Bell–Bohmian theory assumes a pilot vector in the Hilbert space of the whole experiment, evolving purely according to the unitary operator describing the dynamics (i.e. with no application of the projection postulate after measurements). In this it resembles Everettian quantum mechanics, but the metaphysical interpretation is different, as described in Sect. 2. The Hilbert space in question is
$$\begin{aligned} \mathcal {H}_{F_1}\otimes \mathcal {H}_{F_2}\otimes \mathcal {H}_A\otimes \mathcal {H}_W\otimes \mathcal {H}_C\otimes \mathcal {H}_S \end{aligned}$$
where \(\mathcal {H}_C\) and \(\mathcal {H}_S\) are two-dimensional, with orthonormal bases \(\{|\text {head}\rangle ,|\text {tail}\rangle \}\) and \(\{|\uparrow \rangle , |\downarrow \rangle \}\) respectively; and \(\mathcal {H}_{F_1}, \mathcal {H}_{F_2}, \mathcal {H}_A\) and \(\mathcal {H}_W\) are all 3-dimensional, with bases labelled by r, z, x and w, each taking the two values described in Sect. 2 and also a third value 0 to describe the “ready” state of the observer before making any measurement. We take r, z, x and w to be the beables of the system, which always have definite values. Thus the real state vector of the system always lies in one of the 81 viable subspaces
$$\begin{aligned} |r\rangle _{F_1}|z\rangle _{F_2}|x\rangle _A|w\rangle _W\otimes \mathcal {H}_C\otimes \mathcal {H}_S. \end{aligned}$$
and is one of the projections of the pilot vector onto these subspaces.
In order to analyse the experiment, we need to be more precise about the way in which \(F_1\) prepares the spin state after the coin toss at \(t = 0\). I will assume that before the coin toss, the electron spin is prepared in some known initial state \(|0\rangle _S\in \mathcal {H}_S\); after the coin toss, \(F_1\) applies to the electron either a unitary operator which takes \(|0\rangle \) to \(|\downarrow \rangle \) or one which takes \(|0\rangle \) to \(|\rightarrow \rangle \), according to the result of the toss. Then the real state vector before the experiment starts is the same as the pilot state, namely
$$\begin{aligned} |0 \rangle _{F_1}|0 \rangle _{F_2}|0\rangle _A|0\rangle _W\left( \sqrt{\tfrac{1}{3}}|\text {head}\rangle _C + \sqrt{\tfrac{2}{3}}|\text {tail}\rangle _C\right) |0\rangle _S. \end{aligned}$$
At \(t=0\), after \(F_1\)’s measurement of the coin, the pilot vector becomes
$$\begin{aligned} |\Psi (0)\rangle = \left( \sqrt{\tfrac{1}{3}}|\text {head}\rangle _{F_1C} + \sqrt{\tfrac{2}{3}}|\text {tail}\rangle _{F_1C}\right) |0\rangle _S|0\rangle _{F_2}|0\rangle _A|0\rangle _W, \end{aligned}$$
where \(|\text {head}\rangle _{F_1C} = |\text {head}\rangle _{F_1}|\text {head}\rangle _C\) and similarly for “tail”, but the real state vector is one of the two summands in this. We will consider
$$\begin{aligned} |\Phi (0)\rangle = |\text {tail}\rangle _{F_1C}|0\rangle _S|0\rangle _{F_2}|0\rangle _A|0\rangle _W. \end{aligned}$$
At \(t =1\), after \(F_1\) has prepared the electron spin, the pilot state is
$$\begin{aligned} |\Psi (1)\rangle = \left( \sqrt{\tfrac{1}{3}}|\text {head}\rangle _{F_1C}|\downarrow \rangle _S + \sqrt{\tfrac{2}{3}}|\text {tail}\rangle _{F_1C}|\rightarrow \rangle _S\right) |0\rangle _{F_2}|0\rangle _A|0\rangle _W \end{aligned}$$
The real state is one of the two summands in \(|\Psi (1)\rangle \); we take
$$\begin{aligned} |\Phi (1) = \sqrt{\tfrac{2}{3}}|\text {tail}\rangle _{F_1C}|\rightarrow \rangle _S|0\rangle _{F_2}|0\rangle _A|0\rangle _W. \end{aligned}$$
After \(F_2\)’s measurement of S at \(t = 2\), the pilot state becomes
$$\begin{aligned} |\Psi (2)\rangle = \sqrt{\tfrac{1}{3}}\bigg (|\text {head}\rangle _{F_1C}|-\rangle _{F_2S} + |\text {tail}\rangle _{F_1C}|+\rangle _{F_2S} + |\text {tail}\rangle _{F_1C}|-\rangle _{F_2S}\bigg )|0\rangle _A|0\rangle _W, \end{aligned}$$
which has three components with definite values of r, z, x and w (viable components), one of which is
$$\begin{aligned} |\Phi (2)\rangle = \sqrt{\tfrac{1}{3}}|\text {tail}\rangle _{F_1C}|+\rangle _{F_2S}|0\rangle _A|0\rangle _W. \end{aligned}$$
After A’s measurement of \(F_1\) and C at \(t = 3\), the pilot vector becomes
$$\begin{aligned} |\Psi (3)\rangle&= \bigg (\sqrt{\tfrac{1}{6}}\Big ( - |\text {ok}\rangle _{F_1C}|\text {ok}\rangle _A + |\text {fail}\rangle _{F_1C}|\text {fail}\rangle _A\Big )|+\rangle _{F_2S}\rangle \\&\quad +\, \sqrt{\tfrac{2}{3}}|\text {fail}\rangle _{F_1C}|\text {fail}\rangle _A|-\rangle _{F_2S}\bigg )|0\rangle _W \end{aligned}$$
which has six viable components, one of which is
$$\begin{aligned} |\Phi (3)\rangle = \sqrt{\tfrac{1}{12}}|\text {tail}\rangle _{F_1C}|+\rangle _{F_2S}|\text {ok}\rangle _A|0\rangle _W. \end{aligned}$$
After W’s measurement of \(F_2\) and S at \(t = 4\), the pilot vector becomes
$$\begin{aligned} |\Psi (4)\rangle= & {} \sqrt{\tfrac{1}{12}}\Big (|\text {ok}\rangle _{F_1C}|\text {ok}\rangle _A + |\text {fail}\rangle _{F_1C}|\text {fail}\rangle _A\Big )|\text {ok}\rangle _{F_2S}|\text {ok}\rangle _W\\&+ \,\, \sqrt{\tfrac{1}{12}}\Big (- |\text {ok}\rangle _{F_1C}|\text {ok}\rangle _A + 3|\text {fail}\rangle _{F_1C}|\text {fail}\rangle _A\Big )|\text {fail}\rangle _{F_2S}|\text {fail}\rangle _W \end{aligned}$$
which has sixteen viable components, one of which is
$$\begin{aligned} |\Phi (4)\rangle = -\sqrt{\tfrac{1}{24}}|\text {tail}\rangle _{F_1C}|-\rangle _{F_2S}|\text {ok}\rangle _A|\text {ok}\rangle _W. \end{aligned}$$
According to Bell–Bohmian theory, at all times Wigner, his assistant and his two friends are in a single world with definite values of r, z, x and w, the results of their measurements. But Frauchiger and Renner argue that this leads to the contradictory implications (3.1), (3.3), (3.4) and (3.5). We will show, on the contrary, that in Bell–Bohmian theory it is possible that the real state undergoes the transitions
$$\begin{aligned} |\Phi (0)\rangle \longrightarrow |\Phi (1)\rangle \longrightarrow |\Phi (2)\rangle \longrightarrow |\Phi (3)\rangle \longrightarrow \rangle |\Phi (4)\rangle . \end{aligned}$$
It follows that in this theory the implication (3.1) (\(r(1) = \text {tail}\; \Longrightarrow \; w(4) = \text {fail}\)) does not hold: it is possible for \(F_1\) to get the result \(r = \) “tail” (and, incidentally, to remain in a state registering this result) while W gets the result \(w = \)“ok”.
To establish this, we will need to see what transitions between viable states are allowed by Bell’s postulate, and for this we need a model of the processes by which the measurements are made. The following is a general theory of such a process. We consider an experimenter E measuring an observable X on a system S, whose basis of eigenstates of X is \(\{|1\rangle _S,|2\rangle _S\}\), and suppose that the process takes place as follows. The relevant states of the experimenter are taken to be \(|0\rangle _E,|1\rangle _E,|2\rangle _E\), where \(|0\rangle _E\) is the state of the experimenter before the measurement, and \(|1\rangle _E\) and \(|2\rangle _E\) are the states of the experimenter registering the results \(X = 1\) and \(X = 2\). In the course of the measurement the joint state \(|1\rangle _S|0\rangle _E\) evolves to \(|1\rangle _S|1\rangle _E\) and the joint state \(|2\rangle _S|0\rangle _E\) evolves to \(|2\rangle _S|2\rangle _E\). We assume that each of these evolutions is a simple rotation in the joint state space \(\mathcal {H}_E\otimes \mathcal {H}_S\), lasting for a time \(\tau \):
$$\begin{aligned} |k\rangle _S|0\rangle _E \; \longrightarrow \; |\Psi _k(t)\rangle = \cos \lambda t|k\rangle _S|0\rangle _E + \sin \lambda t|k\rangle _S|k\rangle _E \end{aligned}$$
\((k = 1,2; \; 0\le t\le \tau )\) where \(\lambda = \pi /2\tau \). At times outside the interval \([0,\tau ]\), the joint state of the system and the experimenter is assumed to be stationary (with zero energy). This time development is produced by the Hamiltonian
$$\begin{aligned} H = i\hbar \lambda \bigg (|1\rangle \langle 1|_S\otimes \big [|1\rangle \langle 0| - |0\rangle \langle 1|\big ]_E + |2\rangle \langle 2|_S\otimes \big [|2\rangle \langle 0| - |0\rangle \langle 2|\big ]_E\bigg ), \end{aligned}$$
which is switched on at \(t = 0\) and off at \(t = \tau \).
Suppose the system has just one beable M, the observation of the experimenter, with values (0, 1, 2), and suppose the initial state of the joint system is \(\big (a|1\rangle _S + b|2\rangle _S\big )|0\rangle _E\). This has the definite value 0 for the beable M, so it is both the real state vector for the joint system and the pilot vector at \(t = 0\). Then in the time interval \([0,\tau ]\) during which the measurement is proceeding, the pilot state is
$$\begin{aligned} |\Psi (t)\rangle&= a|\Psi _1(t)\rangle + b|\Psi _2(t)\rangle \\&= \cos \lambda t\big (a|1\rangle + b|2\rangle \big )_S|0\rangle _E + \sin \lambda t\big (a|1\rangle _S|1\rangle _E + b|2\rangle _S|1\rangle _E\big ) \end{aligned}$$
and the real state of the joint system at any time in this interval is one of the three states \(|\Psi (0)\rangle = (a|1\rangle _S + b|2\rangle _S)|0\rangle _E\), \(|1\rangle _S|1\rangle _E\) or \(|2\rangle _S|2\rangle _E\). It can make a transition from \(|\Psi (0)\rangle \) to \(|1\rangle _S|1\rangle _E\) or to \(|2\rangle _S|2\rangle _E\) because the (real) matrix elements \((i\hbar )^{-1}\big (\langle k|_S\langle k|_E\big ) H \big (|k\rangle _S|0\rangle _E\big )\) (\(k = 1,2\)) are both positive. It cannot make the reverse transitions because the matrix elements \((i\hbar )^{-1}\big (\langle k|_S\langle 0|_E\big ) H \big (|k\rangle _S|k\rangle _E\big )\) are negative, and it cannot make transitions between \(|1\rangle _S|1\rangle _E\) and \(|2\rangle _S|2\rangle _E\) because the relevant matrix elements of H are zero. Thus at time \(t = 0\) the real state vector and the pilot vector coincide; between \(t = 0\) and \(t = \tau \) the pilot vector \(|\Psi (t)\rangle \) changes smoothly but the real state vector remains at its initial value \(|k\rangle _S|0\rangle _E\) until some undetermined intermediate time at which it changes discontinuously to either \(|1\rangle _S|1\rangle _E\) or \(|2\rangle _S|2\rangle _E\) and remains at that value until \(t = \tau \). A calculation of the final probabilities from the transition probabilities as given by Bell yields the expected values \(|a|^2\) and \(|b|^2\).
To examine the implication (3.1), we will apply this theory to the measurements in the extended Wigner’s friend experiment. We will assume that each of the measurements has duration \(\tau < 1\) before the time assigned to it (e.g. A’s measurement “at time \(t = 3\)” occupies the interval \([3 - \tau , 3])\), and that each measurement consists of a simple rotation as described above.
If the result of \(F_1\)’s measurement at \(t = 0\) is \(r = \)“tail”, then the component of \(|\Psi (0)\rangle \) describing the actual world must be \(|\Phi (0)\rangle \). The pilot vector is still \(|\Psi (0)\rangle \). \(F_1\)’s preparation of the electron spin at \(t=1\) is accomplished by a unitary operator acting only on \(F_1\) and S, such that there are no matrix elements of the Hamiltonian between states with different values of the beables r, x, z, w; therefore the real state at \(t=1\) is \(|\Phi (1)\rangle \). The next measurement, by \(F_2\) at \(t = 2\), is driven by the Hamiltonian \(\mathbf {1}_{F_1C}\otimes (H_2)_{F_2S}\otimes \mathbf {1}_A\otimes \mathbf {1}_W\) where
$$\begin{aligned} H_2&= i\hbar \lambda \Big (|-\rangle _{F_2S}\big (\langle \downarrow |_S\langle 0|_{F_2}\big ) - \big (|\downarrow \rangle _S|0\rangle _{F_2}\big )\langle -|_{F_2S} \nonumber \\&\quad +\, |+\rangle _{F_2S}\big (\langle \uparrow |_S\langle 0|_{F_2}\big ) - \big (|\uparrow \rangle _S|0\rangle _{F_2}\big )\langle +|_{F_2S}\Big ). \end{aligned}$$
(4.1)
The pilot state during the measurement is \(\cos \lambda t|\Psi (1)\rangle + \sin \lambda t|\Psi (2)\rangle \); the real state must therefore be one of the viable components of \(|\Psi (1)\rangle \) or \(|\Psi (2)\rangle \). Since this Hamiltonian has no matrix elements betweeen states containing \(|\text {head}\rangle _{F_1C}\) and states containing \(|\text {tail}\rangle _{F_1C}\), the only possible transitions from \(|\Phi (1)\rangle \) are to the second or third term in \(|\Psi (2)\rangle \), followed by transitions back to \(|\Phi (1)\rangle \) or to other components of \(|\Psi (2)\rangle \). But the Hamiltonian also has no matrix elements between different viable components of \(|\Psi (2)\rangle \), and the only positive matrix elements of \(H/i\hbar \) are those corresponding to transitions in the forward direction, so once a transition has been made to one of the three terms in \(|\Psi (2)\rangle \), there will be no further transitions during this measurement. Thus if the real state after \(F_1\)’s measurement has \(r = \) “tail”, this will still be the case after \(F_2\)’s measurement and the real state will be the second or third term of \(|\Psi (2)\rangle \), and both of these are possible. Thus there is a non-zero probability that the real state evolves as \(|\Psi (0)\rangle \rightarrow |\Phi (1)\rangle \rightarrow |\Phi (2)\rangle \).
A’s measurement of \(F_1\) and C at \(t = 3\) is driven by the Hamiltonian \(H_3\otimes \mathbf {1}_{F_2S}\otimes \mathbf {1}_W\) where \(H_3\), acting in \(\mathcal {H}_{F_1C}\otimes \mathcal {H}_A\), rotates \(|\text {fail}\rangle _{F_1C}|0\rangle _A\) to \(|\text {fail}\rangle _{F_1C}|\text {fail}\rangle _A\) and \(|\text {ok}\rangle _{F_1C}|0\rangle _A\) to \(|\text {ok}\rangle _{F_1C}|\text {ok}\rangle _A\). In terms of the viable states, this is
$$\begin{aligned} H_3&= \tfrac{1}{2}i\hbar \lambda \big (|\text {head}\rangle + |\text {tail}\rangle \big )\big (\langle \text {head}| + \langle \text {tail}|\big )_{F_1C}\otimes \big (|\text {fail}\rangle \langle 0| - |0\rangle \langle \text {fail}|\big )_A\\&\quad +\, \tfrac{1}{2}i\hbar \lambda \big (|\text {head}\rangle - |\text {tail}\rangle \big )\big (\langle \text {head}| - \langle \text {tail}|\big )_{F_1C}\otimes \big (|\text {ok}\rangle \langle 0| - |0\rangle \langle \text {ok}|\big )_A. \end{aligned}$$
This Hamiltonian H has
$$\begin{aligned} \langle \Phi (3)|\frac{H}{i\hbar }|\Phi (2)\rangle > 0, \end{aligned}$$
and there are no positive matrix elements \(\langle \phi |\tfrac{H}{i\hbar }|\Phi (3)\rangle \) for viable states \(|\phi \rangle \), so the transition \(|\Phi (2)\rangle \rightarrow |\Phi (3)\rangle \) is possible, and if it occurs the system remains in the state \(|\Phi (3)\rangle \) until the next measurement.
W’s measurement of \(F_2\) and S at \(t = 4\) is driven by the Hamiltonian \(\mathbf {1}_{F_1C}\otimes \mathbf {1}_A\otimes H_4\) where \(H_4\) is the following operator on \(\mathcal {H}_{F_2S}\otimes \mathcal {H}_W\):
$$\begin{aligned} H_4&= i\hbar \lambda |\text {ok}\rangle \langle \text {ok}|_{F_2S}\big (|\text {ok}\rangle \langle 0|-|0\rangle \langle \text {ok}|\big )_W\\&\quad +\,i\hbar \lambda |\text {fail}\rangle \langle \text {fail}|_{F_2S}\big (\text {fail}\rangle \langle 0| - |0\rangle \langle \text {fail}|\big )_W. \end{aligned}$$
This has
$$\begin{aligned} \langle \Phi (3)|\frac{H}{i\hbar }|\Phi (4)\rangle > 0, \end{aligned}$$
and there are no positive matrix elements \(\langle \phi |\tfrac{H}{i\hbar }|\Phi (4)\rangle \) for viable states \(|\phi \rangle \), so the transition \(|\Phi (3)\rangle \rightarrow |\Phi (4)\rangle \) is possible during W’s measurement, and if it occurs the system remains in the state \(|\Phi (4)\rangle \).
Thus it is possible that W and A both get the result “ok” for their measurements, and this happens even though \(F_1\) records the result \(r =\) “tail”. This contradicts the theorem of Frauchiger and Renner.