# 2018 Workshop on Wigner’s Friend

## On the second FR argument

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Tony Sudbery
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On the second FR argument

The theorem claimed by Frauchiger and Renner in their recent publication in Nature Communications is different from the one they presented in their eprint arXiv:1604.07422, which was discussed in the references to this forum. Both theorems use the same extension of the Wigner’s friend experiment and claim to show that certain assumptions are jointly inconsistent. The assumptions in the 2016 paper included “compliance with standard quantum theory”, and the subsequent discussion highlighted the ambiguity of the phrase “standard quantum thoery” — in particular, does it include an application of the projection postulate, and if so in what circumstances? The 2018 paper replaces this set of assumptions by a more specific, but less powerful, set and argues that these are sufficient to establish a contradiction. I don’t believe that this argument is valid: the authors implicitly rely on some further assumptions which must be added to their set.

The crucial step in the argument is he inference by agent Fbar that W will observe w = fail at time n:31. According to Table 3, this inference proceeds via the assumption (Q), which asserts that knowledge that a system is in an eigenstate of an observable X at time t_0, with eigenvalue xi, licenses an inference that the value of X is xi at a later time t, when the measurement of X is completed. As stated, this assumption appears to apply to all later times t, but this surely cannot be intended; it would assert that no time development is possible. Nevertheless, this is the only reference to time development in the assumptions of the 2018 paper, and the inference which they impute to agent Fbar refers to the final time in the experiment, after the assistant Wbar has made a measurement on Fbar and their lab. The more reasonable interpretation of Assumption (Q) is that t must be a time immediately after the completion of the measurement of X; but in that case there is no warrant for Fbar’s inference of events at the later time t:30. There would be a need for a further assumption about what governs the time develpoment of the system. A minimal form for this assumption might be to the effect that operations on a system cannot affect a spatially separated system, in other words a locality assumption.

But there is a more crucial assumption hidden in FR’s argument. This is

Assumption (P) If an agent A prepares a system S in a pure state \psi at time t, then A believes that the state of S is \psi immediately after t.

This might appear undeniable, but it is not true if A knows that she is in an entangled state, which is the situation in the FR experiment. In that case she believes that after her preparation S is also in an entangled state, and therefore S does not have a pure state.

Thus the theorem established by Frauchiger and Renner should read:
Any theory that satisfies assumptions (Q), (C), (S), (P) and a locality assumption yields contradictory statements when applied to the Gedankenexperiment of Box 1.

Hence one of the assumptions must be dropped. I think the most appropriate one to drop is (P). Denying (P) does seem odd — in the terms of FR’s 2016 paper, (P) is part of standard quantum mechanics — and a way of rescuing it is offered by the relative-state interpretation of Everett and Wheeler. In that interpretation it is true that after A, in the state \alpha, prepares the system S in the pure state \psi, the state of S relative to A and \alpha is indeed the pure state \psi. The use made by Frauchiger and Renner of this assumption is to deduce a belief of Fbar about the absolute state of the spin S. Thus The problem with their argument can be seen as a failure to distinguish between relative and absolute states.

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