Question about Wigner's friend

Question

The Wigner's friend thought experiment can be used to understand non-realism in quantum mechanics. For anyone not familiar, the thought experiment involves two researchers observing an experiment at different times, let's say it's an electron spin. The question is, how should the second researcher to observe the experiment treat the system between his observation and his friend's observation?

In a non-realist interpretation we would say that the second researcher should continue to time evolve the system with the Schrodinger equation until he observes the system himself, but now he must include his friend as part of the quantum system as well. So, the two observers see the collapse of the wavefunctions at different times, which is fine because the time of wavefunction collapse is not given by a linear Hermitian operator so they need not agree about it.

My question concerns what happens if the additional time evolution that the second researcher observes changes the probability. For instance, say there is a 50% probability of spin up when the first researcher observes the experiment. Can this probability change from the second observer's perspective in the time between observations? If it can, how do we explain the fact that if they repeat the experiment 100 times the first researcher would expect to see spin up 50 times and the second researcher would expect to see something else. If it can't, what is the purpose of the additional unitary time evolution that the second researcher uses to describe the situation, couldn't he just use a description where the wavefunction collapses when the first researcher observes the experiment and get the same answer?

Answer 1

I disagree with a central point in your question: the second researcher should not continue to time evolve the system with the Schrodinger equation until he observes the system himself.

Once the first researcher measures the system, the state of the system becomes entangled with the state of it's environment (which includes the first researcher). The information of the measurement result propagates very quickly into the environment (photons from the measurement device spread, etc.), in a process called decoherence, creating a redundancy in which many copies of the result are imprinted in all that surrounds. The total state of the system plus the environment can be effectively described as (assuming the system is a single spin like in your question): $$ \frac{1}{\sqrt{2}} \left( |\uparrow \rangle |\text{environment after spin up result} \rangle + |\downarrow \rangle |\text{environment after spin down result} \rangle \right) $$ The second researcher, which doesn't know the outcome of the measurement yet, would now describe the system as a mixed state, with the density matrix: $$ \rho= \frac{1}{2} \left( |\uparrow \rangle \langle \uparrow| + |\uparrow \rangle \langle \uparrow| \right) $$ The same goes with how the second researcher would describe the first researcher after the measurement: in a mixed state, and not in a coherent superposition of having measured up and having measured down. This is because the information of the result has many copies beside the one in the first researcher's brain, and any such copy will be part of the big state I gave of the system plus the environment. Any description that includes only part of all the environment that was affected by the measurement, will necessarily be of a mixed state and not of a superposition.

All this makes the rest of the question irrelevant. The second researcher now needs to time evolve the system in the same way the first researcher does, only he needs to calculate two possible time evolutions - one if the measurement result was spin up, and one if it was spin down. Each case will be exactly equivalent to what the first researcher, who knows the measurement result, will calculate.

Answer 2

Suppose you have an entangled pair $c_0|0\rangle|0\rangle + c_1|1\rangle|1\rangle$. A local unitary operation $U$ on the left qubit would change this to $c_0U|0\rangle|0\rangle + c_1U|1\rangle|1\rangle$. This will, in general, change the probabilities of the 0/1 outcomes (unless $c_0=c_1$, which is a nice exercise in itself).

But this isn't the kind of "probability change" you want. You probably mean that the state changes to $c'_0|0\rangle|0\rangle + c'_1|1\rangle|1\rangle$. But this requires you to manipulate the whole state as a coherent superposition. For example, you could undo the entanglement, modify the first qubit, and then run it forward again. The second qubit would have no "memory" of the first run, because you undid it. So there would be no contradiction.

As you recognize in your comment on the other answer, any real macroscopic setup would quickly become decoherent, preventing you from manipulating the state in this way.