In restating the argument of [
9,
12] I have changed the notation to try to make it easier to follow. The following table supplies a translation between my notation and that used in [
12].
Readers familiar with Wigner’s original “friend” argument [11] will be primed to attribute extraordinary powers to the experimenter I have named Wigner, and I thought it appropriate to name a second character with such almost “God-like” powers Zeus. This naturally suggested also giving the experimenters charged with less extraordinary tasks names whose initial letters are also at the end of the alphabet, with corresponding labels for their labs and measured observables.
While such changes are merely cosmetic, my restatement deliberately lacks one feature emphasized by the authors of the argument of [12] that they call “consistent reasoning”, illustrate in their Fig. 1, and formalize in their assumption (C). Both in the original and in my restatement it is Wigner (W) whose reasoning is the ultimate focus of the argument. But the authors of the original argument consider it important that Wigner’s reasoning incorporates the reasoning of the other experimenters [via assumption (C)].
It is vital to check whether Wigner’s reasoning is both internally consistent and consistent with the reasoning of the other experimenters in this Gedankenexperiment. My restatement makes it clear how Wigner can consistently apply quantum theory without considering the reasoning of any other experimenters. But are the conclusions of this independent reasoning by Wigner consistent with those of the other experimenters, based on their own applications of quantum theory? Indeed they are, provided each experimenter has applied quantum theory correctly. The problem with the argument of Frauchiger and Renner is that one experimenter (Xena/\(\bar{F} \)) has applied quantum theory incorrectly.
Recall step 4* of the reasoning in my restatement of this argument (see §3). I attributed this reasoning to Wigner, while pointing out that Zeus’s subsequent measurement of
z renders it fallacious. Frauchiger and Renner initially attribute parallel reasoning to Xena/
\(\bar{F}\) and then use assumption (C) to attribute its conclusion also to
Wigner. To see where things go wrong if Xena/
\(\bar{F}\) reasons this way, I quote from [
12].
“Specifically, agent
\(\bar{F}\) may start her reasoning with the two statements
$$\begin{aligned} s_{I}^{\bar{F}}&=``\text {If }r=tails\text { at time }n:10\text { then spin }S\text { is in state }\left| \rightarrow \right\rangle _{S}\text { at time }n:10\text {''}\\ s_{M}^{\bar{F}}&=\text { The value }w\text { is obtained by a measurement of }L\text { w.r.t.}\{\pi _{ok}^{H},\pi _{fail}^{H} \}\text {''}.'' \end{aligned}$$
They conclude that
\(\bar{F}\) can infer from
\(s_{I}^{\bar{F}}\) and
\(s_{M} ^{\bar{F}}\) that statement
\(s_{Q}^{\bar{F}}\) holds:
$$\begin{aligned} s_{Q}^{\bar{F}}= & {} ``\text {If }r=tails\text { at time }n:10\text { then I am certain that }W\text { will observe }\\ w= & {} fail\text { at } n:40\text {''}. \end{aligned}$$
Starting with
\(s_{Q}^{\bar{F}}\), they then apply assumption (C) to the reasoning of the other agents successively, eventually to establish that
Wigner may conclude
$$\begin{aligned} s_{2}^{W}= & {} ``\text { If }\bar{w}=\overline{ok}\text { at time }n:30\text { then I am certain that I will observe }\\ w= & {} fail\text { at }n:40\text {''}, \end{aligned}$$
which (given (S)) is inconsistent with
W’s independent conclusion (based on assumption (Q))
\(s_{Q}^{W}=\)“I am certain that there exists a round \(n\in \mathbb {N} _{\ge 0}\) in which it is
announced that \(\bar{w}=\overline{ok}\) at time n : 30 and \(w=ok\) at n : 40.”
But this chain of reasoning is based on a mistaken starting point, since
\(\bar{F}\) has applied quantum theory incorrectly in asserting statement
\(s_{Q}^{\bar{F}}\). Compare
\(s_{Q}^{\bar{F}}\) with the corresponding conclusion of Wigner’s fallacious reasoning in step 4* of §3:
“If the unique outcome of Xena’s measurement of f on c at \(t=0\) had been “tails”, the unique outcome of my measurement of w on Y at \(t=4\) would have been “fail”.
Agent
\(\bar{F}\)’s reasoning was equally fallacious here. The problem starts with statement
\(s_{I}^{\bar{F}}\):
\(\bar{F}\) is correct to assign state
\(\left| \rightarrow \right\rangle _{S}\) to
S at time
n : 10 for certain purposes but not for others. Suppose, for example, that
\(\bar{F}\) had “flipped the quantum coin
R” by passing that system through the poles of a Stern–Gerlach magnet. By applying unitary quantum theory,
\(\bar{F}\) should conclude that this will induce no physical collapse of
R’s spin state but entangle it with its translational state, and thence with the rest of her lab [
20]. So while
\(\bar{F}\) would be correct then to assign state
\(\left| \rightarrow \right\rangle _{S}\) to
S at time
n : 10 for the purpose of predicting the outcome of a subsequent spin measurement on
S alone, she would be incorrect to assign state
\(\left| \rightarrow \right\rangle _{S}\) to
S at time
n : 10 for the purpose of predicting correlations between
S (or anything with which it subsequently interacts) and her lab
\(\bar{L}\) (or anything with which it subsequently interacts).
By using the phrase ‘is in’, statement \(s_{I}^{\bar{F}}\) ignores the essential relativity of S’s state assignment at time n : 10 to these different applications. By using \(s_{I}^{\bar{F}}\) to infer \(s_{M}^{\bar{F}} \), agent \(\bar{F}\) is, in effect, taking \(\bar{F}\)’s coin flip to involve the physical collapse of R’s state rather than the unitary evolution represented by Eq. (8). So agent \(\bar{F}\) is mistaken to assert \(s_{Q}^{\bar{F}} \), and W would be wrong to incorporate this mistake in his own reasoning by applying assumption (C).
Frauchiger and Renner [12] justify \(\bar{F}\)’s inference from \(s_{I}^{\bar{F}}\) and \(s_{M}^{\bar{F}}\) to \(s_{Q}^{\bar{F}}\) by appeal to assumption (Q). I have argued that \(\bar{F}\) is not justified in asserting \(s_{Q}^{\bar{F}}\), since \(\bar{F}\) is justified in using the state assignment licensed by \(s_{I}^{\bar{F}}\) for the purpose of predicting the outcome of a measurement on S only where S’s correlations with other systems (encoded in an entangled state of a supersystem) may be neglected. But the sequence of interactions in the Gedankenexperiment successively entangle the state of S with those of R, \(\bar{L}\), L and \(\bar{W}\). So in reasoning about the outcome of W’s measurement of w, \(\bar{F}\) must take account of this progressive entanglement of the states of S and \(\bar{W}\).
Specifically, to predict the outcome of W’s measurement of w, \(\bar{F}\) must represent that measurement as the second part of W’s joint measurement on the system \(\bar{W}+L\). This interaction between W and \(\bar{W}\) was represented in §3 as the apparently innocuous Step 1 in which Wigner simply asked Zeus what was the outcome of his measurement. But it is not this interaction but the prior interaction between \(\bar{W}\) and L that undercuts \(\bar{F}\)’s justification for using the state assignment \(\left| \rightarrow \right\rangle _{S}\) in inferring \(s_{Q}^{\bar{F}}\) from \(s_{I}^{\bar{F}}\) and \(s_{M}^{\bar{F}}\). Only by neglecting the prior interaction between \(\bar{W}\) and L can \(\bar{F}\) draw the erroneous conclusion \(s_{Q}^{\bar{F}}\).
Wigner can reason consistently about the unique, physical outcomes of all experiments in the Gedankenexperiment of [9, 12] without any appeal to the reasoning of the other agents involved. Each of these other agents may reason equally consistently. And their collective reasoning is perfectly in accord with assumption (C) as well as the universal applicability of unitary quantum theory and the existence of a unique, physical outcome of every measurement that figures in the Gedankenexperiment of [9, 12].
