ok/fail" basis, one may simultaneously argue that the state has become "h,1" because one of these people has ruled out "ok,0" (and therefore proved "ok,1") and the other has ruled out "t,ok" (and therefore proved "h,ok").

Just like in the previous paper, they basically claim that the state is simultaneously in \(\ket\psi\) and in "ok,ok". But that's not possible because they are two different states. Every observer, and it is true for their four people as well as the remaining 7.5 billion people on Earth, and also for all the non-human observers if there are any, has to measure something about the initial state. Depending on what the given person has learned, the state he uses to predict further measurements may be either \(\ket\psi\) or "ok,ok" – or something else – but not both because these two states are damn different. They yield different predictions. One of the states \(\ket\psi\) allows some outcomes forbidden by "ok,ok", and vice versa. Their "inequality" is a symmetric relationship and there is no way to treat one of these states as a "refinement" of the other state.

So if the state of the two labs \(L,\bar L\) is measured first, "ok/ok" has the probability \(P=1/12\) which is nonzero while "h,1" has the probability zero. On the other hand, if the state is first measured relatively to the "ok/fail,ok/fail" basis, we may get "ok,ok", and "h,1" becomes a possible outcome of the measurement afterwards. If a measurement is performed, the state is changed.

Whether a measurement was performed is a subjective question – the measurement is an act by which an observer (and there may be unequivalent ones) learns about the value of an observable. Does this subjectiveness allow some contradiction, assuming that we allow the people to think how other people think?

Not really because a measurement done by observer O1 has obvious consequences from O1's point of view (the collapse of O1's wave function) but that doesn't mean that it doesn't have consequences from the other observers' viewpoint. The other observers O2,O3,O4 describe the measurement differently – as an interaction of O1's atoms with external objects that creates some entanglement between her brain and the external objects – but even from their viewpoint, the presence or absence of the measurement still changes the experimental situation.

So according to all observers, the "h,1" outcome is impossible at the beginning, after the state \(\ket\psi\) is prepared, but if a measurement of "ok/fail,ok/fail" is first performed, by "oneself" or "another human", this changes the situation and "h,1" becomes possible.

In both papers, Renner and Frauchiger constantly assume that the presence of a measurement does not change anything, at least from some observers' viewpoint, but that's just rubbish and that's the cause of all their difficulties with quantum mechanics.

P.S.: They formulate their wrong theorem as a proof that at least one of the assumptions Q,S,C must fail. The assumptions basically say:
Q: When quantum mechanics allows an observer to calculate the probability \(P=1\) of an outcome, he may say the proposition with certainty.

S: When something is measured \(L=\lambda_1\), one may say that \(L\neq \lambda_1\) is impossible.

C: If an observer O1 may determine that the observer O2, by using quantum mechanics, may determine she is certain about something, then O1 may be certain about it, too.
Now, Q is some "validity of quantum mechanics", so it's violated by some (four, in their table!) theories that really disagree with the predictions of quantum mechanics. S seems like basic logic but they frame it as the "single outcome" axiom, so it's violated in the many worlds picture (contradictory outcomes are all true – somewhere).

And it's "C" (compassion?) that is said to be violated by the "Copenhagen Interpretation". And that's the claim they adapted for the title. But it's simply not true that the axiom "C", as they have written it down, is invalidated by proper (Copenhagen) quantum mechanics. If you can legitimately prove that another observer may make a conclusion to be certain, given assumptions that you also know to be correct, then you may make the very same conclusion yourself.

Their invalid "proof" that the "Copenhagen Interpretation" requires to abandon "C" boils down to their incorrect assumption that it doesn't matter, from some observers' viewpoints, whether an observable was measured (by someone else). But the measurement of a quantity whose outcome isn't certain at the moment always changes the situation – and it changes the situation from all observers' viewpoint.

People who still try to prove an inconsistency of quantum mechanics in 2018 are cretins.

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