A paper by Renato Renner and Daniela Frauchiger poses an interesting conundrum. It shows how a series of observations made on a specific quantum system can lead to an apparent contradiction. The paper is titled "Quantum theory cannot consistently describe the use of itself".
Here, I will briefly point out an error in the logic of the paper.
The error in the argument
The authors present a sequemce of observations, inferences and conclusions in Table 3. Each one depends on the prior one. There is an error in the first one which invalidates the whole chain of logic.
The first observation is: agent /F reads r as 'tails' at time n:01.
According to the procedure in Box 1, she then sends a qubit to agent F. Table 2 tells us that agent F reads the qubit, then her state gets measured by agent W. Agent /F knows all of this, and concludes the following;
"I am certain that W will observe w = 'fail' at time n:31.”
This conclusion assumes that the qubit reaches agent F in exactly the state prepared by agent /F.
But in fact, agent /F is in a superposition. There are two versions of her, one who read r='tails' and one who read r='heads'. Both versions of agent /F will prepare the qubit differently. The two versions of the qubit are coherent, so agent F will read their combined value. This value does not inevitably lead to agent W observing 'fail'.
So the first conclusion in the table is false because agent /F ignores her own quantum superposition.
The authors mention, in equation 7, that there is a non-zero probability of measuring the system in a specific state. Then, using the chain of logic in Table 3, they attempt to show that this state is impossible to reach.
Since Table 3 contains an error (described already) then there is no contradiction. Equation 7 is correct. The system, when measured by agents W and /W, will sometimes be found in the "impossible" state.