Quantum theory - Nature Paper 18 Sept

[Demystifier]

I fact, I think I have a new definition of the minimal interpretation of quantum mechanics. Like many other interpretations, it consists of
a) Computation rules for probabilities of measurement outcomes.
b) Explanation of what a) means.
However, other interpretations contain something weird in b). The minimal interpretation, by contrast, contains nothing weird in b). That's why the minimal interpretation is so great, unlike other interpretations it contains nothing weird in b). The only price paid for this absence of weirdness is that some of non-weird claims in b) are in logical contradiction with each other. But logical contradiction is only a philosophical problem, which does not affect the really important fact that the claims in a) are logically consistent and in agreement with experiments. So whenever someone points to a logical inconsistency in non-weird claims in b), you ignore b) entirely and concentrate on a). That's the essence of minimal interpretation.

 

[Demystifier]

The minimal ontology might be something like this:

1. There is only one world.
2. Quantum mechanics works the same on any system (microscopic or macroscopic, measurement devices or not)
3. The probabilities of measurement results are given by the Born rule.

I don't think that 2. and 3. should be called ontology.

You can't have the Born rule being true in general if there is nothing special about measurements.

I agree.

 

[PeterDonis]

In world $W_1$, $F$ believes that he is in world $W_2$.

Why?

 

[stevendaryl]

Why?

The universal wave function is $\sqrt{\frac{1}{3}} |\overline{h}\rangle |\frac{-1}{2}\rangle + \sqrt{\frac{1}{3}} |\overline{t}\rangle |\frac{-1}{2}\rangle + \sqrt{\frac{1}{3}} |\overline{t}\rangle |\frac{+1}{2}\rangle$ If you project for $F$ measuring $+1/2$, you get (after adjusting the normalization):

$|\overline{t}\rangle |\frac{+1}{2}\rangle$

That's the relative state for the world corresponding to the second component having value +1/2. That's world $W_2$.

In a "collapse" interpretation, $F$ measuring +1/2 would collapse the wave function of the world to collapse to the state $|\overline{t}\rangle |\frac{+1}{2}\rangle$. In a many-worlds interpretation, $F$ still thinks that the world is described by $|\overline{t}\rangle |\frac{+1}{2}\rangle$, but he allows for alternative possible worlds in which (for example) the other lab got result $\overline{h}$ rather than $\overline{t}$.

So when $F$ is in the state $|\frac{+1}{2}\rangle$ (which is the state of his having observed +1/2), his brain believes the world to be $W_2$.

But world $W_1$ is the state described by the wave function $|\overline{ok}\rangle |\frac{+1}{2}\rangle = \sqrt{\frac{1}{2}} |\overline{h}\rangle |\frac{+1}{2}\rangle - \sqrt{\frac{1}{2}} |\overline{t}\rangle |\frac{+1}{2}\rangle$. In this world, $F$ is in the state $|\frac{+1}{2}\rangle$. So we have already argued that $F$ believes that the world is $W_2$. "Believes" might not be the appropriate word, here, because it's not a matter of his wrong about that---there is no world that has the status of being "real", and so there is no right or wrong about what world we are in.

 

[PeterDonis]

when $F$ is in the state $|\frac{+1}{2}\rangle$ (which is the state of his having observed +1/2), his brain believes the world to be $W_2$.

That part I get, yes.

world $W_1$ is the state described by the wave function...

This part I don't get, because this wave function is not the projection of the universal wave function you give for $\bar{W}$ measuring ok.

 

[stevendaryl]

This part I don't get, because this wave function is not the projection of the universal wave function you give for $\bar{W}$ measuring ok.

Yes, it is. The definition of the state $|\overline{ok}\rangle$ is $\sqrt{\frac{1}{2}} |\overline{h}\rangle + \sqrt{\frac{1}{2}} |\overline{t}\rangle$. The state $|\overline{fail}\rangle$ is $\sqrt{\frac{1}{2}} |\overline{h}\rangle - \sqrt{\frac{1}{2}} |\overline{t}\rangle$.

[edit: I got the minus signs wrong. It should be:]
$|\overline{ok}\rangle = \sqrt{\frac{1}{2}} |\overline{h}\rangle - \sqrt{\frac{1}{2}} |\overline{t}\rangle$
$|\overline{fail}\rangle = \sqrt{\frac{1}{2}} |\overline{h}\rangle + \sqrt{\frac{1}{2}} |\overline{t}\rangle$

So in terms of the basis $|\overline{ok}\rangle$, $|\overline{fail}\rangle$, the state of the world is:

$\sqrt{\frac{2}{3}} |\overline{fail}\rangle |\frac{-1}{2}\rangle + \sqrt{\frac{1}{6}} |\overline{fail}\rangle |\frac{+1}{2}\rangle - \sqrt{\frac{1}{6}} |\overline{ok}\rangle |\frac{+1}{2}\rangle$

So the projection onto the subspace where the first component is $\overline{ok}$ yields: $|\overline{ok}\rangle |\frac{+1}{2}\rangle$. That's $W_1$

 

[DarMM]

What is at stake here is the Born rule in the subsystem. Is probability of a subsystem given by $|\psi|^2$?

Thanks for the response. I assume you saw their Table 4 where they mention that HV theories like Bohmian Mechanics drop Consistency when applied to subsystems, but the Born rule when applied to the universe.

 

[PeterDonis]

in terms of the basis $|\overline{ok}\rangle$, $|\overline{fail}\rangle$, the state of the world is:

I get something different than you; here are the steps of my calculation:

$$
\sqrt{\frac{1}{3}} |\overline{h}\rangle |\frac{-1}{2}\rangle + \sqrt{\frac{1}{3}} |\overline{t}\rangle |\frac{-1}{2}\rangle + \sqrt{\frac{1}{3}} |\overline{t}\rangle |\frac{1}{2}\rangle
$$

$$
\sqrt{\frac{2}{3}} |\overline{ok}\rangle |\frac{-1}{2}\rangle + \sqrt{\frac{1}{3}} |\overline{t}\rangle |\frac{1}{2}\rangle
$$

$$
\sqrt{\frac{2}{3}} |\overline{ok}\rangle |\frac{-1}{2}\rangle + \sqrt{\frac{1}{6}} |\overline{ok}\rangle |\frac{1}{2}\rangle - \sqrt{\frac{1}{6}} |\overline{fail}\rangle |\frac{1}{2}\rangle
$$

The projection of this onto the $\overline{ok}$ subspace has components for both $+ 1/2$ and $- 1/2$ for $F$.

 

[stevendaryl]

I get something different than you; here are the steps of my calculation:

$$
\sqrt{\frac{1}{3}} |\overline{h}\rangle |\frac{-1}{2}\rangle + \sqrt{\frac{1}{3}} |\overline{t}\rangle |\frac{-1}{2}\rangle + \sqrt{\frac{1}{3}} |\overline{t}\rangle |\frac{1}{2}\rangle
$$

$$
\sqrt{\frac{2}{3}} |\overline{ok}\rangle |\frac{-1}{2}\rangle + \sqrt{\frac{1}{3}} |\overline{t}\rangle |\frac{1}{2}\rangle
$$

I made a mistake (corrected now): The state $|\overline{ok}\rangle$ is defined with a minus sign: $|\overline{ok}\rangle = \sqrt{\frac{1}{2}} |\overline{h}\rangle - \sqrt{\frac{1}{2}} |\overline{t}\rangle$

 

[PeterDonis]

I made a mistake (corrected now): The state $|\overline{ok}\rangle$ is defined with a minus sign

Ah, ok.

 

[zonde]

I do not exactly agree with the example of polarizing filters. I would say that a polarizing filter is not an observation. Seeing that a photon passed through a filter is an observation.

Of course my example does not represent the experiment fairly. But that's because the thought experiment is problematic. It seems to me that this thought experiment violates no-cloning theorem. Observer $\overline{F}$ is measuring $|\overline{h}\rangle$ or $|\overline{t}\rangle$, then depending on result he sends to $F$ either $|\frac{-1}{2}\rangle$ or $|fail\rangle$. But phase relationship between observer $\overline{F}$ memory states $|\overline{h}\rangle$ and $|\overline{t}\rangle$ is still there and observer $F$ gets copy of that phase relationship with the system he gets. So there are two copies of arbitrary state. That's the source of the contradiction I would say.
If the paper fairly represents reasoning of MWI then it is a problem of MWI. But I'm not sure about that. When the worlds split is it required that each system in separate world remains in coherent superposition with it's complementary copy in the other world? The world as a whole can be in coherent superposition with the other world, but the subsystems of worlds are then only entangled.

 

[vanhees71]

Reading the marvelous blog by Aaronson posted in #44, I come to the conclusion that once more the authors (and the referees of the paper except one) got lost in philosophy. It's a bit sad that Aaronson doesn't explain its states. If I understand it right he has a 2D Hilbert space (spin 1/2) with one basis he labels with $|0 \rangle$ and $1 \rangle$. Taking the usual spin-$z$ eigenstates, these are the eigenvectors of the operators $2 \hat{s}_z +1$ with eigenvalues $1$ and $0$. Then, I figured out that the other basis is that of the spin operator in $\pm 45^{\circ}$ direction, i.e.,
$$|\pm \rangle=\frac{1}{\sqrt{2}} (|1 \rangle \pm |0 \rangle),$$
and the Hadamard transform is
$$\hat{H}=|+ \rangle \langle 1|+ |- \rangle \langle 0 |.$$

Then, there's indeed obviously no more "weirdness" than the usual "weirdness" and thus, within the minimal interpretation, no weirdness at all. To boil down the argument from an experimental point of view, what's wrong in the author's argument is the assumption that the various "labs are in isololation" and then nevertheless assuming the various agents measuring each other and the labs still are in isolation. The apparent contradiction is thus a classical example for the old rule "ex falso quodlibet".

So the assumption of isolated labs which are nevertheless measured is contradicting QT right in the assumptions and one of the reasons for much "quantum weirdness", namely that it is impossible to measure something microscopic without disturbing it, because one needs at least another microscopic thing of the same order of magnitude as the measured one. The most obvious example are charges of elementary particles. It makes no sense to talk about the Coulomb field of a single electron, say, since an electric field is defined by the action of this field on a test charge, which is so small that it doesn't affect the measured field significantly. That's of course impossible if the measured field is that of an electron since one needs at least another charged test particle. To keep the classical notion of the Coulomb field you'd need a charge much smaller than the electron's charge which doesn't exist in Nature (given that quarks with there 1/3 and 2/3 elementary charges are confined and thus not available, besides the fact that even this charge is not "negligibly" small against the electron charge).

 

[zonde]

To boil down the argument from an experimental point of view, what's wrong in the author's argument is the assumption that the various "labs are in isololation" and then nevertheless assuming the various agents measuring each other and the labs still are in isolation.

I don't see that this is assumed. Observer $\overline{F}$ is isolated up to the point when he is measured by $\overline{W}$ (except that he sends a system to $F$ before that). After $\overline{W}$ measured $\overline{F}$, there is no need for $\overline{F}$ to be isolated, he just has to keep his memories. And $\overline{F}$ is not measuring $\overline{W}$, he just knows what the setup is supposed to be and uses this knowledge in his reasoning.
It's similar for $F$ and $W$ pair.

 

[zonde]

The apparent contradiction is thus a classical example for the old rule "ex falso quodlibet".

Didn't knew that contradictory statement have such a disastrous consequences for logical reasoning: Principle of explosion.

 

[Demystifier]

it is impossible to measure something microscopic without disturbing it, because one needs at least another microscopic thing of the same order of magnitude as the measured one. The most obvious example are charges of elementary particles. It makes no sense to talk about the Coulomb field of a single electron, say, since an electric field is defined by the action of this field on a test charge, which is so small that it doesn't affect the measured field significantly. That's of course impossible if the measured field is that of an electron since one needs at least another charged test particle.

It's an off-topic, but how then the charge of a single electron is measured?

 

[stevendaryl]

Of course my example does not represent the experiment fairly. But that's because the thought experiment is problematic. It seems to me that this thought experiment violates no-cloning theorem. Observer $\overline{F}$ is measuring $|\overline{h}\rangle$ or $|\overline{t}\rangle$, then depending on result he sends to $F$ either $|\frac{-1}{2}\rangle$ or $|fail\rangle$. But phase relationship between observer $\overline{F}$ memory states $|\overline{h}\rangle$ and $|\overline{t}\rangle$ is still there and observer $F$ gets copy of that phase relationship with the system he gets. So there are two copies of arbitrary state. That's the source of the contradiction I would say.

No, I don't think there is any cloning of states going on. What I would think state cloning would amount to, in a two-component system, is something like this:

$(\alpha |\overline{h}\rangle + \beta |\overline{t}\rangle) |0\rangle \Rightarrow (\alpha |\overline{h}\rangle + \beta |\overline{t}\rangle) (\alpha |\frac{+1}{2}\rangle + \beta |\frac{-1}{2} \rangle)$ (where $|0\rangle$ is some initial state of the $F$ system).

That would violate linearity. But there is nothing like that going on in this example.

I think that the point about phase relationships is relevant here, though. If $|\overline{h}\rangle$ and $|\overline{t}\rangle$ are macroscopically different states, then the phase relationship between them is probably unmeasurable. But that's not no-cloning (which applies to microscopic systems, as well).

If the paper fairly represents reasoning of MWI then it is a problem of MWI. But I'm not sure about that. When the worlds split is it required that each system in separate world remains in coherent superposition with it's complementary copy in the other world? The world as a whole can be in coherent superposition with the other world, but the subsystems of worlds are then only entangled.

The only problem that it presents for MWI is with the intuitive idea that in a particular world, all macroscopic facts are determinate (Schrodinger's live cat and his dead cat are in different worlds). But there is no guarantee of that. A world may contain macroscopic superpositions. Maybe that exotic possibility detracts from the appeal of MWI, but it doesn't (to me) seem to contradict MWI.

Actually, if you analyze this thought-experiment through MWI, I think you will get something more like the many-minds theory. Each observer is in his own possible world.

 

[stevendaryl]

So the assumption of isolated labs which are nevertheless measured is contradicting QT right in the assumptions and one of the reasons for much "quantum weirdness", namely that it is impossible to measure something microscopic without disturbing it, because one needs at least another microscopic thing of the same order of magnitude as the measured one.

It depends on what you mean by "measure". If you have two microscopic variables that are entangled, then you can learn something about one variable by measuring the other.

 

[vanhees71]

It's an off-topic, but how then the charge of a single electron is measured?

You do scattering experiments and deduce the coupling constant from the measured cross section. That's how all 20+x constants of the Standard Model + neutrino masses and mixing parameters have to be measured.

 

[vanhees71]

It depends on what you mean by "measure". If you have two microscopic variables that are entangled, then you can learn something about one variable by measuring the other.

Sure, but then you have to adapt your state according to the information gained. If the issue is really as simple as reviewed in the blog by Aaronson, there's nothing new with this gedanken experiment but good (or bad depending on your philosophical prejudices) old contextuality of QT.

 

[zonde]

No, I don't think there is any cloning of states going on. What I would think state cloning would amount to, in a two-component system, is something like this:

$(\alpha |\overline{h}\rangle + \beta |\overline{t}\rangle) |0\rangle \Rightarrow (\alpha |\overline{h}\rangle + \beta |\overline{t}\rangle) (\alpha |\frac{+1}{2}\rangle + \beta |\frac{-1}{2} \rangle)$ (where $|0\rangle$ is some initial state of the $F$ system).

That would violate linearity. But there is nothing like that going on in this example.

I think that the point about phase relationships is relevant here, though. If $|\overline{h}\rangle$ and $|\overline{t}\rangle$ are macroscopically different states, then the phase relationship between them is probably unmeasurable. But that's not no-cloning (which applies to microscopic systems, as well).

Ok, if there is only one copy of the state than the experiment is no more mysterious than three polarizers experiment. When $\overline{W}$ measures the state and gets $|\overline{ok}\rangle$ the state of the system he gets is updated from $|fail\rangle$ to $|\frac{+1}{2}$ and then $W$ measuring this system obviously can get $|ok\rangle$ half of the time.
I'm not sure however what happens with $\overline{F}$ and $F$ observers and their memories.

 

[stevendaryl]

Ok, if there is only one copy of the state than the experiment is no more mysterious than three polarizers experiment. When $\overline{W}$ measures the state and gets $|\overline{ok}\rangle$ the state of the system he gets is updated from $|fail\rangle$ to $|\frac{+1}{2}$ and then $W$ measuring this system obviously can get $|ok\rangle$ half of the time.
I'm not sure however what happens with $\overline{F}$ and $F$ observers and their memories.

The way that the thought experiment is described, it seems that if $\overline{W}$ gets $\overline{ok}$ and $W$ gets $ok$, then both $F$ and $\overline{F}$ are in indefinite, Schrodinger's Cat type states, a superposition of two different memory states.

 

[Strilanc]

Where does performing their two measurements require uncomputing their thoughts and measuring a single qubit. I would have thought for each of them independently they are simply measuring an alternate outcome basis for the respective labs under their control. However they themselves are left alone.

When I said "Alice and Bob" I was referring to the people being placed under superposition, not the people triggering the measurements. Sorry if that was confusing.

If you are referring to $\overline{F}$ and $F$ instead, even there is one uncomputing their thoughts? I would have thought for example that $W$ measures some strange observable $Q$ that places the $L$ lab containing $F$ into the $\{|okay\rangle,|fail\rangle\}$ basis. However I'm not sure how that corresponds to uncomputing $F$'s thoughts measuring a qubit and recomputing them. Again most likely I am missing something.

Technically speaking, you could implement the measurement described by the paper in many ways. But in these kinds of confusing paradoxical situations, it is important to be concrete about the details. The devil is literally in the details (that's a Maxwell's demon pun).

I happen to think the simplest way to implement the described measurement is to uncompute back to the state where the relevant information is in a trivial form (just one qubit), do the measurement there, then recompute. After all, you're already loading Alice and Bob into a quantum computer capable of simulating time forward while maintaining complete coherence. In that situation it's trivial to run simulated time backwards: take your forward-time circuit, invert each gate, and run them in the reverse order.

That being said, even if you use a more complicated strategy to perform the measurement it's still equivalent to uncomputing+measuring+recomputing. So I think it's fine to use an argument based on that interpretation. It's a specific case of "the measurement perturbs the system, so your previous conclusions become invalid.".

 

[DarMM]

I happen to think the simplest way to implement the described measurement is to uncompute back to the state where the relevant information is in a trivial form (just one qubit), do the measurement there, then recompute. After all, you're already loading Alice and Bob into a quantum computer capable of simulating time forward while maintaining complete coherence. In that situation it's trivial to run simulated time backwards: take your forward-time circuit, invert each gate, and run them in the reverse order.

I see now, thanks. I think this captures Bub's objection. He would say that if $F$ and $\bar{F}$ are really in a quantum computer then no "measurements" occur, i.e. $F$ doesn't obtain anything. In essence it would mean they lie behind the Heisenberg cut, no actual macroscopic events are associated with them.

 

[Strilanc]

I see now, thanks. I think this captures Bub's objection. He would say that if $F$ and $\bar{F}$ are really in a quantum computer then no "measurements" occur, i.e. $F$ doesn't obtain anything. In essence it would mean they lie behind the Heisenberg cut, no actual macroscopic events are associated with them.

If they don't "lie behind the Heisenburg cut", it's impossible to implement the measurement that is described by the paper (or rather, the system will have decohered in a way that changes the statistics of the output).

 

[DarMM]

If they don't "lie behind the Heisenburg cut", it's impossible to implement the measurement that is described by the paper (or rather, the system will have decohered in a way that changes the statistics of the output).

Exactly and thus when they do lie behind the cut he would claim you can't say such a thing as "$F$ obtained $z=+\frac{1}{2}$" and thus all such reasoning is invalid.

In other words he is saying the existence of "conclusions" is only valid in a scenario where the measurement can't be implemented.

He's not saying the measurement could be implemented even when they aren't behind the cut. He's more saying in order to have conclusions, you need to not lie behind the cut, which then prevents the measurement described for $\bar{W}$.

Although Bub's reasoning is not meant to be "interpretation neutral" where Aaranson's is to a larger degree. It's an additional objection based on a Neo-Copenhagen viewpoint, with that interpretation's specific explanation of decoherence.

 

[S.Daedalus]

OK, there's a bit I don't get, I think. So the final state is:

$|final\rangle = \frac{1}{\sqrt{3}} |\overline{h}\rangle |-\frac{1}{2}\rangle + \frac{1}{\sqrt{3}} |\overline{t}\rangle |\frac{1}{2}\rangle + \frac{1}{\sqrt{3}} |\overline{t}\rangle |-\frac{1}{2}\rangle$

Expressed in the ok/fail-basis of $\overline{W}$, this is:

$|final\rangle = \frac{2}{\sqrt{6}}|\overline{fail}\rangle|-\frac{1}{2}\rangle + \frac{1}{\sqrt{6}}|\overline{fail}\rangle|\frac{1}{2}\rangle - \frac{1}{\sqrt{6}}|\overline{ok}\rangle|\frac{1}{2}\rangle$

After $\overline{W}$ measures and obtains the $\overline{ok}$-outcome, only the term $|\overline{ok}\rangle|\frac{1}{2}\rangle$ survives. If we express that again in the $\{|\overline{h}\rangle,|\overline{t}\rangle\}$-basis, we get:

$|\overline{w}=\overline{ok}\rangle=\frac{1}{\sqrt{2}}(|\overline{h}\rangle|\frac{1}{2}\rangle - |\overline{t}\rangle|\frac{1}{2}\rangle)$

But this is not a state in which we can claim that $\overline{F}$ predicts that W observes 'fail', as either outcome is possible. Consequently, if $\overline{F}$ applies quantum mechanics to themselves, they should rather reason that if $\overline{W}$ applies their measurement and observes the $\overline{ok}$-outcome, then either measurement outcome of W's measurement is possible, so no contradiction arises. Alternatively, $\overline{W}$ is wrong to believe that their inference from the measurement outcome still holds after the measurement has been performed, since that inference depends on the structure of the pre-measurement state.

The only way of making this seem reasonable, it looks to me, is if $\overline{F}$ supposes that their measurement (of the coin) 'collapses' the state to 'tails'; but that's just the same as the original Wigner's Friend-thought experiment, where if the friend were to make that assumption, they'd predict (wrongly) that a suitable experiment involving the entire lab would show no interference. So if F supposes that W observes 'fail', that just seems to me to be a misapplication of quantum mechanics in the same way as Wigner's Friend's prediction of no interference would be.

 

[PeterDonis]

He would say that if $F$ and $\bar{F}$ are really in a quantum computer then no "measurements" occur

Another way of putting this is that, if all of the interactions are reversible, then no measurement occurs; a measurement requires an irreversible interaction.

 

[msumm21]

Trying to understand some definitions in this paper:
https://www.nature.com/articles/s41467-018-05739-8

Specifically page 3, box 1: "At n:20 Wbar measures Lbar wrt a basis containing okbar"

okbar is defined in table 2 in terms of hbar and tbar which are not defined as far as I can tell. I'd guessed those were something like the heads and tails states of R, but that doesn't seem to pan out. Anyone know what okbar is?

 

[zonde]

okbar is defined in table 2 in terms of hbar and tbar which are not defined as far as I can tell.

Under Fig.2 there is description from which one can guess that $|\bar{h}\rangle$ and $|\bar{t}\rangle$ are states of the coin:
$\bar{F}$ tosses a coin and, depending on the outcome r, polarises a spin particle S in a particular direction.

I'd guessed those were something like the heads and tails states of R, but that doesn't seem to pan out.

You will have to say something more. What does not seem to pan out?

 

[atyy]

Trying to understand some definitions in this paper:
https://www.nature.com/articles/s41467-018-05739-8

Specifically page 3, box 1: "At n:20 Wbar measures Lbar wrt a basis containing okbar"

okbar is defined in table 2 in terms of hbar and tbar which are not defined as far as I can tell. I'd guessed those were something like the heads and tails states of R, but that doesn't seem to pan out. Anyone know what okbar is?

Maybe try stevendaryl's post #14 and see if that helps you out.

 

[stevendaryl]

OK, there's a bit I don't get, I think. So the final state is:

$|final\rangle = \frac{1}{\sqrt{3}} |\overline{h}\rangle |-\frac{1}{2}\rangle + \frac{1}{\sqrt{3}} |\overline{t}\rangle |\frac{1}{2}\rangle + \frac{1}{\sqrt{3}} |\overline{t}\rangle |-\frac{1}{2}\rangle$

Expressed in the ok/fail-basis of $\overline{W}$, this is:

$|final\rangle = \frac{2}{\sqrt{6}}|\overline{fail}\rangle|-\frac{1}{2}\rangle + \frac{1}{\sqrt{6}}|\overline{fail}\rangle|\frac{1}{2}\rangle - \frac{1}{\sqrt{6}}|\overline{ok}\rangle|\frac{1}{2}\rangle$

After $\overline{W}$ measures and obtains the $\overline{ok}$-outcome, only the term $|\overline{ok}\rangle|\frac{1}{2}\rangle$ survives. If we express that again in the $\{|\overline{h}\rangle,|\overline{t}\rangle\}$-basis, we get:

$|\overline{w}=\overline{ok}\rangle=\frac{1}{\sqrt{2}}(|\overline{h}\rangle|\frac{1}{2}\rangle - |\overline{t}\rangle|\frac{1}{2}\rangle)$

But this is not a state in which we can claim that $\overline{F}$ predicts that W observes 'fail', as either outcome is possible. Consequently, if $\overline{F}$ applies quantum mechanics to themselves, they should rather reason that if $\overline{W}$ applies their measurement and observes the $\overline{ok}$-outcome, then either measurement outcome of W's measurement is possible, so no contradiction arises. Alternatively, $\overline{W}$ is wrong to believe that their inference from the measurement outcome still holds after the measurement has been performed, since that inference depends on the structure of the pre-measurement state.

The only way of making this seem reasonable, it looks to me, is if $\overline{F}$ supposes that their measurement (of the coin) 'collapses' the state to 'tails'; but that's just the same as the original Wigner's Friend-thought experiment, where if the friend were to make that assumption, they'd predict (wrongly) that a suitable experiment involving the entire lab would show no interference. So if F supposes that W observes 'fail', that just seems to me to be a misapplication of quantum mechanics in the same way as Wigner's Friend's prediction of no interference would be.

I think you're right. The paradox just boils down to: noncommuting operators can't all be said to have definite values at the same time.

But as I summarized in one of the earlier posts:

1. $\overline{W}$ can conclude: "If I measure $\overline{ok}$, then $F$ must measure $+1/2$.
2. $F$ can conclude: "If I measure $+1/2$, then that implies that $\overline{F}$ measured $\overline{t}$"
3. $\overline{F}$ can conclude: "If I measure $\overline{t}$, then $W$ will measure $fail$"

The problem is chaining these claims together. Each of the conclusions is true under the assumption that we're starting in the state $|final\rangle$. But under a "collapse" model, after the first observation, the state collapses to something other than $|final\rangle$, so the reasoning no longer applies. If you don't assume collapse, then it's a little more complicated to reason about it, but I think that all interpretations agree that things will work out as if the state collapses upon making an observation.

 

[msumm21]

Sorry if I'm re-stating what people have already said earlier in this thread, but isn't there a basic error in the analysis. The author assumes Fbar measures R (the "coin" heads or tails) and subsequently acts on that result, ... but later assumes the state of R was unaffected (back in the init state) when Wbar measures. That's not QM is it? QM says R is in the state pure heads OR pure tails, not the init superposition, right?

Correcting for these mistakes, the point of the paper doesn't pan out, right? E.g. the claim that OKbar implies S is spin up wouldn't hold, so the main conclusion wouldn't hold.

The explanation by stevendaryl earlier (#14) in this thread much appreciated, very clear & concise.

 

[vanhees71]

I think you're right. The paradox just boils down to: noncommuting operators can't all be said to have definite values at the same time.

But as I summarized in one of the earlier posts:

1. $\overline{W}$ can conclude: "If I measure $\overline{ok}$, then $F$ must measure $+1/2$.
2. $F$ can conclude: "If I measure $+1/2$, then that implies that $\overline{F}$ measured $\overline{t}$"
3. $\overline{F}$ can conclude: "If I measure $\overline{t}$, then $W$ will measure $fail$"

The problem is chaining these claims together. Each of the conclusions is true under the assumption that we're starting in the state $|final\rangle$. But under a "collapse" model, after the first observation, the state collapses to something other than $|final\rangle$, so the reasoning no longer applies. If you don't assume collapse, then it's a little more complicated to reason about it, but I think that all interpretations agree that things will work out as if the state collapses upon making an observation.

So to make a long story short: Much ado about nothing! Just check all such "philosophical claims" within the minimal interpretation, i.e., first through out the unobservable balast and then check what to be expected in physics experiments. I think it simply boils down to the suspicion I had after first reading the paper: The assumption of the various systems being isolated from each other but nevertheless measured by each other is simply an oxymoron. The very point of QT is that you cannot neglect the influence of measurements on microscopic systems and thus also in general you are able to prepare states such that incompatible observables have determined values (although there are exceptions, e.g., if preparing the angular momenum of a system in the state $j=0$, all three components of angular momentum (or thus any component of angular momentum) take the determined value $m=0$).

 

[DarMM]

I was thinking about this a bit more, probably doing something dumb, but considering agent $F$, if they get the result $z = +\frac{1}{2}$ they can conclude that $\bar{F}$ got $r = \bar{t}$. Fine.

Now if they think of themselves from $\bar{F}$'s reasoning and apply unitary evolution without any collapse to themselves they get that the state of their lab is:
$$|\psi\rangle = \frac{1}{\sqrt{2}}\left(|+\frac{1}{2}\rangle_L + |-\frac{1}{2}\rangle_L\right)$$
which is $W$'s $|fail\rangle$ state. Thus $W$ must get $fail$.

However if they think of themselves from what they see in their own lab then either $z = -\frac{1}{2}$ or $z = +\frac{1}{2}$ results correspond to a superposition in the $\{|okay\rangle,|fail\rangle\}$ basis. Thus they would reason that $W$ could get $okay$.

So isn't there already a contradiction with just three observers?

$F$ will reach one conclusion about $W$'s result if he just considers his own experiences in his lab and another conclusion if he considers himself from $\bar{F}$'s model of him as a unitarily evolving superposition without decoherence.

 

[DarMM]

For anybody interested Renato Renner's argument has three forms. The form in the original paper from 2016 (explained in standard language by Bub here: https://arxiv.org/abs/1804.03267), the form from the nature article we have been discussing and a third form due to Luis Masanes.

It is harder to pinpoint errors of this third form in my opinion. Expositions of it are found in section 4 of this paper by Richard Healey:
https://arxiv.org/abs/1807.00421

And also this talk by Matthew Pusey (18:35):


And this lecture by Matthew Leifer (39:44):