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Several question on the Physics SE have mentioned a 2018 paper by D. Frauchiger and R. Renner with the title "Quantum theory cannot consistently describe the use of itself" (doi, arXiv). Slightly abridged abstract:

[...] [W]e propose a Gedankenexperiment to investigate the question whether quantum theory can, in principle, have universal validity. The idea is that, if the answer was yes, it must be possible to employ quantum theory to model complex systems that include agents who are themselves using quantum theory. Analysing the experiment under this presumption, we find that one agent, upon observing a particular measurement outcome, must conclude that another agent has predicted the opposite outcome with certainty. The agents' conclusions, although all derived within quantum theory, are thus inconsistent. This indicates that quantum theory cannot be extrapolated to complex systems, at least not in a straightforward manner.

The actual paper is difficult to follow because of the complexity of the argument and the idiosyncratic notation. Is it possible to obtain the same result from a simpler experiment, or at least the same experiment in simpler notation?

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Answering my own question: I'll rephrase the argument, or at least crucial parts of it, in the language of quantum computing.

The experiment

We start with a qubit prepared in the state $\sqrt\frac23\,|0\rangle + \sqrt\frac13\,|1\rangle$.

Alice (called $\bar{\mathrm F}$ in the paper, with F standing for "friend") measures the qubit in the computational basis. This measurement is supposed to happen in a perfectly insulated lab. I'll model it as the making of a large number of copies of the qubit, $a$ of them to be precise. The state is then

$$\sqrt{\tfrac23}\,|0^a\rangle + \sqrt{\tfrac13}\,|1^a\rangle.$$

$|0^a\rangle$ and $|1^a\rangle$ model, among other things, Alice's memory of the experimental outcome and any beliefs she forms as a consequence of it. Note that I'm assuming that Alice's measurement doesn't cause objective wave function collapse. F&R also make that assumption; see below.

Alice then prepares another qubit in the state $|{+}\rangle = \sqrt\frac12\big(|0\rangle+|1\rangle\big)$ if she measured $0$, or $|1\rangle$ if she measured $1$. She sends that qubit to Bob (called $\mathrm F$ in the paper), who measures it in the computational basis, in a lab that is perfectly insulated from Alice's and the world (aside from a channel used only to send the qubit). The state is then

$$\big( |0^a0^b\rangle + |0^a1^b\rangle + |1^a1^b\rangle \big) / \sqrt3.$$

Next, someone – who is called $\bar{\mathrm W}$ in the paper (W standing for Wigner), but who I'll call "the demon" for reasons that will be clear later – measures Alice's lab in a basis containing $\big\{ |{+}_a\rangle, |{-}_a\rangle \big\}$, where $|{\pm}_a\rangle \triangleq (|0^a\rangle \pm |1^a\rangle)/\sqrt2$, and we postselect on the outcome $|{-}_a\rangle$. Note that $|{\pm}_a\rangle \ne |{\pm}\rangle^{\otimes\,a}$, which is why I didn't use a superscript.

(That isn't actually how F&R describe that measurement. To give you a sense of what I had to wade through, their description of it is $π^{n:00}_{(\bar w,z)=(\overline{\mathrm{ok}},-\frac12)} = \big[(U^{00\rightarrow10}_{R\rightarrow\bar LS})^\dagger |\overline{\mathrm{ok}}\rangle_{\bar L}|{\downarrow}\rangle_S\big] \big[\cdot\big]^\dagger$. As best as I can tell from the accompanying text, this is supposed to be a Heisenberg-picture measurement of the state of Alice's first qubit at an earlier time before she sends the second prepared qubit to Bob. Such measurements are allowed in principle if there is only unitary evolution between the time of the measurement and the time effectively being measured. But that measurement can't be performed with access to Alice's lab only, since her lab is entangled with Bob's. My $\big\{ |{+}_a\rangle, |{-}_a\rangle \big\}$ measurement represents my best guess as to what they mean here, despite not entirely understanding their notation. It leads to a probability of $1/12$ for the halting condition, which is correct according to F&R, and it leads to a paradox like theirs, so it is "close enough" at least.)

The outcome of the demon's measurement is announced publicly, which means it is irreversible and can be modeled as a wavefunction collapse. After the measurement, if the postselection condition holds, the state is

$$|{-}_a0^b\rangle = \big( |0^a0^b\rangle - |1^a0^b\rangle \big)/\sqrt2.$$

Note that this contains a term, $|1^a0^b\rangle$, that wasn't in the state before. This term represents a world in which Alice measures $1$, and therefore prepares Bob's qubit in the state $|1\rangle$, but Bob measures $0$ – a clear contradiction.

This is followed by a measurement of Bob's lab in a basis containing $\big\{ |{+}_b\rangle, |{-}_b\rangle \big\}$, publicly announced and with postselection on $|{-}_b\rangle$. The halting condition I mentioned earlier is that both postselection conditions hold. F&R then spend a long time deriving a contradiction involving a dozen different logical inferences by their four agents, but as far as I can tell this is all unnecessary, since we already found a contradiction that is visible to all agents when the experiment ends, so I won't bother with it.

Three assumptions

F&R say that their no-go result implies that you must abandon at least one of three assumptions, called Q, S, and C. They also say that their no-go theorem is neutral with regard to which assumption must be abandoned, but that doesn't seem to be true at all.

  • Q is informally the assumption that "the agents can use quantum mechanics". This is a pure thought-experiment; it can't be conducted in the real world because the demon's measurements are physically impossible to perform. F&R assume QM is correct in their analysis. It makes little sense to me to suggest that the agents might need to use a non-quantum theory to model the outcome of an experiment that by construction is correctly modeled by quantum mechanics. But perhaps that's what the title of the paper is meant to imply.

    Formally, Q is supposed to be the assumption that you can learn information about the system at an earlier time by Heisenberg-picture measurements. As I said earlier, I don't think the demon's measurements are actually measurements of that kind, so it's not clear the conclusions F&R have their agents draw from Q are really valid (not because QM is wrong, but because they misapplied it).

  • S is the assumption that measurements have a definite outcome (i.e., there aren't many worlds). F&R's own analysis of the experiment, which is supposed to be definitive, violates S, since they say that the probability of the halting condition is $1/12$, while if Alice and Bob's measurements caused wave function collapse, it would be $1/4$. The inhabitants of the F&R universe can do experiments that show S is false (such as this very experiment), so it makes no sense for them to believe it.

  • C is the assumption that agents will never reach contradictory conclusions. The experiment above shows, and I'll show more simply below, that C is inconsistent with the existence of a being with the demon's measurement powers.

A simpler experiment

As far as I can tell, this simpler experiment has all of the interesting properties of the F&R experiment.

  • Alice prepares a qubit in the state $|0\rangle$, then measures it in the computational basis. The resulting state of her lab is $|0^a\rangle$, which includes her reaction to the result, which may be "well, obviously: I just prepared it in that state."

  • The demon measures Alice's lab in a basis containing $\big\{ |{+}_a\rangle, |{-}_a\rangle \big\}$, which leaves it in one of those states.

  • The demon then measures the lab in a basis containing $\big\{ |0^a\rangle, |1^a\rangle \big\}$, and if the result is $|0^a\rangle$ (50% chance), repeats both measurements until the lab's state is $|1^a\rangle$. That state includes Alice's reaction to measuring $1$, which may be "wait, that's impossible; is quantum mechanics wrong?"

In place of $|1^a\rangle$, you can use any vector normal to $|0^a\rangle$, or, with minor changes, any vector. The demon can therefore make Alice remember and believe anything at all. So I think F&R's paradox boils down to the observation that it's impossible to be sure of anything when faced with an adversary who can arbitrarily reprogram your brain.

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  • $\begingroup$ This is the clearest explanation of that paper I've ever seen, I hope this gets the attention it deserves! $\endgroup$
    – knzhou
    May 6 at 23:27
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    $\begingroup$ I came to pretty much the exact same conclusion at the end, as did Scott Aaronson: scottaaronson.blog/?p=3975. The real trick is in those crooked-basis measurements. $\endgroup$ May 7 at 0:53
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    $\begingroup$ @The_Sympathizer Oh, of course Scott Aaronson decoded the paper before me. I don't know why I didn't check his blog. The post title is a perfect summary, too. He doesn't go nearly as far as me in tearing it apart, though. He doesn't seem to think it was wrong to publish it in this form (comment #6), which I do, because there seem to be outright errors that were missed by the reviewers – particularly that the argument separately violates each named assumption in various ways, unless I missed something. $\endgroup$
    – benrg
    May 7 at 1:47
  • $\begingroup$ In your simpler experiment, why should Alice be surprised to measure $1$ when the lab is in the state $\lvert1^a\rangle$? In this state, wouldn't it be the case that Alice prepared the system in the state $\lvert1\rangle$? $\endgroup$
    – Sandejo
    May 10 at 19:11
  • $\begingroup$ @Sandejo I omitted certain things from the state vectors I wrote down, which in this experiment includes Alice's memory of the initial preparation. I agree it's confusing and I'll think about rewriting it. $\endgroup$
    – benrg
    May 11 at 19:43
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I see that the question doesn't really need an answer anymore, so here goes a comment :) There is an accessible review paper (which I co-authored, feel free to ask here as well) on thought experiments in quantum mechanics, including the one by Frauchiger and Renner, and the criticisms surrounding them. It is in open access on arXiv arXiv:2106.05314. The experiments are simplified to short protocols portrayed as quantum circuits. If you are confused by the original paper, I'd recommend to check it out!

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  • $\begingroup$ Hello highfae and welcome to Physics SE. I understand that you don' have enough reputation to write comments on other people's posts but please refrain from posting answers containing comments. This is not how StackExchange works and won't help if people do that in the long run. In time to come you'll acquire enough reputation and then you can come back to leave the comment with your information. $\endgroup$
    – ZaellixA
    May 12 at 16:41

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