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:

- The only copy of Wigner sees $A_-,F2_-$ at t=0:40.
- 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.
- 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.
- 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.*

Contradiction.

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.