If your interpretation of quantum mechanics has a single world but no collapse, you have a problem

To inaugurate this blog I want to talk about Daniela Frauchiger and Renato Renner’s polemical new paper, Single-world interpretations of quantum theory cannot be self-consistent. Since lots of people want to understand what the paper is saying, but do not want to go through its rather formal language, I thought it would be useful to present the argument here in a more friendly way.

To put the paper in context, it is better to first go through a bit of history.

Understanding unitary quantum mechanics is tough. The first serious attempt to do it only came in 1957, when Everett proposed the Many-Worlds interpretation. The mainstream position within the physics community was not to try to understand unitary quantum mechanics, but to modify it, through some ill-defined collapse rule, and some ill-defined prohibition against describing humans with quantum mechanics. But this solution has fallen out of favour nowadays, as experiments show that larger and larger physical systems do obey quantum mechanics, and very few people believe that collapse is a physical process. The most widely accepted interpretations nowadays postulate that the dynamics are fundamentally unitary, and that collapse only happens in the mind of the observer.

But this seems a weird position to be in, to assume the same dynamics as Many-Worlds, but to postulate that there is anyway a single world. You are bound to get into trouble. What sort of trouble is that? This is the question that the paper explores.

That you do get into trouble was first shown by Deutsch in his 1985 paper Quantum theory as a universal physical theory, where he presents a much improved version of Wigner’s friend gedankenexperiment (if you want to read something truly insane, take a look at Wigner’s original version). It goes like this:

Wigner is outside a perfectly isolated laboratory, and inside it there is a friend who is going to make a measurement on a qubit. Their initial state is

\[ \ket{\text{Wigner}}\ket{\text{friend}}\frac{\ket{0}+\ket{1}}{\sqrt2} \]

After the friend does his measurement, their state becomes

\[ \ket{\text{Wigner}}\frac{\ket{\text{friend}_0}\ket{0} + \ket{\text{friend}_1}\ket{1}}{\sqrt2} \]

At this point, the friend writes a note certifying that he has indeed done the measurement, but without revealing which outcome he has seen. The state becomes

\[ \ket{\text{Wigner}}\frac{\ket{\text{friend}_0}\ket{0} + \ket{\text{friend}_1}\ket{1}}{\sqrt2}\ket{\text{I did the measurement}} \]

Now Wigner undoes his friend’s measurement and applies a Hadamard on the qubit (i.e., rotates them to the Bell basis), mapping the state to

\[ \ket{\text{Wigner}}\ket{\text{friend}}\ket{0}\ket{\text{I did the measurement}} \]

Finally, Wigner and his friend can meet and discuss what they will get if they measure the qubit in the computational basis. Believing in Many-Worlds, Wigner says that they will see the result 0 with certainty. The friend is confused. His memory was erased by Wigner, and the only thing he has is this note in his own handwriting saying that he has definitely done the measurement. Believing in a single world, he deduces he was either in the state $\ket{\text{friend}_0}\ket{0}$ or $\ket{\text{friend}_1}\ket{1}$, and therefore that the qubit, after Wigner’s manipulations, is either in the state $\frac{\ket{0}+\ket{1}}{\sqrt2}$ or $\frac{\ket{0}-\ket{1}}{\sqrt2}$, and that the result of the measurement will be either 0 or 1 with equal probability.

So we have a contradiction, but not a very satisfactory one, as there isn’t an outcome that, if obtained, falsifies the single world theory (Many-Worlds, on the other hand, is falsified if the outcome is 1). The best one can do is repeat the experiment many times and say something like: I obtained N zeroes in a row, which means that the probability that Many-Worlds is correct is $1/(1+2^{-N})$, and the probability that the single world theory is correct is $1/(1+2^{N})$.

Can we strengthen this contradiction? This is one of the things Frauchiger and Renner want to do. Luckily, this strengthening can be done without going through their full argument, as a simpler scenario suffices.

Consider now two experimenters, Alice and Bob, that are perfectly isolated from each other but for a single qubit that both can access. The state of everyone starts as

\[ \ket{\text{Alice}}\frac{\ket{0}+\ket{1}}{\sqrt2}\ket{\text{Bob}} \]

and Alice makes a first measurement on the qubit, mapping the state to

\[ \frac{\ket{\text{Alice}_0}\ket{0}+\ket{\text{Alice}_1}\ket{1}}{\sqrt2}\ket{\text{Bob}} \]

Now focus on one of Alice’s copies, say Alice$_0$. If she believes in a single world, she believes that Bob will definitely see outcome 0 as well. But from Bob’s point of view both outcomes are still possible. If he goes on to do the experiment and sees outcome 1 it is over, the single world theory is falsified.

This argument has the obvious disadvantage of not being testable, as Alice$_0$ and Bob$_1$ will never meet, and therefore nobody will see the contradiction. Still, I find it an uncomfortable contradiction to have, even if hidden from view. And as far as I understand, this is all that Frauchiger and Renner have to say against Bohmian mechanics.

The full version of their argument is necessary to argue against a deeply personalistic single-world interpretation, where one would only demand a single world to exist for themselves, and allow everyone else to be in Many-Worlds. This would correspond to taking the point of view of Wigner in the first gedankenexperiment, or the point of view of Alice$_0$ in the second. As far as I’m aware nobody actually defends such an interpretation, but it does look similar to QBism to me.

To the argument, then. Their scenario is a double Wigner’s friend where we have two friends, F1 and F2, and two wigners, A and W. The gedankenexperiment starts with a quantum coin in a biased superposition of heads and tails:

\[ \frac1{\sqrt3}\ket{h} + \sqrt{\frac23}\ket{t} \]

At time t=0:10 F1 measures the coin in the computational basis, mapping the state to

\[ \frac1{\sqrt3}\ket{h}\ket{F1_h} + \sqrt{\frac23}\ket{t}\ket{F1_t} \]

To avoid clutter, I will redefine the degrees of freedom of this coin to be part of F1’s degrees of freedom, and write simply

\[ \frac1{\sqrt3}\ket{F1_h} + \sqrt{\frac23}\ket{F1_t} \]

Now, F1 prepares a qubit in the state $\ket{0}$ if she saw heads, or the state $\ket{+}$ if she saw tails, mapping the state to

\[ \frac1{\sqrt3}\ket{F1_h}\ket{0} + \sqrt{\frac23}\ket{F1_t}\ket{+} \]

F1 sends this qubit to F2, who measures it in the computational basis at time t=0:20, mapping the state to (I’m writing the state of the qubit inside F2’s state to avoid clutter, as before)

\[ \frac1{\sqrt3}\ket{F1_h}\ket{F2_0} + \frac1{\sqrt3}\ket{F1_t}\ket{F2_0} + \frac1{\sqrt3}\ket{F1_t}\ket{F2_1}\]

Note that the term in the superposition that has $\ket{F2_1}$ has also $\ket{F1_t}$, and no other term in the superposition has $\ket{F2_1}$. Based on that, F2 reasons: If there is a copy of F2 that sees 1 at t=0:20, there must be a copy of F1 that saw tail at t=0:10.

F1, on her side, knows that this is happening, and furthermore she knows that W will at time t=0:40 measure F2 in the basis \[\{\ket{F2_+},\ket{F2_-}\} = \left\{\frac{\ket{F2_0} + \ket{F2_1}}{\sqrt2},\frac{\ket{F2_0} – \ket{F2_1}}{\sqrt2}\right\}\]
On that basis, the current state reads
\[ \frac1{\sqrt6}\ket{F1_h}\ket{F2_+} + \frac1{\sqrt6}\ket{F1_h}\ket{F2_-} + \sqrt{\frac23}\ket{F1_t}\ket{F2_+} \]
Based on this, F1 reasons: If there is a copy of F1 that sees tail at t=0:10, there must be a copy of W that sees $\color{red} F2_+$ at t=0:40.

Now it is A’s turn to act. She measures at time t=0:30 the state of F1 in the basis
\[\{\ket{F1_+},\ket{F1_-}\} = \left\{\frac{\ket{F1_h} + \ket{F1_t}}{\sqrt2},\frac{\ket{F1_h} – \ket{F1_t}}{\sqrt2}\right\}\]
In this basis (and the computational basis for F2) the state of F1 and F2 is
\[ \frac2{\sqrt6}\ket{F1_+}\ket{F2_0} + \frac1{\sqrt6}\ket{F1_+}\ket{F2_1} – \frac1{\sqrt6}\ket{F1_-}\ket{F2_1} \]
Based on this state, A reasons: If there is a copy of A that sees – at t=0:30, there must be a copy of F2 that saw 1 at t=0:20.

Finally, Wigner acts. He measures at time t=0:40 the state of A and the state of F2, both in the superposition basis. The full state is, before the measurement (with F1 included within A)

\[ \frac3{\sqrt12}\ket{A_+}\ket{F2_+} + \frac1{\sqrt12}\ket{A_-}\ket{F2_+} + \frac1{\sqrt12}\ket{A_+}\ket{F2_-} – \frac1{\sqrt12}\ket{A_-}\ket{F2_-} \]
so Wigner obtains the result $A_-,F2_-$ with probability $1/12$. Let us assume he did so (otherwise we just repeat the experiment a few times, and this result will happen with exponentially high probability). Furthermore, we assume that Wigner is the superobserver for whom there is a single world, so there is only one copy of him. Collecting this statement together with the statements in red, we have:

  1. The only copy of Wigner sees $A_-,F2_-$ at t=0:40.
  2. If there is a copy of A that sees – at t=0:30, there must be a copy of F2 that saw 1 at t=0:20.
  3. If there is a copy of F2 that sees 1 at t=0:20, there must be a copy of F1 that saw tail at t=0:10.
  4. If there is a copy of F1 that sees tail at t=0:10, there must be a copy of W that sees $F2_+$ at t=0:40.

Following the chain of implications, we have
The only copy of Wigner sees $A_-,F2_-$ at t=0:40.
There is a copy of A that saw – at t=0:30.
There is a copy of F2 that saw 1 at t=0:20.
There is a copy of F1 that saw tail at t=0:10.
There is a copy of W that sees $F2_+$ at t=0:40.

What should we conclude from this? Is this kind of reasoning valid? The discussions about this paper that I have witnessed have focussed on two questions: Are the red statements even valid, in isolation? Assuming that they are valid, is it legitimate to combine them in this way?

Instead of giving my own opinion, I’d like to state what different interpretations make of this argument.

Collapse models: I told you so.

Copenhagen (old style): Results of measurements must be described classically. If you try to describe them with quantum states you get nonsense.

Copenhagen (new style): There exist no facts of the world per se, there exist facts only relative to observers. It is meaningless to compare facts relative to different observers.

QBism: A measurement result is a personal experience of the agent who made the measurement. An agent can not use quantum mechanics to talk about another agent’s personal experience.

Bohmian mechanics: I don’t actually know what Bohmians make of this. But since Bohmians know about the surrealism of their trajectories, know that “empty” waves have an effect on the “real” waves, know that their solution to the measurement problem is no better than Many-Worlds’, and still find Bohmian mechanics compelling, I guess they will keep finding it compelling no matter what. In this point, I agree with Deutsch: pilot-wave theories are parallel-universes theories in a state of chronic denial.

What do you think?

Update: Rewrote the history paragraph, as it was just wrong. Thanks for Harvey Brown for pointing that out.
Update 2: Changed QBist statement to more accurately reflect the QBist’s point of view.

10 thoughts on “If your interpretation of quantum mechanics has a single world but no collapse, you have a problem

  1. Congratulations on your first post! I hope that more will follow.

    The red statements are indeed the more debatable parts of the arguments; it’s nice of you to highlight them :). Each of them, taken in isolation, seem like a perfectly fine statement about the global state of the system consisting of all observers. However, these statements apply at three different times, and I find it strange to combine them. If measurement is a physical process, why should we be allowed to conclude that there is a contradiction, when we are combining instantaneous facts about the (evolving) state of the system at three different times? It seems that no reasonable interpretation would actually follow this line of reasoning, although I am quite ignorant about Bohmian mechanics.

  2. Thanks for the comment!

    Strictly speaking you are correct, but I think one can perfectly well combine statements about the system at different times if one checks that they remain true under the specified time evolution (as trivially there will always be a time evolution that makes them false). And Frauchiger and Renner didn’t actually check that they remain true, so this is something they should do, at least in the Many-Worlds interpretation.

    But more generally I think that Copenhagen’s blanket ban on combining statements is just obscurantism. About Bohmian mechanics, I just don’t know. I wish I could find a Bohmian to tell me what they think about this, I’m genuinely curious.

  3. As an author of the paper, let me clarify that the statements we make in the paper *do* include the time when the values are observed.

    For example, the first highlighted statement is actually, in its full version:

    “If there is a copy of F2 that sees 1 at time t=0:20, there must be a copy of F1 that saw tail at time t=0:10.”

    Similarly, the complete version of the second one starts with: “If there is a copy of F1 that sees tail at time t=0:10, …”

    I guess that Mateus just shortened them to simplify the presentation. But indeed, without the timing information there would be some ambiguity.

  4. Thanks for chiming in, Renato. I had replaced the exact time with the verbal tenses “saw” and “sees” as only the ordering is important, not the exact times. But that was clearly a bad idea, since people are getting confused by it, so I updated the post to include the time information.

    But my point was that you did not mention the time evolution when making the statements. Since the evolution is trivial in statements 1,2, and 3, this needs to be done only for statement 4, that is, one needs to check that the unitary that is applied to F1 at time t=0:30 does not affect his prediction about what W will observe. It clearly doesn’t, at least in the Many-Worlds interpretation, but some crazy interpretations might say it does, I don’t know.

  5. Thanks for the clarifying reply.

    I have one other comment, which concerns your analysis of the Deutsch version of the Wigner’s friend gedankenexperiment. As you are writing, the contradiction one obtains from it is not a very satisfactory one, because it is only probabilistic. I would like to point to another important reason why this gedankenexperiment is not sufficient to rule out single-world theories in general.

    The reason is that the analysis you are describing relies on “self-referencing”, i.e., it assumes that the friend can reason about his own quantum state. Such self-referencing (also known as “self-measurement”, see Section 4 of http://plato.stanford.edu/entries/qm-relational/) is explicitly forbidden by many interpretations of quantum mechanics (notably interpretations according to which a system’s state is defined only relative to an observer, e.g., QBism or Rovellis’ “Relational Quantum Mechanics”). Other interpretations (e.g., “new style Copenhagen”) do not forbid it explicitly, but are ambiguous about how one should deal with it.

    These considerations forced us to “extend” Wigner’s Friend experiment to two friends (in the way described in our preprint). As one can easily verify, its analysis does not involve any self-referencing (i.e., each experimenter can make their predictions without reasoning about the quantum state of theirself).

  6. I’m aware that they forbid it, but I don’t think this prohibition is defensible: it is just obscurantism.

    But I get your point: by avoiding it you make your argument hit even people who take this prohibition seriously, so in this sense it does strengthen Deutsch’s argument.

  7. Dear Mateus: thanks for the very clear and nice blog. I am a SUAC type of scientist (and a chemist by background) so I probably miss some subtleties. But in the first experiment I would guess that the paradox is solved de facto by making impossible to perform a Hadamard gate in the state of the friend without touching at the same time the state of the note. If I am not mistaken, following decoherence, it is not possible to perform coherent manipulations on a macro state that moreover has left some imprints on the environment, like writing a note. Even if I do not write what I saw, it is really a different me that emerges depending on the outcome of the first experiment.

  8. Thanks =)

    Performing a Hadamard in the state of the friend without touching the note is no more difficult than doing the Hadamard in the state of the friend in the first place. Both require an absurd level of control over extremely complex quantum systems (and the note can be replaced by a qubit).

    This argument does require the assumption that it is in principle possible to such an experiment, or some analogous one (for example with a quantum computer taking the place of the friend). Denying this assumption is believing that the proper description is some sort of collapse model, and this is a very unpopular choice.

  9. Mateus: Thank you for the reply.

    As I see it, accepting that one can erase the memory of an observer (which basically implies there was no increase of entropy and no psychological time and lots of confusion) makes the whole idea of a contradiction no really that powerful or surprising. There is no contradiction for the external observer and the internal observer did not really work as a reliable observable – his brain was manipulated from the outside and he is in a state of total confusion.

    But on the other hand, if “collapse” theories are not considered appropriate (or popular), how would one explain the final result if no Hadamard is performed and Wigner measures a single outcome that a priori had 50% of probabilities?

    (I apologize if I reach the “dull” or annoying level. I understand we all have to do other stuff.)

  10. If you wish the friend can make his prediction before Wigner erases his memory: regardless of the outcome he observed, he will predict that both final outcomes will happen with the same probability, and he will be proven wrong. He can also write down this prediction on the piece of paper, that will not be erased.

    Well, your second question is exactly what this whole discussion is about: if you assume that there is no collapse, can you explain the gedankenexperiment with anything other than Many-Worlds?

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