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.