The Renner-Frauchiger experiment is a very simple one, but the authors' paper can be difficult to follow. Here, I will step through it and spell it out plainly, without superfluous details. I will demonstrate a mistake in the authors' conclusions.
In my analysis of the experiment, I won't rely on any specific interpretation of QM. But I will assume that quantum states are relative to the observer.
Discarding much of Renner-Frauchiger's notation, I will use simplified names for the parts of the experiment.
Pure or mixed states?
Renner and Frauchiger describe their apparatus as two human observers in perfectly isolating "labs". They call it a "complex system" because it contains observers thinking about other observers' observations.
But, if you examine their equations, you find that the two "labs" are mere qubits. They cannot be humans or other macro objects because they are measured in arbitrary bases and they have superpositions. They don't decohere into mixed quantum states. Indeed, if you recalculate everything using mixed states, the experiment doesn't work.
So Renner and Frauchiger's "complex system" is in fact extremely simple. There is no need for human observers inside their system to make observations because we, the people reading about the experiment, can calculate what they would observe.
(Renner and Frauchiger wonder aloud if it's possible for one observer, bewitched by quantum magic, to reliably calculate what another one sees. But if this can't be done, then Renner and Frauchiger themselves cannot imagine how their experiment will play out, and their paper must be discarded.)
So, in my analysis, I will do as Renner and Frauchiger did. I will mathematically treat the two "labs" as qubits, while speaking of them as people.
Beginning the experiment
At first, FBAR reads a quantum "coin" and goes into a superposition of states |HEADS> and |TAILS>. This superposition is relative to any observer outside her lab, including F and WBAR.
The probability (under the Born rule) of her two states being measured is |HEADS> 1/3 , |TAILS> 2/3. I am keeping track of measurement probabilities because they will reveal the point where a mistake was made by Renner and Frauchiger.
Entangling F with FBAR
FBAR then emits a temporary qubit called R, which shall be measured in the UP/DOWN basis. We now have a 2-qubit system in a superposition of three states :
Then, F measures R and becomes entangled with its state. We now redefine the system to comprise FBAR and F. The system state, EQUATION A, is unchanged.
F is now in a superposition of the states |UP> and |DOWN>, relative to external observers. Their probabilities are 1/3 and 2/3 respectively.
Thanks to the way R was configured, F and FBAR are now partially entangled. That means if you measure one, you will obtain at least some information about the other. As an example, let's see what happens if you measure F in the UP/DOWN basis.
The overall system state, relative to you, will become either :
By "collapsing" F into either the |UP> or |DOWN> state, you have "collapsed" the system (two entangled qubits) into one of these joint states. By measuring F, you have obtained some information about FBAR. In particular, if you measured F as |UP>, then you got complete information about FBAR because it's necessarily in the state |TAILS>.
Points of view
One aspect of the Renner-Frauchiger experiment that confuses people, is the different points of view that the observers have. At this point in the procedure, F has measured FBAR (via R), so F has seen a "collapse", a change of the system state. But external observers like WBAR have not measured anything. The system state is still EQUATION A relative to them. How can we reconcile these two points of view?
The key to understanding this, is to imagine that F is now in a superposition. For an outside observer, there are two copies of F coexisting inside the "lab". One of them measured |UP> and the other |DOWN>. And they have different opinions about the state of FBAR. In fact, for those two copies of F, the system state is now EQUATION B or EQUATION C respectively.
Both of these superposed copies of F feel very real to themselves, but for an outsider, neither of them certainly exists. If the outsider were to open the "lab" and look inside, each version of F has only a certain probability of being found. Those probabilities are 1/3 and 2/3, as I mentioned already.
Here is the reconciliation: if you weight EQUATION B and EQUATION C by their probabilities, and add them, you get the original system state EQUATION A.
In other words: the measurement made by F, which seemed very real to her, is not real to an outside observer. To them, it's a possible future measurement. It's just one way of potentially "slicing" the system state into two parts. There are other ways to "slice" it; e.g. they can measure FBAR instead, or they can use other bases to measure in.
Whatever measurement they make, it will divide the system state into two portions, one of which will be the result. The two parts will always add up to the whole. And if an internal observer (like F) has already measured the system, then that observer is in a superposition relative to the outside, and her superposed weighted measurement results add up to the whole.
I hope you can now see that Quantum Mechanics describes "nested observer" experiments in a logical, self-consistent way.
The external measurement
Renner and Frauchiger now proceed to create their "paradox". They tell us that WBAR measures FBAR, but he uses a new basis called OK/FAIL.
As I discussed, this will collapse the joint system state into one of two values. To find them, we rewrite EQUATION A, putting FBAR into the new basis :
This tells us that WBAR has a 1/6 probability of measuring |OK>, and the other 5/6 goes to |FAIL>.
Renner and Frauchiger tell us to assume the |OK> result. So we pick out the terms of EQUATION D that contain |OK> and we normalise, to get :
This is interesting, because WBAR didn't measure F, but nevertheless he is absolutely certain that she is now in the |UP> state (relative to him).
Now, look at the system state relative to WBAR. The "lab" called F is in the |UP> state and FBAR is in the |OK> state. But EQUATION B says that when F is in the |UP> state, FBAR is in the |TAILS> state.
EQUATION B doesn't represent a "counterfactual" measurement. We know that F really did make a measurement. It put her into a superposition of two results, but WBAR has now "collapsed" that, and she's |UP>.
It seems that these two system states, EQUATION E and EQUATION B, should both be real now, with respect to WBAR. They contain F in the |UP> state and different states for FBAR. But FBAR can't be in two states at the same time with respect to one observer. That is Renner and Frauchiger's "paradox".
In their paper, they don't point out the "paradox" right here. Instead, they carry on with more measurements and more calculations and another observer called W, and eventually this "paradox" causes a discrepancy between W and WBAR. But that extra work is unnecessary. We can already see the "paradox".
I won't follow Renner and Frauchiger's remaining steps. I will resolve the discrepancy between EQUATION B and EQUATION E.
FBAR cannot be in two different states, relative to F, at the same time. These equations seem to break that rule. Or do they? EQUATION B was calculated before WBAR made his measurement, and EQUATION E was calculated after it. That is not literally "the same time". Can that explain the contradiction?
Actually, no it can't. The observer F can consider the entire world outside her lab to be a quantum system. Measurements made by observers within that system should not change its state relative to her. So if FBAR was |TAILS>, relative to F, then WBAR's measurement should not change that fact.
The resolution of the "paradox" is subtle. Let's recall the "a priori" probabilities of these states:
From EQUATION D, there's a 1/6 chance of WBAR measuring FBAR as |OK> and thus "collapsing" F into the state |UP>. But from EQUATION A, there's overall a 1/3 chance of F being |UP>. That mismatch is a clue that points to the solution.
Let's rewrite EQUATION B as a superposition of two states :
When WBAR "collapses" FBAR into the |OK> state, and also "collapses" F into |UP>, he is really "collapsing" this joint state into |UP OK>. This joint state had 1/3 probability, so each part of it had 1/6 probability. Now, the probabilities match.
Explaining the resolution
The equations are telling us this: the oberver F, who "sees" FBAR in the |TAILS> state, can be regarded as a superposition of two observers. They are both F herself, they are both in the |UP> state, but FBAR has different states relative to them.
Relative to WBAR, the original F was real at first. But when he measured FBAR, he "collapsed" that F into one of the two copies.
So, there is no "paradox". It's not true that FBAR had two different states relative to the same observer at the same time, because these two versions of F are not "the same observer".
This explanation will baffle some. The observer F has not been touched, and nothing inside her "lab" has changed between EQUATION B and EQUATION E. How can I say she's "not the same observer" in both?
In cat terms
Let's look at the much simpler "Schrodinger's Cat" experiment. Relative to Schrodinger, the cat is in a superposition of |ALIVE> and |DEAD>. In this trivial experiment I can set up a "paradox" similar to that of Renner and Frauchiger:
"Relative to the live cat, Schrodinger is in state |S>. Relative to Schrodinger in state |S>, the cat is in a superposition. But the cat cannot be both fully alive AND in a superposition. "
The resolution of this paradox is the same. We must admit that Schrodinger, in state |S>, is a superposition of two Schrodingers, both in that same state |S>, but there is a different cat state relative to each of them.
The correct mathematical way to handle this situation, is to deal with the joint state of the two objects. For example, |ALIVE S> and |DEAD S> are two joint states of the Schrodinger apparatus, each with 50% probability, and they add to yield the full system state.
In quantum mechanics, we often calculate joint states in cases of entanglement. But Schrodinger and the cat are not entangled. It's our decision to consider the state of one, relative to the other, that makes it necessary to calculate the joint state. Renner and Frauchiger failed to do that.
Renner and Frauchiger have not discovered a problem in Quantum Mechanics. They have not proven that one of three "assumptions" is faulty.
They did not recognise that an observer can be a superposition of multiple observers, all seemingly identical, but all seeing different things.