I What Is the Frauchiger-Renner Theorem?

[DarMM]

Since there seems to be a bit of confusion on this, I thought I'd just post a brief summary.

Just some terminology:
Superobserver: Somebody who measures another observer, i.e capable of resolving the complete quantum state of another observer and performing measurements on it.
Hyperobserver: Like a Superobserver, but also capable of using a unitary evolution to reverse the state of an observer to its pre-measurement form. (I just made this term up to distinguish the two cases, it's not standard)

In essence the Frauchiger-Renner theorem derives a contradiction between:

1. Validity of Probability One predictions of quantum theory, i.e. if QM says something has 100% chance of occurring it is certain.
2. Single World, i.e. experiments have one objective outcome
3. Inter-agent reasoning, i.e. I can obtain my predictions by reasoning about how you would use quantum theory.
4. Intervention insensitivity for Classical Objects/Measurement results. As a superobserver your reasoning about measuring an observer is not affected by subsequent measurements by superobservers spacelike separated from you. In short this says that observers aren't to be considered as being entangled/Bell-inequality violating by superobservers.

This is equivalent to the following reformulation by Richard Healey which I think is easier to grasp:

1. Quantum Mechanics applies objectively to all systems/is universal
2. Single World
3. Superobservers should use superposed states to describe observers, prior to their measurements of them
4. Intervention insensitivity

Most criticism of the FR paper is because they don't mention (4) as an assumption and thus you can escape dropping the other three assumptions by dropping it. However note that dropping (4) does mean that observers cannot be considered as purely Classical, so a very strict form of Copenhagen is blocked.

Also note that the Frauchiger-Renner theorem does not use HyperObservers so it doesn't assume measurements are reversible.

There is an alternate form of the theorem due to Luis Masanes, which is in truth a separate theorem which derives a similar contradiction, but replaces (4) with:

(4*) It is possible to unitarily reverse a measurement, i.e. HyperObservers exist.

Again here you might deny (4*) if you wanted a certain type of Copenhagen interpretation. However since from the point of view of a HyperObserver they are licensed to use superposed states (via (3)) they'd have no reason to suppose some unitaries don't have physically realisable inverses, so this would have to take the form of an ad-hoc restriction of QM when used by such observers.

Of course one might deny (3) and (4*), observers shouldn't be modeled with superpositions and you can't reverse measurements. This would be objective collapse like the Ghirardi–Rimini–Weber theory.

 

[Demystifier]

However note that dropping (4) does mean that observers cannot be considered as purely Classical, so a very strict form of Copenhagen is blocked.

Exactly! The version of Copenhagen that insists that observers are classical is wrong.

 

[David Byrden]

Regarding the Frauchiger-Renner paper, at https://arxiv.org/abs/1604.07422v1

People in various forums have pointed out that this paper is flawed, but they talk about the flaw in different ways. And, to be fair, there are different ways of looking at the flaw. Some of them are "holistic", involving portions of the paper. Some consider its methods and some invoke the relevant parts of QM theory. Most of these comments, while correct, are not immediately convincing.

So I will now tell you, using the FR paper's own wording, the precise point at which this paper makes a mistake that invalidates its conclusion :

In section 5, in "Analysis of Experiment F2", the authors point out that z=+1/2 is a possible measurement result at time n:20. In that case, they conclude that a state of ψS existed at n:10 which had a nonzero projection on the Hilbert-space measurement operator for z=+1/2.

This is correct, but remember: there are many possible states of ψS that meet this requirement. Any state with a nonzero projection will do.

The authors then assume that only one specific state of ψS could meet the requirement. That assumption is unexplained and unjustified by FR.

Then, using that assumption, they plough on to derive a "contradiction" dependent on their assumption.

If you need, I can give you an analysis of their thought experiment, showing that another state of ψS was responsible, and that everything works out consistently, and there's no reason here to question Quantum Mechanics at all.

David

 

[DarMM]

This is correct, but remember: there are many possible states of ψS that meet this requirement. Any state with a nonzero projection will do.

The authors then assume that only one specific state of ψS could meet the requirement. That assumption is unexplained and unjustified by FR.

A few things:

1. Doesn't that section say "at least one exists", not that one specific one exists? Which is all you need for the r = tails conclusion
2. Remember the forms of $\psi_S$ that $\bar{F}$ will prepare are agreed in advance from a set of two states
3. I would stick to reading Version 2 of the paper (which is basically identical to the one published in Nature) as it is much easier to follow and written in more standard terminology

 

[David Byrden]

I would stick to reading Version 2 of the paper ...as it is much easier to follow and written in more standard terminology

But then I could be accused of not finding a real flaw, because Version 2 might contain simplified or altered logic.

Doesn't that section say "at least one exists", not that one specific one exists?

I'll spell it out.
In between equations 25 and 26 they say that a state exists with a nonzero probability of causing the measurement z=+1/2.
They say that because, in this "round", that measurement was made.
So, yes, in that line they imply that "at least one exists".

But remember, it's a projection. Any state that's not orthogonal to the operator can yield that measurement.
It could be a superposition state, with any ratio of "up" and "down" in the qubit, except for fully "down".

Now, let's continue...

Which is all you need for the r = tails conclusion

Nope.
As I just said, almost any superposition of the qubit states can give the same result.
If you pick one, then your conclusion is that the randomiser measurement "r" was a superposition of "heads" and "tails" in some ratio (except for fully "heads").

But the authors proceed to equation 26 where they conclude that "r=tails".
They ignore all the possible superposition states.

What's their justification for this? They invoke "constraint 13".
But that's nothing more than the mapping from "r" to the qubit. It does not constrain either of those objects to not be in superposition.
The conclusion is not justified.

Remember the forms of $\psi_S$ that $\bar{F}$ will prepare are agreed in advance from a set of two states

I know that.
But I don't see any reason to disallow a superposition of these two states.
Especially since the authors have superpositions elsewhere in the experiment.
(e.g. in "Analysis of Experiment F1" they put lab "L" in a superposition of two states, guaranteeing the "fail" measurement)

The starting analysis of the expected result ( "OK, OK") depends on the two labs being in superpositions, it depends on the system being in a pure state, it will not happen if we disallow superpositions for either of the F agents.
But then, as I pointed out, deep in the middle of their "proof", the authors disallow a superposition state for /F, without reason or comment.

David

 

[Demystifier]

People in various forums have pointed out that this paper is flawed, but they talk about the flaw in different ways. And, to be fair, there are different ways of looking at the flaw.

So I think we can agree that the paper is flawed in a rather subtle and nontrivial way. Such a flawed paper contributes more to understanding of QM than most correct papers do. Nothing can be so illuminating as a deep subtle error.

 

[DarMM]

So I think we can agree that the paper is flawed in a rather subtle and nontrivial way. Such a flawed paper contributes more to understanding of QM than most correct papers do. Nothing can be so illuminating as a deep subtle error.

I think the original paper is flawed for not stating the fourth assumption. However the Foundations community seems to be using Luis Masanes's version which doesn't have a flaw in its logic, as far as I can tell.

 

[DarMM]

But then I could be accused of not finding a real flaw, because Version 2 might contain simplified or altered logic.

I wasn't planning on accusing you. It's just written in a neater way, but it's no problem if you want to stick with v1, I've read both.

Nope.
As I just said, almost any superposition of the qubit states can give the same result.
If you pick one, then your conclusion is that the randomiser measurement "r" was a superposition of "heads" and "tails" in some ratio (except for fully "heads").

But the authors proceed to equation 26 where they conclude that "r=tails".
They ignore all the possible superposition states.

What's their justification for this? They invoke "constraint 13".
But that's nothing more than the mapping from "r" to the qubit. It does not constrain either of those objects to not be in superposition.
The conclusion is not justified.

I don't agree.

$\bar{F}$ measures the state $\sqrt{\frac{1}{3}}|heads\rangle + \sqrt{\frac{2}{3}}|tails\rangle$. If they see "heads" they prepare $|\downarrow\rangle$ and send it to $F$, if they see tails they prepare $|\rightarrow\rangle = \sqrt{\frac{1}{2}}\left(|\uparrow\rangle + |\downarrow\rangle\right)$ and send it to $F$.

This is the agreed procedure between $\bar{F}$ and $F$.

Thus if $F$ measures a spin up outcome he'll know that he must have been sent the state $|\rightarrow\rangle$ and not the only other alternative, i.e. $|\downarrow\rangle$. And since $\bar{F}$ agreed to only send $|\rightarrow\rangle$ in the event of r = tails we know that r = tails.

 

[DarMM]

Just changed the wording of the Single World assumption:

Single World, i.e. experiments have one objective outcome

I added objective because I think "objective outcome vs subjective outcome" makes the distinction clearer than the "outcome vs experience" labels some use.

Most Copenhagen related interpretations (e.g. Brukner, QBism, Rovelli's Relational View) are making the move (or have already made the move long ago) to experimental outcomes being relative to the observer, not a fully objective feature of the world. Brukner has the most explicit paper on this:
https://arxiv.org/abs/1804.00749

Note Brukner's paper does not actually prove the need for subjective outcomes for Copenhagen style interpretations with reversible measurements like Masanes version of Frauchiger-Renner does. However I found it useful as an illustration of what a "subjective detector click" might even be!

 

[David Byrden]

DarMM:

The randomiser is initially in a pure superposition state - the state that you wrote.

Then, /F measures it. What that implies, can depend on your interpretation of "measure".

In "collapse" interpretations, it means that the lab /L collapses the randomiser into a single state, and this heads-or-tails state exists objectively.

In other interpretations, the lab splits into two "worlds", each containing one result. For observers outside the lab, its contents are in superposition, and measurements will yield probabilistic results.

If we use the "collapse" interpretation, there's a problem. The experiment will not work. We will not get the final result "ok,ok" for one twelfth of all rounds.

Because FR's paper explicitly states that the system state will read "ok,ok" one time in twelve, they cannot be talking about a system where the hidden F agents "collapse" their labs into a single objective state.

To convince yourself of this, look at /F's thinking when she has tails. She sends a qubit in z superposition of up and down (as you wrote!), then she assumes that lab L will read it and go into a superposition also. And that superposition state vector coincides with the "fail" vector of W in his chosen basis. That superposition state is what guarantees the "fail" measurement.

If agent F were to "collapse" the qubit in the same way that you just proposed for agent /F, then a "fail" reading would not be guaranteed. Agent W would get meaningless results, randomly fail or ok, because his measurement basis state vector is at 45 degrees to the collapsed state vector of the lab in Hilbert space.

It's unfortunate that FR did not state this at the outset, but their experiment can't work with macroscopic observers inside the labs. They require the labs to go into superposition pure states. Only with quantum devices can this experiment be realized.

David

 

[Demystifier]

However the Foundations community seems to be using Luis Masanes's version which doesn't have a flaw in its logic, as far as I can tell.

I guess you linked this version somewhere before, but I missed it. Can you give a link again?

 

[atyy]

So I think we can agree that the paper is flawed in a rather subtle and nontrivial way. Such a flawed paper contributes more to understanding of QM than most correct papers do. Nothing can be so illuminating as a deep subtle error.

So you don't think it's like trying to find the flaw in a perpetual motion machine?

 

[Demystifier]

So you don't think it's like trying to find the flaw in a perpetual motion machine?

I don't. Do you?

 

[stevendaryl]

Regarding the Frauchiger-Renner paper, at https://arxiv.org/abs/1604.07422v1

People in various forums have pointed out that this paper is flawed, but they talk about the flaw in different ways. And, to be fair, there are different ways of looking at the flaw. Some of them are "holistic", involving portions of the paper. Some consider its methods and some invoke the relevant parts of QM theory. Most of these comments, while correct, are not immediately convincing.

So I will now tell you, using the FR paper's own wording, the precise point at which this paper makes a mistake that invalidates its conclusion :

In section 5, in "Analysis of Experiment F2", the authors point out that z=+1/2 is a possible measurement result at time n:20. In that case, they conclude that a state of ψS existed at n:10 which had a nonzero projection on the Hilbert-space measurement operator for z=+1/2.

This is correct, but remember: there are many possible states of ψS that meet this requirement. Any state with a nonzero projection will do.

The authors then assume that only one specific state of ψS could meet the requirement. That assumption is unexplained and unjustified by FR.

Then, using that assumption, they plough on to derive a "contradiction" dependent on their assumption.

If you need, I can give you an analysis of their thought experiment, showing that another state of ψS was responsible, and that everything works out consistently, and there's no reason here to question Quantum Mechanics at all.

David

I'm not sure if I understand your objection, but it certainly occurred to me that the setup was making possibly inconsistent abstractions. The argument depends on there being essentially 2 possible states for each of the observers. Then by carefully choosing superpositions of these two states, the authors derive their contradiction. In reality, there are many, many microstates that correspond to the same macroscopic description: "So-and-so got measurement result such-and-such". So the argument makes an enormous simplification by considering only a small, discrete number of states of the observers. But it wasn't clear to me whether a more realistic treatment would invalidate their conclusions, or not. Certainly, a more realistic treatment of observers would make the key step of putting them into a superposition of states with precise phase relationships impossible in practice. But can we use an impossible-in-practice thought experiment to show something about the nature of QM? I'm not sure.

 

[David Byrden]

The argument depends on there being essentially 2 possible states for each of the observers.

There are 2 states for each lab, yes. In essence, the whole apparatus is a 2-qubit system. It's much simpler than the paper's abundance of notation might lead you to believe.

Then by carefully choosing superpositions of these two states, the authors derive their contradiction.

No.
They put the two labs into carefully chosen superpositions, yes.
They derive certain results (e.g. "W must measure Fail" ) by taking those superposition states into account.
But in one case ( "r must haveTails" ) they completely ignore the possibility of superposition.
The superposition that they've already set up, already used in their calculations; they ignore it.

They derive a contradiction by making a mistake.

In reality, there are many, many microstates that correspond to the same macroscopic description

Microstates of macroscopic objects are not at fault here.
You're correct to say that the experiment cannot be realized with human observers; (decoherence would ruin the very necessary superposition states).
But that's not why a contradiction appeared to result from the experiment.

 

[stevendaryl]

They derive a contradiction by making a mistake.

I wouldn't say that they made a mistake. They proved that a certain "transitivity" property fails, and it really does fail.

In the original paper, there are 4 observers, $W, F, \overline{F}, \overline{W}$:

• $W$ performs a measurement that has two possible results, $ok$ or $fail$.
• $F$ performs a measurement that has two possible results, $+\frac{1}{2}$ or $-\frac{1}{2}$.
• $\overline{F}$ performs a measurement that has two possible results, $\overline{t}$ or $\overline{h}$
• $\overline{W}$ performs a measurement that has two possible results, $\overline{ok}$ or $\overline{fail}$

Now let me introduce a kind of logical implication, $A \leadsto B$, where $A$ and $B$ are statements of the form "So-and-so measures such-and-such and got result this-or-that". The meaning of this kind of implication is that, relative to some initial state $\psi$, if you let the initial state evolve in time to the point where the measurement corresponding to $A$ is made, and then project the state onto the subspace corresponding to the corresponding result, then the state will be such that it is certain that a later measurement will have the corresponding result. So an example from EPR is: $A$ corresponds to Alice measuring spin-up for one particle along the z-axis, and $B$ corresponds to Bob measuring spin-down for the other particle along the z-axis. If Alice really does measure spin along the z-axis gets spin-up, and later Bob measures spin along the z-axis, he is guaranteed to get spin-down. So $A \leadsto B$.

So for the thought-experiment under discussion, we have:

1. $\overline{W}$ measures $\overline{ok}$ $\leadsto$ $F$ measures $+\frac{1}{2}$
2. $F$ measures $+\frac{1}{2}$ $\leadsto$ $\overline{F}$ measures $\overline{t}$
3. $\overline{F}$ measures $\overline{t}$ $\leadsto$ $W$ measures $fail$.

But what we don't have is: $\overline{W}$ measures $\overline{ok}$ $\leadsto$ $W$ measures $fail$.

So unlike regular implication, $\leadsto$ is not transitive. That's just a fact. Whether you expect it to be transitive or not depends on your interpretation of quantum mechanics.

On the other hand, I claim that for all practical purposes, $\leadsto$ is transitive, because a setup violating transitivity requires making superpositions between macroscopic states with precise phase relationships, which is not possible in practice.

 

[David Byrden]

I wouldn't say that they made a mistake.

Because you can't see the mistake, because you are making the exact same mistake, as I will now show.

They proved that a certain "transitivity" property fails

No, they didn't prove anything.

So for the thought-experiment under discussion, we have:

1. $\overline{W}$ measures $\overline{ok}$ $\leadsto$ $F$ measures $+\frac{1}{2}$

Yes.
If you do the math, the system state has zero projection on { /ok , -1/2 }.
Therefore the /ok measurement implies that z=+1/2.

1. $F$ measures $+\frac{1}{2}$ $\leadsto$ $\overline{F}$ measures ¯t
   

No.
This is exactly the mistake that Renner and Frauchiger make.
A measurement of z=+1/2 does not imply that /F is in the state of having measured "tails".
It does imply that /F is in any superposition of "heads" and "tails", in any ratios, except for "heads only".
Any of those superpositions has a probability of yielding a z=+1/2 measurement.
Not a 100% guarantee, I admit, but that's not important post hoc.
It happened.
Every possible lead-up to the measurement should be taken into account.

To assert that /F must have been precisely "tails", is to assert that the first lab is not in a superposition state.
But the rest of the experiment depends on both labs being in superposition states.
The calculations, and the final outcome, depend critically on superposition (not mixed) states existing in both labs.

So, Renner and Frauchiger made an unjustified assumption here - that agent /F is not in a superposition relative to agent F.
But earlier, they assumed that she is in a superposition relative to agent /W.
They can't have it both ways.

David

 

[stevendaryl]

This is exactly the mistake that Renner and Frauchiger make.
A measurement of z=+1/2 does not imply that /F is in the state of having measured "tails".
It does imply that /F is in any superposition of "heads" and "tails", in any ratios, except for "heads only".

Why do you say that? They stipulated the assumptions. You might want to say that those assumptions are not possible to realize, but you can't just say that the assumptions were wrong. A proof starts with assumptions and derives a conclusion. It's correct if the conclusion follows from the assumptions.

Unraveling the words and trying to make it mathematical, their thought-experiment amounts to assuming that after their measurements, the two observers, $\overline{F}$ and $F$ are in the composite state
$$|final\rangle = \frac{1}{\sqrt{3}} (|\overline{h}\rangle |\frac{-1}{2}\rangle + |\overline{t}\rangle |\frac{+1}{2}\rangle + |\overline{t}\rangle |\frac{-1}{2}\rangle)$$
Now, $\overline{W}$ performs a measurement on the first component that has an outcome $\overline{ok}$ (corresponding to the state $|\overline{ok}\rangle = \frac{1}{\sqrt{2}} (|\overline{h}\rangle - |\overline{t}\rangle)$) or $\overline{fail}$ (corresponding to the state $|\overline{fail}\rangle = \frac{1}{\sqrt{2}} (|\overline{h}\rangle + |\overline{t}\rangle)$)

To aid in projecting on this alternative basis, we can rewrite $|final\rangle$:

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

So
$|final\rangle = \frac{1}{\sqrt{6}} [ |\overline{fail}\rangle |\frac{-1}{2}\rangle + |\overline{ok}\rangle |\frac{-1}{2}\rangle + |\overline{fail}\rangle |\frac{+1}{2}\rangle - |\overline{ok}\rangle |\frac{+1}{2}\rangle + |\overline{fail}\rangle |\frac{-1}{2}\rangle - |\overline{ok}\rangle |\frac{-1}{2}\rangle ]$
$= \sqrt{\frac{2}{3}} |\overline{fail}\rangle |\frac{-1}{2}\rangle + \sqrt{\frac{1}{6}}|\overline{fail}\rangle |\frac{+1}{2}\rangle + \sqrt{\frac{1}{6}} |\overline{ok}\rangle |\frac{+1}{2}\rangle$

If $\overline{W}$ measures that $\overline{F}$ is in state $|\overline{ok}\rangle$, then that implies that $F$ is in the state $|\frac{+1}{2}\rangle$

This is exactly like in EPR when Alice measures that her particle is in the state spin-up in the z-direction, she knows that Bob's particle is in the state spin-down in the z-direction.

So we conclude: $\overline{W}$ measures "ok" $\leadsto$ $F$ measures +1/2.

 

[stevendaryl]

(continued)

Going back to the state
$$|final\rangle = \frac{1}{\sqrt{3}} (|\overline{h}\rangle |\frac{-1}{2}\rangle + |\overline{t}\rangle |\frac{+1}{2}\rangle + |\overline{t}\rangle |\frac{-1}{2}\rangle)$$
We can rewrite that as
$$|final\rangle = \frac{1}{\sqrt{3}} |\overline{t}\rangle |\frac{+1}{2}\rangle + \sqrt{\frac{2}{3}} |\overline{fail}\rangle |\frac{-1}{2}\rangle$$
If we project onto the subspace in which $F$ measures $+1/2$, then in that subspace, $\overline{F}$ is in state $|\overline{t}\rangle$

 

[David Byrden]

Your math is correct and I commend you for it.
However you missed something.

In this 4 dimensional Hilbert space, there is not just a single state vector corresponding to "F measures +1/2".
There is an entire plane.

When you projected the original system state onto the subspace, you landed on that plane, at one specific state vector.
When you started from "/W measures /OK", you landed on that plane again, but elsewhere with a different state vector.

(This will become obvious if you draw the diagram of the vectors in Hilbert space.)

So, your assumption is that two distinct states are in fact the same state.
But they're not.

And this is exactly the error of Renner and Frauchinger.

But, don't take my word for this. Go back to your equations.

Take your state equation from post #19.
Agent F has measured +1/2, but what does agent /W measure now?

Or, take your equation from post 18.
Agent /W has measured /ok, but what is the state of agent /F ?
It's a superposition.

These are two different system states.
By jumping illegally from one to the other, Renner and Frauchiger created an artificial "contradiction".

David

 

[stevendaryl]

To assert that /F must have been precisely "tails", is to assert that the first lab is not in a superposition state

I think you're delving into explaining their contradiction, rather than showing that it's not a contradiction.

Once again, relative to an initial state $\psi$, we can define the meaning of $A \leadsto B$ where $A$ and $B$ are both statements of the form "observable X has value Y" as follows:

$A \leadsto B$ if the projection of the state $\psi$ onto the subspace corresponding to $A$ produces a state in which measurement result $B$ is certain.

More precisely, corresponding to a pair $\Lambda, \lambda$ where $\Lambda$ is an observable and $\lambda$ is one of its eigenvalues, then we can say that (relative to state $\psi$)

$(\Lambda_1, \lambda_1) \leadsto (\Lambda_2, \lambda_2)$ if the projection of the state $\psi$ onto the subspace in which operator $\Lambda_1$ has value $\lambda_1$ results in a state in which operator $\Lambda_2$ has definite value $\lambda_2$.

So for the thought experiment in question, there are 4 operators of interest:

1. $o/f$, which has eigenvalues $ok$ and $fail$
2. $p/m$, which has eigenvalues $+1/2$ and $-1/2$ (spin-up and spin-down)
3. $h/t$, which has eigenvalues $h$ and $t$ (heads and tails)
4. $\overline{o}/\overline{f}$, which has eigenvalues $\overline{ok}$ and $\overline{fail}$

The initial state $\psi$ can be written in four different ways:

1. $|\psi\rangle = \sqrt{\frac{1}{3}} (|\overline{h}\rangle |\frac{-1}{2}\rangle + |\overline{t}\rangle |\frac{-1}{2}\rangle + |\overline{t}\rangle |\frac{+1}{2}\rangle$
2. $|\psi\rangle = \sqrt{\frac{2}{3}} |\overline{fail}\rangle |\frac{-1}{2}\rangle + \sqrt{\frac{1}{6}} |\overline{fail}\rangle |\frac{+1}{2}\rangle - \sqrt{\frac{1}{6}} |\overline{ok}\rangle |\frac{+1}{2}\rangle$
3. $|\psi\rangle = \sqrt{\frac{2}{3}} |\overline{fail}\rangle |\frac{-1}{2}\rangle + \sqrt{\frac{1}{3}} |\overline{t}\rangle |\frac{+1}{2}\rangle$
4. $|\psi\rangle = \sqrt{\frac{1}{3}} |\overline{h}\rangle |\frac{-1}{2}\rangle + \sqrt{\frac{2}{3}} |\overline{t}\rangle |fail\rangle$

where
$|ok\rangle = \frac{1}{\sqrt{2}} (|\frac{+1}{2}\rangle - |\frac{-1}{2}\rangle)$,
$|\overline{ok}\rangle = \frac{1}{\sqrt{2}} (|\overline{h}\rangle - |\overline{t}\rangle)$,
$|fail\rangle = \frac{1}{\sqrt{2}} (|\frac{+1}{2}\rangle + |\frac{-1}{2}\rangle)$,
$|\overline{fail}\rangle = \frac{1}{\sqrt{2}} (|\overline{h}\rangle + |\overline{t}\rangle)$

With the definition of $\leadsto$ we can just read off the facts:

1. $\overline{o}/\overline{f} = \overline{ok}$ $\leadsto$ $p/m = +1/2$
2. $p/m = +1/2$ $\leadsto$ $h/t = t$
   
3. $h/t = t$ $\leadsto$ $o/f = fail$

It just follows from the definitions. But it's not the case that $\overline{o}/\overline{f} = \overline{ok}$ $\leadsto$ $o/f = fail$

 

[David Byrden]

Could you check your math please, numbers 3 and 4 look wrong to me. Thank you.

David

 

[stevendaryl]

Could you check your math please, numbers 3 and 4 look wrong to me. Thank you.

David

Yes, I screwed it up. Fixed now.

 

[stevendaryl]

In any kind of philosophical or mathematical or scientific argument, you can split things up into roughly three parts:

1. Assumptions and definitions
2. A proof
3. Interpretation of the conclusions of the proof

Being charitable, you can rework the argument so that the proof is correct, even if the assumptions and/or interpretation may be implausible. I think with my $\leadsto$ relation, you can rework the result of the paper so that it basically just says that $\leadsto$ is not a transitive relation. That is true, a mathematical fact about projections. What exactly this transitivity failure tells us about quantum mechanics is up to the interpretation part, which I have no firm opinions about.

 

[David Byrden]

Let me put it this way:

It's a 2 qubit system.
F-R start by putting both qubits into a known state, where they are entangled.

Then they measure the first qubit, getting " /ok ".
This changes the system state. This affects both qubits.
They make this very clear, saying that the second qubit is now "up", and it can no longer have the value "down" which was formerly possible.
So, that measurement moved the system state to a new vector in Hilbert space.

Then they say: the second qubit is "up", so what does that imply?
But they answer that question wrongly.
They take the original system state vector, and work from there.
They don't use the current state vector.

So, it all leads to this ridiculous sequence of statements :
They say: what did /W measure? He got /OK.
Meaning that agent /F is now ( heads - tails ) / √2
And then they proceed to prove that agent /F is ( tails ) when they just said she was something else!

The "contradiction" is visible in those few lines, the other 99% of the paper achieves nothing.

And it's not really a "contradiction", it's simply that they changed the state vector and then pretended they didn't.

David

 

[DarMM]

I guess you linked this version somewhere before, but I missed it. Can you give a link again?

I should have!

Richard Healey has the best exposition:
https://arxiv.org/abs/1807.00421

 

[stevendaryl]

Let me put it this way:

It's a 2 qubit system.
F-R start by putting both qubits into a known state, where they are entangled.

Then they measure the first qubit, getting " /ok ".

Yes, that was an issue that was mentioned the last time this paper was discussed, here:

https://www.physicsforums.com/threads/quantum-theory-nature-paper-18-sept.955748

If you interpret a measurement as affecting the thing measured, then the state changes after the first measurement, and so the various $\leadsto$ facts that I listed are no longer applicable, since they all are only true in the initial state.

 

[DarMM]

So, Renner and Frauchiger made an unjustified assumption here - that agent /F is not in a superposition relative to agent F.
But earlier, they assumed that she is in a superposition relative to agent /W.

This isn't a contradiction in FR, it's part of what generates (under other assumptions) the contradiction in interpretations targetted by the paper.

$F$ can view $\bar{F}$ as not being in superposition because they have interacted, $F$ has received a state from $\bar{F}$ and measured/interacted with it. $\bar{W}$ however is a superobserver for whom $\bar{F}$ lies behind the quantum-classical cut to use conventional Copenhagen language. In some interpretations it is valid for one observer to consider a system as collapsed/not in superposition and for another to consider it as in a superposition.

As far as I can see, you're saying this is wrong. FR says "yes in fact that is wrong, unless..." and there are some possibilities for the "unless..."

 

[David Byrden]

If you interpret a measurement as affecting the thing measured

That's a maddening thing about the paper. It's open to interpretation on important points.
We can safely say that it's not possible to measure a living person in a basis such as "OK". But when the paper talks about doing so, do they mean;

1. - the real implementation of the experiment will use qubits, not people
   
2. - the real implementation will use people, who will encode their state in qubits, which we can measure in arbitrary bases

That goes to the question of whether measurement will alter the system state.
I don't believe (I'm not sure!) that option [2] is workable - won't the people decohere into mixed states, whereas the experiment requires pure states?

David

 

[stevendaryl]

That's a maddening thing about the paper. It's open to interpretation on important points.
We can safely say that it's not possible to measure a living person in a basis such as "OK". But when the paper talks about doing so, do they mean;

1. - the real implementation of the experiment will use qubits, not people
   
2. - the real implementation will use people, who will encode their state in qubits, which we can measure in arbitrary bases

That goes to the question of whether measurement will alter the system state.
I don't believe (I'm not sure!) that option [2] is workable - won't the people decohere into mixed states, whereas the experiment requires pure states?

David

Yes, I think I can extract some actual correct content to the paper, but it seems to me that it only leads to an actual contradiction if you make assumptions that basically nobody would ever make, according to any known interpretation of QM.

 

[David Byrden]

$F$ can view $\bar{F}$ as not being in superposition because they have interacted, $F$ has received a state from $\bar{F}$ and measured/interacted with it.

If they were not supposed to be in superpositions relative to each other, then they'd be in the same lab.
The purpose of having two labs is precisely so that these agents can be in superposition states relative to each other.

Example: /F reads "tails" and sends a "horizontal" qubit to F, then she assumes that F is in a superposition which I will write as
( up + down ) / √2
and that is necessary to the paper's result.

Agent F has received a state in a qubit from /F, that's true, but (as I've been pointing out) many superpositions of states in /F can result in agent F making the same measurement. In simple terms; if F measures "down", can she deduce the state of /F with certainty?

David

 

[DarMM]

Yes, I think I can extract some actual correct content to the paper, but it seems to me that it only leads to an actual contradiction if you make assumptions that basically nobody would ever make, according to any known interpretation of QM.

I think the Masanes version lines up pretty closely with things assumed by objective outcome $\psi$-epistemic interpretations, e.g. forms of Copenhagen, Zeilinger and Brukner's original views, what's in Haag's local quantum physics.

To be clear I am talking about views where $\psi$ is epistemic and there are no hidden variables. Terminology isn't locked down on these, some calling them $\psi$-doxastic, $\psi$-epistemic or $\psi$-epistemic type II.

Views where $\psi$ is epistemic but there are hidden variables are variously called $\psi$-epistemic, $\psi$-statistical or $\psi$-epistemic type I.

 

[DarMM]

Agent F has received a state in a qubit from /F, that's true, but (as I've been pointing out) many superpositions of states in /F can result in agent F making the same measurement. In simple terms; if F measures "down", can she deduce the state of /F with certainty?

If $F$ views $\bar{F}$ as being in superposition, then their state (from the agreed procedure) is:
$$\sqrt{\frac{1}{3}}\left(|\downarrow,h\rangle + |\downarrow,t\rangle + |\uparrow,t\rangle\right)$$

So if they measure spin up it must be tails right?

 

[David Byrden]

So if they measure spin up it must be tails right?

Modern interpretatiions of QM acknowledge that the state of a quantum system is relative to the observer.
This goes to the heart of the Renner-Frauchiger experiment, where the various observers hold different subsets of information about the system.
For each observer there can be a different system state.

If Agent F measures "spin up", then she can infer something about Agent /F, but it depends on the system state vector.
The original vector, as you pointed out, tells her that /F can only be "tails".

But Agent F doesn't know what the system state is.
She knows it in her mind, to be sure, but her environment doesn't contain knowledge of it, acquired through the Wave Function.

So, all that she really can say, is that she measured "spin up".

David

 

[Demystifier]

She knows it in her mind, to be sure, but her environment doesn't contain knowledge of it, acquired through the Wave Function.

I don't think that there is a difference between knowledge "in the mind" and knowledge "contained in the environment". After all, the state of the mind is in fact a state of the brain described by a wave function of the brain.

 

[Demystifier]

Richard Healey has the best exposition:
https://arxiv.org/abs/1807.00421

Concerning the Masanes's argument, that is the "Third Argument" in the Healey's paper, it seems to me that the crucial assumption responsible for the appearance of inconsistency is the assumption of Lorentz invariance. Would you agree? If so, and given that the assumption of Lorentz invariance is closely related to the assumption of locality, isn't the "Third Argument" just a restatement of the good old Bell theorem that the existence of unique objective outcomes in QM is incompatible with locality?

 

[DarMM]

Concerning the Masanes's argument, that is the "Third Argument" in the Healey's paper, it seems to me that the crucial assumption responsible for the appearance of inconsistency is the assumption of Lorentz invariance. Would you agree? If so, and given that the assumption of Lorentz invariance is closely related to the assumption of locality, isn't the "Third Argument" just a restatement of the good old Bell theorem that the existence of unique objective outcomes in QM is incompatible with locality?

Just to let you know I need to do some thinking on this, there are others who have said similar and it seems to be a subtle observation. 2-3 days I estimate, I don't want to just blurt out a lazy response.

 

[RUTA]

See also:
On Formalisms and Interpretations
Veronika Baumann, Stefan Wolf
https://arxiv.org/abs/1710.07212

I detail in The Quantum Mystery of Wigner's Friend how Baumann and Wolf's paper bears on FR's paper.

 

[David Byrden]

I don't think that there is a difference between knowledge "in the mind" and knowledge "contained in the environment".

Here's an example showing the difference; an uneducated person is told that we're firing particles at two slits (Young's experiment). In his mind, he knows that each particle will pass through a single slit. But the environment knows that they pass through both slits.
The environment wins out, and it forms interference patterns.

 

[DarMM]

Modern interpretatiions of QM acknowledge that the state of a quantum system is relative to the observer.

Well some do. It's not a universal feature of all modern interpretations.

But Agent F doesn't know what the system state is.
She knows it in her mind, to be sure, but her environment doesn't contain knowledge of it, acquired through the Wave Function.

So, all that she really can say, is that she measured "spin up".

You're basically saying that all the agent can say is their own personal experience of the outcome occurred? Even after being told the initial state and the exact way the experiment will be performed, you're saying they could still just assign "some" random state inconsistent with that information? If the quantum state forms an objective description of the statistics of the experiment then:
$$\sqrt{\frac{1}{3}}\left(|\downarrow,h\rangle + |\downarrow,t\rangle + |\uparrow,t\rangle\right)$$
is the only valid state for them to use, the correct one.

If on the other hand, there is no such thing as $|\psi\rangle$ and it's just in the agent's head and subjective to their situation, i.e. they could pick any $|\psi\rangle$ and just use the observation "spin is up" to update it via Lüders Rule, then you are rejecting the "objective" part of Assumption 1 I gave in the OP.

You're just assuming an interpretation that gets out of the contradiction from the beginning, it's not a flaw in the argument.

The actual flaw in the Frauchiger-Renner paper is that they don't notice they have an assumption about observer's not being entangled with each other from a superobserver's perspective. This is sort of the more developed version of Scott Aaronson's issue with the paper and is given more coverage in Richard Healey's paper.

To be clear, the contradiction FR find can be escaped by:

1. Saying QM is a single-user subjective theory, like Relational QM or QBism
2. Saying QM will be wrong in certain experiments (this seems to be the response of some Bohmians)
   
3. Multiple Worlds, like the Everett interpretation
4. Wave functions objectively undergo collapse as a physical process
5. Accept that Observers can be entangled, they're not purely classical like basic Copenhagen, i.e. accept that even after a measurement you could always be on the Quantum side of a very powerful Observer's Heisenberg cut. There isn't a pure Quantum-Classical divide.

The problem is FR don't mention the assumption that permits (5.) as a way out.

The Masanes's version removes (5.) as a way out, but introduces an alternate way out:
5*. Not all unitaries are reversible in principle.

 

[atyy]

I don't. Do you?

Hmmm, I don't know. I haven't had the time to read it carefully. It does seem to me as mistaken as Ballentine's argument that Copenhagen is wrong, and ignores previous work like Hay and Peres's https://arxiv.org/abs/quant-ph/9712044.

 

[1977ub]

I don't think that there is a difference between knowledge "in the mind" and knowledge "contained in the environment". After all, the state of the mind is in fact a state of the brain described by a wave function of the brain.

If one has a problem with other minds, then other people don't have minds. This is not a contradiction of anything empirical. Also it may be that each particle goes through one slit or the other and something mysterious keeps a tally to make it as if it had gone through both, no?

 

[DarMM]

ignores previous work like Hay and Peres's

I had a read of Hay and Peres's paper, what aspect of it do they ignore. The subject topic is similar, but I don't see them ignoring anything there.

It does seem to me as mistaken as Ballentine's argument that Copenhagen is wrong

What part seems mistaken?

 

[Demystifier]

Just to let you know I need to do some thinking on this, there are others who have said similar and it seems to be a subtle observation. 2-3 days I estimate, I don't want to just blurt out a lazy response.

I made some additional thinking too and now I think I know exactly what is wrong with the Healey's "Third Argument". In the critique of the Frauchiger and Renner’s argument, Healey correctly objects that this argument rests on the unjustified assumption of intervention insensitivity. But in the "Third Argument", Healey argues that the "Third Argument" does not assume the intervention insensitivity, and therefore that the "Third Argument" is correct. I think it's wrong. I think the "Third Argument" also uses the assumption of intervention insensitivity and is therefore incorrect.

Let me explain. The main culprit is Eq. (31). Healey does not explain how exactly this inequality is obtained, but as far as I can see, this inequality is nothing but a variant of CHSH inequality (see e.g. https://en.wikipedia.org/wiki/CHSH_inequality). On the other hand, it is well known that CHSH inequality is a consequence of the assumption of non-contextuality (which in the literature is usually justified by the assumption of locality). Hence the contradiction derived by Healey seems to originate from the assumption of non-contextuality, which is more-or-less the same as the assumption of intervention insensitivity. Since this assumption is unjustified, the "Third Argument" seems wrong.

 

[stevendaryl]

In my effort to understand what exactly was proved by the Frauchiger and Renner argument, I have come to the conclusion that what they show is impossible is not something that anybody believed, anyway. I already said this, but I feel that it's worth repeating:

The way I described it is through the $\leadsto$ relation: That should actually have a subscript, $\psi$, because it's relative to an initial state.

I write $(A, \alpha) \leadsto_\psi (B, \beta)$ to mean: If a measurement of observable $A$ in state $\psi$ produces result $\alpha$, then it is certain that a later measurement of $B$ will produce result $\beta$. This is the sort of reasoning using in EPR experiments. With anti-correlated spin-1/2 twin pairs, if Alice measures the spin of her particle along axis $\vec{a}$ and gets result $\alpha = \pm \frac{1}{2}$ and later Bob measures the spin of his particle along that same axis, he will get result $-\alpha$ with certainty.

So the FR argument just seems to me to be the claim that the $\leadsto$ relation is not transitive:

If $(A, \alpha) \leadsto (B, \beta)$ and $(B, \beta) \leadsto (C, \gamma)$, it's not necessarily true that $(A, \alpha) \leadsto (C, \gamma)$.

It's hard to see why anyone would think it was transitive. A measurement potentially modifies the state, so with a chain of measurements,

1. Initially, the state is $\psi$
2. A measurement of $A$ produces result $\alpha$.
3. Now, the state is $\psi' \neq \psi$
4. A measurement of $B$ produces result $\beta$.
5. But we don't have $(B, \beta) \leadsto_{\psi'} (C, \gamma)$ so we can't conclude that a measurement of $C$ will produce result $\gamma$.

(Note: the statement $(A, \alpha) \leadsto (B, \beta)$ must assume that there is no interference in the state that is relevant to the measurement of $B$ that takes place after the measurement of $A$. Going back to the EPR case, obviously, if between Alice's measurement and Bob's measurement, someone interacts with Bob's particle, then it will no longer be guaranteed that his result will be anticorrelated with Alice's)

If I'm not missing something, then it seems to me that the thing proved impossible by the argument is not something that anyone would believe, anyway.

Now, I think that an argument for transitivity could be made in certain circumstances. The measurement of $A$ may change the state, but in a way that is irrelevant to the final measurement of $C$. But that requires a separate argument, it seems to me.

 

[DarMM]

If I'm not missing something, then it seems to me that the thing proved impossible by the argument is not something that anyone would believe, anyway.

I don't think you are missing anything. As you say $\leadsto_\psi$ simply couldn't be transitive (in slightly different language this paper points out what you do: http://philsci-archive.pitt.edu/15552/).

I think the only thing blocked by the result are certain forms of Copenhagen where you say observers/measuring devices are completely classical, where they'd say you should be able to use a transitive form of $\leadsto_\psi$ or something like it. However I don't think forms of Copenhagen this strong made much sense anyway, since although it is fine for the observer to treat themselves as classical, asking that superobservers treat observers as classical is essentially just objective collapse.

However your objection doesn't seem to apply to Masanes's version. Though perhaps it has other problems as @Demystifier mentions.

 

[DarMM]

The main culprit is Eq. (31). Healey does not explain how exactly this inequality is obtained, but as far as I can see, this inequality is nothing but a variant of CHSH inequality (see e.g. https://en.wikipedia.org/wiki/CHSH_inequality). On the other hand, it is well known that CHSH inequality is a consequence of the assumption of non-contextuality

Eq. 31 comes about via an assumption of objective outcomes, i.e. actual objective facts about the four measurements exist. Do trillions of runs of the Masanes experiment and there will be a frequency distribution from the data $p(a,b,c,d)$. Via Fine's theorem this gives the inequality quoted for the marginals and ultimately one has the contradiction.

Or in brief the contradiction is QM is saying that the $E(i,j)$ should break the CHSH inequality, but the fact that they are marginals of the distribution for the ontic/objective device clicks means they shouldn't break the inequality.

This is avoided in a normal CHSH experiment because you can only get outcomes for a pair of variables in a single run, not all four as here. Hence the $E(i,j)$ are simply the probabilities predicted by QM. Reversing the measurements in Masanes thought experiment however causes them to also be marginals of a classical (i.e. Kolmogorov) probability distribution.

The QBist way out of this is that $p(a,b,c,d)$ is simply meaningless, there are no distributions for objective facts, only facts for agents. Since nobody can experience all of $a,b,c,d$, it is meaningless to speak of $p(a,b,c,d)$.

Retrocausal theories permit one of the marginals, e.g. $E(a,d)$, to deviate significantly from QM allowing consistency between the marginals and $p(a,b,c,d)$.

Interested to hear your thoughts.

 

[Demystifier]

Eq. 31 comes about via an assumption of objective outcomes, i.e. actual objective facts about the four measurements exist.

I don't see how Eq. (31) comes from that assumption alone. I would like to see an explicit derivation. I will check the Fine's paper and comment it later.

 

[Demystifier]

I don't see how Eq. (31) comes from that assumption alone. I would like to see an explicit derivation. I will check the Fine's paper and comment it later.

I checked the Fine's paper and now I am even more convinced that I am right. Eq. (31) in the Healey's paper is
$$|corr(a,b)+corr(b,c)+corr(c,d)-corr(a,d)|\leq 2$$
after which he writes: "Note that no locality assumption is required to derive this inequality here, since it is mathematically equivalent to the existence of a joint distribution over the actual, physical outcomes whose existence has been assumed [17]."

The Ref. [17] is the paper by Fine. The central result is the Fine's Theorem 7 which says:
"Observables $A_1$ , ... ,$A_n$ satisfy (the joint distribution condition) if and only if all pairs commute."
To apply this to Eq. (31) above, the relevant "pairs" are pairs of spin operators in different directions. But spin operators in different directions do not commute, so Theorem 7 implies that the joint distribution condition is not satisfied. Hence the quoted Healey's claim above that the "existence of a joint distribution ... has been assumed" is simply an expression of a wrong assumption. It looks as if Healey have not understood the content of Ref. [17] that he cites as an alleged support of his claim. It's not only that Ref. [17] does not support his claim, but just the opposite, it explains why his claim after Eq. (31) is wrong.

 

[DarMM]

Please note I'm not too certain about this myself...

It looks as if Healey have not understood the content of Ref. [17] that he cites as an alleged support of his claim

Just to be clear, this would mean many people don't understand the content of Fine's theorem as Healey's exact presentation of the argument has been used by Matt Leifer and Matthew Pusey. That's not to say you are wrong, just telling you the "scope" of the error if you are right.

To apply this to Eq. (31) above, the relevant "pairs" are pairs of spin operators in different directions. But spin operators in different directions do not commute

Okay, but is this not just another presentation of the contradiction, i.e. we have two results:

1. The $E(i,j)$ are the probabilities predicted by QM.
2. The $E(i,j)$ are marginals of a distribution $p(a,b,c,d)$, coming from simply the occurrence of the four outcomes in each run.

I agree that (2.) then implies the observables would commute, which is wrong. However to me that is simply the contradiction stated another way:

1. Given the experimental set up, they are marginals
2. As marginals they have to commute
3. However from QM they obviously don't commute
4. Thus a contradiction

I don't see the fact that they need to commute as contradicting Healey, as the contradiction is between conditions on them from being marginals conflicting with their properties from QM.

In a way the whole point of Masanes set up is to force the existence of $p(a,b,c,d)$ to give these contradictions.

Hence the quoted Healey's claim above that the "existence of a joint distribution ... has been assumed" is simply an expression of a wrong assumption

It's a fact of the experiment I would have said, $a,b,c,d$ occur in a single run, thus there must be a frequency of their occurance $p(a,b,c,d)$. How could there not be?

$p(a,b,c,d)$ is not coming from QM as such, just a fact of objective experimental outcomes existing.