# FeaturedI Summary of Frauchiger-Renner

1. Jan 7, 2019

### 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.

Last edited: Jan 9, 2019
2. Jan 8, 2019

### Demystifier

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

3. Jan 8, 2019

### 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

Last edited: Jan 8, 2019
4. Jan 8, 2019

### DarMM

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

5. Jan 9, 2019

### David Byrden

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

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...

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 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

Last edited: Jan 9, 2019
6. Jan 9, 2019

### Demystifier

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.

7. Jan 9, 2019

### DarMM

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.

8. Jan 9, 2019

### DarMM

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.

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.

9. Jan 9, 2019

### 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!

10. Jan 10, 2019

### 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 realised.

David

11. Jan 10, 2019

### Demystifier

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

12. Jan 10, 2019

### atyy

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

13. Jan 10, 2019

### Demystifier

I don't. Do you?

14. Jan 10, 2019

### stevendaryl

Staff Emeritus
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.

15. Jan 10, 2019

### David Byrden

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.

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.

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

16. Jan 10, 2019

### stevendaryl

Staff Emeritus
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.

17. Jan 10, 2019

### David Byrden

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

No, they didn't prove anything.

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.

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

18. Jan 10, 2019

### stevendaryl

Staff Emeritus
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.

Last edited: Jan 10, 2019
19. Jan 10, 2019

### stevendaryl

Staff Emeritus
(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 $|\overline{t}\rangle$

20. Jan 10, 2019

### 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

Last edited: Jan 10, 2019