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Frauchiger-Renner: trivial to see that QM has no contradictions

Click at the pirate icon above the title for a no-nonsense mobile version of this blog post.

Maken has pointed out the new paper

In Defense of a "Single-World" Interpretation of Quantum Mechanics
by Jeffrey Bub which negates a 2016 paper by Frauchiger and Renner (see a superficial comment at TRF or a PI journal club by barefoot and hot Lídia del Rio; she's the first girl from the obnoxiously PC Renner's political video).

Bub is right about the main claims – there is a single world, there is no contradiction, quantum mechanics is consistent etc. – and he presents a wonderfully concise explanation of the alleged Frauchiger-Renner paradox. But I am still dissatisfied with Bub's paper as well. He doesn't really address some incorrect formulations by Frauchiger and Renner – about "stories" etc. – and he adds some unfortunate new non-quantum sentences involving super-observers (quantum mechanics only has observers and all of them follow the same rules), measurements of wave functions (wave functions cannot be measured), and others.

But let me focus on the claims about the "contradiction within quantum mechanics" as summarized by Bub. The bulk of his explanation is one-page-long and I will make it even more concise. The Frauchiger-Renner experiment involves two qubits, with several bases on that 4-dimensional space, and several observers, and is framed as a hybrid of Hardy's paradox and Wigner's friend thought experiment.

But all these complications are almost completely irrelevant. OK, first, let's look at the experiment from the perspective of the most precise observers (sometimes misleadingly called super-observers), namely Wigner and Friend Ltd. They have the following state of the two qubits \(A,B\):\[

\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

\] These four terms are orthogonal to each other in the usual way. The states "fail" and "ok" for qubits \(A,B\) may also be written in terms of other orthogonal bases,\[

\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}}

\] where the upper sign holds for "fail" and the lower sign holds for "ok". In the first expression for \(\ket\psi\), you may substitute and replace either the "ok,fail" basis of \(B\) by the "0,1" basis, and/or you may replace the "ok,fail" basis of \(A\) by the "\(h,t\)" basis (head, tails) in some terms, to get two additional forms of \(\ket\psi\):\[

\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

\] I have verified the formulae for you – all four expressions for \(\ket\psi\) (the main one will be written soon) may be obtained from each other by switching in between the two bases for both qubits. In each of the three formulae for \(\ket\psi\) as a sum (and also in the fourth one below), the individual terms are orthogonal to each other.

Now, the experiment as seen by Wigner and Friend Ltd had some sub-experiments. Alice prepared the first qubit in the state\[

\sqrt{\frac{1}{3}} \ket{h}_A +\sqrt{\frac{2}{3}} \ket{t}_A

\] and measured the observable \(h\) vs \(t\). If she sees \(h\), she prepares the second qubit \(B\) in the state \(\ket {0}_B\), if she sees \(t\), she transforms it to \((\ket {0}_B + \ket {1}_B)/\sqrt{2}\) which is not orthogonal to the first one. This hard work by Alice makes it clear that the two-qubit state \(\ket\psi\) may also be written using the fourth expression\[

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

\] And Bob measures \(0\) or \(1\) on the qubit \(B\) afterwards. Alice, Bob, their two qubits, apparatuses etc. are observed by the very precise Wigner and Friend Ltd. observer who describe the state of everything (after Alice and Bob did their job) with the pure state \(\ket\psi\). Wigner and Friend Ltd. are the usual "super-observers" who don't study what Alice and Bob saw during the experiment – they only make their final measurements at the very end.

What is the contradiction? The contradiction is supposed to be that the first form of \(\ket \psi\) allows all four combinations of "ok" and "fail" for both qubits \(A,B\). In particular, the combination "ok,ok" is allowed with the probability \(1/12\). However, an argument based on the other two forms of \(\ket\psi\) implies that if the state is "ok,ok", then it has to be "\(h,1\)" according to the totally opposite basis – one which is natural from the viewpoint of Alice's and Bob's observations. And that's a contradiction because the combination "\(h,1\)" is forbidden: recall that if Alice sees \(h\), she prepares the other qubit in \(0\) which has no admixture of \(1\).

Now, you don't need to study any details to be immediately certain that the proof of the contradiction has to be wrong. If the state is "ok,ok", then you cannot possibly have a valid proof that the state is equal to "\(h,1\)" at the same moment – because these two states are different elements of the Hilbert space. To correctly prove that \(\ket{\rm ok,ok}=\ket{h,1}\) is as impossible as to prove that \(4=5\). You just can't do that.

So when you go through the "proof" that "ok,ok" has to "be" "\(h,1\)", you're guaranteed to find a mistake. OK, how do Frauchiger and Renner "prove" that "ok,ok" has to "be" "\(h,1\)"? Let's quote Bub:
From the second expression, the pair "ok,0" has zero probability, so "ok,1" is the only possible pair of values for the super-observables "\(X,B\)" if \(X\) has the value "ok".

From the third expression, the pair "\(t\),ok" has zero probability, so "\(h\),ok" is the only possible pair of values for the super-observables "\(A,Y\)" if \(Y\) has the value "ok".
LOL, that's funny. So in the "proof" that "ok,ok" is the same thing as "\(h,1\)", how did they change the two objects in the pair? Well, it's easy: they first changed the second "ok" to \(1\); and then they changed the first "ok" to "\(h\)". You could have guessed that. ;-) Clearly, both steps are completely analogous to one another and each of them is equally wrong.

So without a loss of generality, it's enough to look at one of these two analogous incorrect steps, for example the first argument:
From the second expression, the pair "ok,0" has zero probability, so "ok,1" is the only possible pair of values for the super-observables "\(X,B\)" if \(X\) has the value "ok".
Not so fast, comrades. ;-) Wigner and Friend have actually made a measurement – which had the probability \(1/12\) before it happened – whose outcome was "ok,ok". So is it true that the pair "ok,0" has zero probability, as claimed above? It's obviously not true. If their state is "ok,ok", the probability that it's found in the state "ok,0" is the squared inner product of these two vectors (the absolute value of it), namely \(1/2\), not zero! These two states aren't orthogonal to each other which means that if the system is brought to the first state, the probability that it's found in the second state is given by Born's rule! And be damn sure that it's not zero.

They ludicrously conclude that "ok,ok" "is" "ok,1" because "ok,0" has zero probability according to the second expression for \(\ket\psi\). But this reasoning is completely incorrect. If you have the two-qubit system in the state "ok,ok", you may easily check that "ok,0" doesn't have the advertised zero probability. It has the probability \(1/2\) as well – given by Born's rule. (Both bases for \(A\) and both bases for \(B\) are obtained by the usual rotations by 45° from one another.) The squared inner product of "ok,ok" and "ok,0" is equal to \(1/2\) which settles the question. The probability is certainly not zero as claimed.

OK, so what's wrong about their argument involving "the second expression"? You know the punchline that I will tell you now, don't you? The point is that the "second expression" is only relevant if a corresponding measurement is made by Wigner and Friend Ltd. So we started by assuming that Wigner and Friend Ltd. made a measurement to get one of the four states "ok/fail,ok/fail" included in the first expression. And they got "ok,ok".

They could have made another measurement of observables that would pick one of the two basis vectors in the second expression (or the third expression). They could have made this different measurement either instead of the measurement of "ok/fail,ok/fail", or after the measurement of "ok/fail,ok/fail". But that would have changed the experiment by Wigner and Friend Ltd. It would simply be a completely different situation. There doesn't need to be any "consistency" in the results because the histories in which Wigner and Friend Ltd. decide to measure different observables are different histories. That doesn't mean that all of them take place – like in some types of "many worlds". It's totally OK to assume that Wigner and Friend only do one of these measurements (or sequences of measurements).

So when they know the state to be "ok,ok", they must have measured it, and the probability that the state is "\(h,1\)" is surely nonzero after that measurement. But because the original \(\ket\psi\) is orthogonal to "\(h,1\)", the probability that the state was "\(h,1\)" was zero before that measurement by Wigner and Friend Ltd.! That agrees with Alice's and Bob's intuition, of course.

If and when Wigner et al. happen to measure "ok/fail,ok/fail" first and get "ok,ok", and then they make another measurement to get "\(h,1\)", one doesn't need to invent any "human story" about what Alice and Bob did to produce "\(h,1\)". There is no such story. The measurement producing "\(h,1\)" is just some dull measurement of two electrons' spins or something that is as simple as that. Those electrons' spins have been entangled with complicated "states of minds" of Alice and Bob before that measurement (the qubits were copied into many copies of classical bits by some entanglement), but the measurement of "ok/fail,ok/fail" had to pick some particular definitions of those qubits and the entanglement of these qubits with the Alice's and Bob's brains was broken. It had to be broken because entanglement is monogamous: the newer measurement created a new entanglement of the measured spins with the newly used apparatus, so the entanglement between the spins and Alice's and Bob's brains (i.e. with the "copies" of that bit) was "rewritten". Alice and Bob don't have to invent complicated explanations why their activity led to the forbidden "\(h,1\)", after all because it didn't: "\(h,1\)" is a consequence of a measurement done by Wigner and Friend Ltd., not an achievement of Alice and Bob. That later measurement by Wigner et al. changed the state and the previous adventures of Alice and Bob became largely irrelevant when that later measurement was made.

It's that simple. So the whole point that Frauchiger and Renner overlook is that the measurement(s) made by Wigner and Friend Ltd. change the state of the two qubits. Frauchiger and Renner simply err in this trivial point. It's still the same point that Richard Feynman stressed to his undergraduate students about 30 times. It matters whether you make the measurement or not! The measurement whose result isn't 100% guaranteed a priori always changes the state of the system. The measurement changes the subjective knowledge from the observer's viewpoint – it collapses the wave function; but even from the viewpoint of other humans and potential observers, a measurement has to affect the measured object object and the course of the experiment is changed physically – i.e. according to these additional, external observers – too. Measurements always affect the course of the experiment although the description by the true observer who learns about the outcome (collapse) is different than the description by another observer who doesn't learn it and who is more precise (in his picture, entanglement is created between the human who measures and the measured object).

Again, in \(1/12\) of the cases, the measurement of "ok/fail,ok/fail" by Wigner and Friend Ltd. changed the state \(\ket\psi\) – which was orthogonal to "\(h,1\)" and therefore had a vanishing probability to be found in "\(h,1\)" – to the state "ok,ok" – which is not orthogonal to "\(h,1\)" and has the probability equal to \(1/4=1/2\times 1/2\) to be found in "\(h,1\)".

To make their mistake sound really simple, Frauchiger and Renner assume their initial state to be \(\ket\psi\) and \(\ket{\rm ok,ok}\) at the same moment. But this assumption is simply inconsistent – the situation in which both things are precisely true at the same moment cannot occur. If the system is observed in the state \(\ket\psi\), then it cannot be assumed to be in \(\ket{\rm ok,ok}\), and vice versa!

Equivalently, they're just being sloppy about the key point whether a measurement has taken place or not. But that's too serious and elementary a mistake.

What happens is the standard trivial good old change of the state (or a "collapse" associated with the measurement) in quantum mechanics. The state was changed when a measurement was made. One can find more trivial examples of the same thing. Measure the electron's spin \(s_z\) and then \(s_z\) again. You get the same result. If you insert a measurement of \(s_x\) in between, the initial and final measurements of \(s_z\) don't have to agree. Whether the measurement of \(s_x\) is inserted in the middle matters for the subsequent predictions. It's always the case in quantum mechanics.

Claims that thought experiments like that may prove many worlds or find any contradiction within the orthodox quantum mechanics – a theory articulated in Copenhagen – are just proofs of the writers' complete idiocy. Idiocy of writers who keep on producing ever more contrived but still fundamentally wrong papers with simple experiments involving several qubits and trivial mistakes – Renners, Frauchigers, Puseys, Pussies (that's the only one that was okayed by my spellchecker), Barretts, Rudolphs, Maudlins, Deutsches, and hundreds of other idiots. There obviously can't be any contradiction like that. A contradiction between two measurements would only occur if the two outcomes contradicted each other. But they would contradict each other only if the two post-measurement states were orthogonal to each other. But when they're orthogonal to each other, quantum mechanics predicts the probability to be zero by Born's rule!

Quantum mechanics guarantees that there can't ever be any similar contradiction. It guarantees that all Frauchiger-Renner papers are just plain moronic. It guarantees it by a simple fact: a contradiction needs the probability of the arrangement of outcomes to be zero. But the arrangement has zero probability, it means that the theory predicts that this arrangement never occurs. A nonzero number cannot be zero at the same moment! That's it.

When you bring your physical system into the state \(\ket\psi\) by a measurement, there is a nonzero probability to find it in the state \(\ket\gamma\) – any state that isn't orthogonal to \(\ket\psi\) – after the subsequent complete set of measurements. The probability is simply \(|\bra\psi\gamma\rangle|^2\) i.e. given by Born's rule. So as long as the states have a nonzero inner product, these two outcomes cannot be said to contradict each other.

However, that doesn't mean that you may replace the initial state \(\ket\psi\) with any state, e.g. \(\ket{\rm ok,ok}\), that isn't orthogonal to \(\ket\psi\). Even though these two states wouldn't contradict one another as results of two different measurements that follow each other, they are different assumptions about the initial state and lead to different predictions for subsequent measurements. In other words, the initial state \(\ket\psi\) prohibits different outcomes of subsequent measurements than the initial state \(\ket{\rm ok,ok}\). It's common sense. Different assumptions produce different predictions; different vectors have different sets of vectors that are orthogonal to them. You need to be precise about the initial state if you want correct predictions – but quantum mechanics always gives you just probabilistic predictions for those subsequent measurements i.e. Nature refuses to be precise in the future despite the precision it demands from you when it comes to the initial state.

(In classical physics, probabilistic mixtures of "pure states" – probabilistic distributions \(\rho(x_i,p_j)\) on the phase space with an extended support – always allow more (are compatible with a higher number of outcomes in the future) than a pure state in the mixture allows separately. But in quantum mechanics, different pure states aren't probabilistic mixtures of each other. For example, states \(\ket\psi\) and \(\ket{\rm ok}\ket{\rm ok}\) have a nonzero overlap, are equally pure, their role is symmetric, and each allows some outcomes that the other forbids. This is probably another trivial fact that the anti-quantum zealot deliberately fail to realize – or intentionally deny.)

None of these things has anything to do with many worlds or other buzzwords of the anti-quantum zealots. Every proof that this implies the existence of many worlds, dangerous global warming, discrimination of women, or Second Coming of David Bohm or Hugh Everett is just silly. And the suggestion that what Wigner and Friend are doing requires some incredible futuristic technology is also wrong. Wigner and Friend are just some guys – perhaps the sponsors of Alice and Bob – who just give tasks to Alice and Bob and who measure the state of two employees and/or two electrons' spins. There's nothing mysterious about it. The only thing that precise observers like Wigner and Friend could be able to do – which Alice and Bob can't – is to predict the interference between microstates of Alice's and Bob's brains (because Alice and Bob treat their brains, or at least the center of their consciousness, classically – that's what makes them conscious observers). But no such "brain inference" is needed to discuss the setup of Frauchiger and Renner.

Bonus: (non-existent) paradox assuming that "ok/fail,ok/fail" is never actually measured by Wigner and Friend

If they never assumed that "ok/fail,ok/fail" is measured and they always took the superposition \(\ket\psi\) as the foundation of their reasoning, it would be correct to say "ok,1" is the only possible result of "ok/fail,0/1" measurement that has the form "ok,0/1"; and it would be correct to say that "\(h\),ok" is the only possible outcome of the "\(h/t\),ok/fail" measurement that has the form "\(h/t\),ok".

But these statements are statements about two possible complete sets of measurements that we may make: either about the "ok/fail,0/1" measurement (which makes the second expression for \(\ket\psi\) relevant) or about the "\(h/t\),ok/fail" measurement (which makes the third expression for \(\ket\psi\) relevant).

However, the problem with the logic would then be that these two arguments can't be applied in a single derivation because the measurements of "ok/fail,0/1" and "\(h/t\),ok/fail" can't be done at the same moment – because the corresponding observables don't commute. They don't commute for a simple reason: the bases of four eigenstates are different.

That's why, in this precise understanding of the paradox, the problem is that the two parts of the reasoning
From the second expression, the pair "ok,0" has zero probability, so "ok,1" is the only possible pair of values for the super-observables "\(X,B\)" if \(X\) has the value "ok".

From the third expression, the pair "\(t\),ok" has zero probability, so "\(h\),ok" is the only possible pair of values for the super-observables "\(A,Y\)" if \(Y\) has the value "ok".
simply violate the uncertainty principle. Using these two arguments simultaneously incorrectly assumes that two non-commuting measurements may be made simultaneously! If quantum mechanics is properly applied, the first part of the quoted argument above is only relevant in some situation, for one choice of the first complete measurement done by Wigner and Friend; and the second part above is relevant for another choice of their first complete measurement. These two parts of the arguments can never be simultaneously relevant in any derivation because they make mutually exclusive assumptions about what Wigner and Friend decide to measure first!

We may say that their argument based on the two implications exploiting the "second expression" and the "third expression" is a proposed method to measure "ok/fail,ok/fail" by a combination of two measurements of "fail-0/\(t\)-1/#/#" and "\(h\)-0/\(t\)-fail/#/#" where, in both cases, #/# are two extra vectors needed to complete an orthonormal basis. The strategy is to extract the information about the first "ok/fail" from one measurement and about the second "ok/fail" from the other measurement. However, these two measurements can't be done simultaneously because of the (good old, usual, and standard) uncertainty principle – the operators don't commute. So the measurement of "ok/fail,ok/fail" is distinct from (and cannot be replaced by) any combination of other measurements such as "ok/fail,0/1" and "\(h/t\),ok/fail" or the two even more rotated measurements that exploit the second and third expression for \(\ket\psi\). This fact is completely analogous to the fact that you can't reduce the measurement of the angular momentum \(\vec L\) to the measurements of \(\vec x\) and \(\vec p\) – which don't commute – despite the fact that \(\vec L = \vec x \times \vec p\).

(Note that Alice and Bob are together making a measurement of "\(h/t,0/1\)" which also doesn't commute with the later measurements by Wigner and Friend, e.g. the "ok/fail,ok/fail" measurement. But that's OK because these measurements aren't pretended to be made at the same moment. They're done after each other. Alice and Bob simply prepare the state \(\ket\psi\) which produces probabilities \(1/3,1/3,1/3,0\) for the options "\(h/t\),0/1" where "\(h,1\)" is the forbidden one; and probabilities \(1/12,1/12,1/12,1/3\) for the later measurement of "ok/fail,ok/fail" where "fail,fail" is the most likely outcome.)

At any rate, whatever is the exact way how you interpret their claim about a "paradox" that they have found, their derivation of the paradox violates some really basic rules of quantum mechanics that should be understood by undergrads who have made it through a good enough course. They're sloppy about the question whether the complete measurement of "ok/fail,ok/fail" or "ok/fail,0/1" or "\(h/t\),ok/fail" was made as the first measurement by Wigner and Friend (which violates Feynman's point that it matters whether a measurement is made because it affects the situation); equivalently, they assume that these complete measurements may be assumed to be done simultaneously and/or the corresponding observables have simultaneously well-defined values (which contradicts Heisenberg's uncertainty principle because the observables don't commute).

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