Wigner's Friend and Incompatibility

[Demystifier]

Also, is there a precise definition of macroscopically distinct states?

I seems to me that the most precise definition could be given in terms of Bohmian mechanics. But it is somewhat subjective because it depends on what one means by "precise".

 

[Demystifier]

My understanding is that the two states $|{\rm up}\rangle$ and $|{\rm down}\rangle$ are macroscopically distinct if there exist projectors $\Pi_{\rm up}$ and $\Pi_{\rm down}$ onto macroscopic subspaces such that

\begin{eqnarray*}\Pi_{\rm up}|{\rm up}\rangle &=& |{\rm up}\rangle\\
\Pi_{\rm down}|{\rm down}\rangle &=& |{\rm down}\rangle\\
\Pi_{\rm down}|{\rm up}\rangle &=& 0|{\rm up}\rangle\\
\Pi_{\rm up}|{\rm down}\rangle &=& 0|{\rm down}\rangle\end{eqnarray*}

But I'm not sure what a being as supernatural as Wigner would consider a macroscopic subspace.

Since you didn't define what "macroscopic subspace" means, it is somewhat circular and begs the question.

 

[Morbert]

Since you didn't define what "macroscopic subspace" means, it is somewhat circular and begs he question.

Sorry, by macroscopic subspaces I mean the very large subspaces associated with pointer properties. But this just raises a similar question about pointer properties. Both Wigner and his friends use different pointer properties, each suitable for their own measurement purposes.

Wigner's friend wants to measure the spin of the particle, so he uses a measurement apparatus that can exhibit pointer properties $D_{\rm up}$ and $D_{\rm down}$, such that
\begin{eqnarray}
|\uparrow_z\rangle\langle\uparrow_z| &=& J^\dagger\Pi_{D_{\rm up}}J\\
|\downarrow_z\rangle\langle\downarrow_z| &=& J^\dagger\Pi_{D_{\rm down}}J
\end{eqnarray}
where $J$ is the appropriate measurement isometry. But Wigner has no use for these pointer projectors. He has his own measurement device and is interested in the pointer projectors $\Pi_{D_\mathcal{X}=1}$ and $\Pi_{D_\mathcal{X}=0}$, such that
\begin{eqnarray}|\mathcal{X}=1\rangle\langle\mathcal{X}=1| &=& J^\dagger\Pi_{D_\mathcal{X}=1}J\\
|\mathcal{X}=0\rangle\langle\mathcal{X}=0| &=& J^\dagger\Pi_{D_\mathcal{X}=0}J
\end{eqnarray}
Would Wigner apply a normative collapse rule based on Wigner's definition of macroscopically distinct properties or his friends?

I seems to me that the most precise definition could be given in terms of Bohmian mechanics. But it is somewhat subjective because it depends on what one means by "precise".

I'll check it out thanks. Would you recommend any particular article/book?

 

[Demystifier]

Would Wigner apply a normative collapse rule based on Wigner's definition of macroscopically distinct properties or his friends?

Both! Whenever there are macro distinct properties a collapse rule must be applied, for otherwise the collapse rule is inconsistent. I think that's the main message to learn from the FR theorem and from this simple example by @DarMM . In subjective interpretations such as QBism, this seemingly objective collapse rule can be justified by intersubjective reasoning of the form "Wigner knows that friend knows that ...".

 

[Demystifier]

Would you recommend any particular article/book?

See e.g. my "Bohmian mechanics for instrumentalists" linked in my signature below, especially Sec. 3.1 and Eq. (4). It is attempted to be more intuitive than precise, but it seems to me that it can also be rewritten in a more precise way.

 

[Morbert]

Both! Whenever there are macro distinct properties a collapse rule must be applied, for otherwise the collapse rule is inconsistent. I think that's the main message to learn from the FR theorem and from this simple example by @DarMM . In subjective interpretations such as QBism, this seemingly objective collapse rule can be justified by intersubjective reasoning of the form "Wigner knows that friend knows that ...".

Hmm, I guess the issue is Wigner predicts his measurement outcome with certainty, which would not be the case if he used a collapsed state or a mixture. Richard Healey argues that Wigner can use the pure state for setting credence about his own measurement, and a mixed state to set credence about his friend's measurement (for reasons related to equation 4 in your paper).

As an aside: In the consistent histories formalism, collapse is history-specific. E.g. If a history contains a pair of time-ordered events like $\cdots\odot[\uparrow_z + \downarrow_z]\odot[\uparrow_z]\odot\cdots$, that is a collapse. And Wigner might use histories with collapse, or he might not, depending on what he wants to describe. An incorrect oversimplifcation

 

[Demystifier]

Hmm, I guess the issue is Wigner predicts his measurement outcome with certainty,

My point is that such a prediction is wrong. In principle (if not in practice) such an experiment could be performed and I claim that Wigner with a such a prediction would turn out to be wrong in 50% cases.

 

[Demystifier]

Richard Healey argues that Wigner can use the pure state for setting credence about his own measurement, and a mixed state to set credence about his friend's measurement (for reasons related to equation 4 in your paper).

And I claim that Healey is wrong. Measurement does not only change the knowledge about the system. It also changes the system itself.

 

[Demystifier]

One important difference between the setup in this thread and the setup in the FR theorem is that the setup in this thread does not involve any undoing of measurement. The purpose of undoing the measurement is to annihilate the effect of collapse. Since this setup does not involve the undoing of measurement, it looks as if the presence of the collapse effect is more obvious than in the FR setup. Does it mean that this setup is more "trivial" and hence less interesting than the FR setup? @DarMM what do you think?

 

[Demystifier]

Summary: Wigner's friend seems to lead to certainty in two complimentary contexts

This is probably pretty dumb, but I was just thinking about Wigner's friend and wondering about the two contexts involved.

The basic set up I'm wondering about is as follows:

The friend does a spin measurement in the $\left\{|\uparrow_z\rangle, |\downarrow_z\rangle\right\}$ basis, i.e. of $S_z$ at time $t_1$. And let's say the particle is undisturbed after that.

For experiments outside the lab Wigner considers the lab to be in the basis:
$$\frac{1}{\sqrt{2}}\left(|L_{\uparrow_z}, D_{\uparrow_z}, \uparrow_z \rangle + |L_{\downarrow_z}, D_{\downarrow_z}, \downarrow_z \rangle\right)$$

He then considers a measurement of the observable $\mathcal{X}$ which has eigenvectors:
$$\left\{\frac{1}{\sqrt{2}}\left(|L_{\uparrow_z}, D_{\uparrow_z}, \uparrow_z \rangle + |L_{\downarrow_z}, D_{\downarrow_z}, \downarrow_z \rangle\right), \frac{1}{\sqrt{2}}\left(|L_{\uparrow_z}, D_{\uparrow_z}, \uparrow_z \rangle - |L_{\downarrow_z}, D_{\downarrow_z}, \downarrow_z \rangle\right)\right\}$$
with eigenvalues $\{1,-1\}$ respectively.

At time $t_2$ the friend flips a coin and either he does a measurement of $S_z$ or Wigner does a measurement of $\mathcal{X}$

However if the friend does a measurement of $S_z$ he knows for a fact he will get whatever result he originally got. However he also knows Wigner will obtain the $1$ outcome with certainty.

However $\left[S_{z},\mathcal{X}\right] \neq 0$. Thus the friend seems to be predicting with certainty observables belonging to two separate contexts. Which is not supposed to be possible in the quantum formalism.

What am I missing?

I've just realized, a version of this experiment can actually be performed. The friend prepares an atom in the state
$$|\uparrow_x\rangle=\frac{|\uparrow_z\rangle+|\downarrow_z\rangle}{\sqrt{2}}$$
and then measures its spin in the $z$-direction. After that he sends to Wigner the following message: "Hi, I just prepared the atom in the state $|\uparrow_x\rangle$ and then measured its spin in the $z$-direction. But I will not tell you what the result of my measurement was." After that Wigner decides to measure the spin in the $x$-direction. What the result of Wigner's measurement will be?

I think it's pretty obvious that there is only 50% chance that the result will be $|\uparrow_x\rangle$, as well as 50% chance that the result will be $|\downarrow_x\rangle$. Furthermore, as far as I can see, this version of the experiment is completely equivalent to the version by @DarMM above.

 

[Morbert]

Ok so it sounds like it ultimately boils down to a matter of interpretation regarding the nature of collapse. Anyway:

One important difference between the setup in this thread and the setup in the FR theorem is that the setup in this thread does not involve any undoing of measurement. The purpose of undoing the measurement is to annihilate the effect of collapse. Since this setup does not involve the undoing of measurement, it looks as if the presence of the collapse effect is more obvious than in the FR setup.

The histories formalism iiuc would imply direct analogue between the FR experiment and the WF experiment. In the WF experiment we have an isolated system that evolves into $|{\rm up}\rangle + |{\rm down}\rangle$ In the FR experiment we have an isolated system that evolves into a state represented by equation 1 or, equivalently, 4 here. In the WF experiment, we can construct a family of histories to talk about Wigner's measurement, or a family to talk about his friend's measurement. In the FR experiment, we can construct a family to talk about Wigner-1's measurement and Wigner-2's measurement, or we can construct a family to discuss Friend-1's measurement, or a family to discuss Friend-2's measurement.

 

[DarMM]

I think it's pretty obvious that there is only 50% chance that the result will be |↑x⟩|\uparrow_x\rangle, as well as 50% chance that the result will be |↓x⟩|\downarrow_x\rangle. Furthermore, as far as I can see, this version of the experiment is completely equivalent to the version by @DarMM above.

We have to be careful here as if Wigner performs a trace over other states of the lab he gets the mixed state (due to a simplistic form of decoherence):
$\rho = \frac{1}{2}|\uparrow \rangle\langle \uparrow | + \frac{1}{2}|\downarrow \rangle\langle \downarrow |$
So he wouldn't expect to witness interference on the particle itself, but only for superobservables involving the atomic state of the entire lab.

 

[DarMM]

One important difference between the setup in this thread and the setup in the FR theorem is that the setup in this thread does not involve any undoing of measurement. The purpose of undoing the measurement is to annihilate the effect of collapse. Since this setup does not involve the undoing of measurement, it looks as if the presence of the collapse effect is more obvious than in the FR setup. Does it mean that this setup is more "trivial" and hence less interesting than the FR setup? @DarMM what do you think?

I'll say a bit about their link shortly, just need to think about it.

 

[DarMM]

Ok so it sounds like it ultimately boils down to a matter of interpretation regarding the nature of collapse.

That's often the issue with Wigner's friend, is there subjective or objective collapse.

Objective collapse solves all issues for Wigner's friend considered in isolation, but you're left with restricting the Unitary evolution of quantum systems. Wigner must know not to use unitary evolution whenever something crosses the "macroscopic" threshold, but then we are faced with the ambiguity of where this threshold is.

Some like Bub consider this to simply be part of QM, the kinematic structure of the theory itself prevents a dynamical understanding of measurement and thus yes Wigner must collapse the state (in the form of replacing the state that results from unitary evolution with a density matrix) when he knows a macroevent has occurred.

Omnes takes a similar view, but instead takes the view that unitary evolution itself when analysed correctly gives a mixed state on most observables, with interference only being apparent in highly detailed observables (atomic state of whole lab) that have no operational meaning.

Pure unitary evolution always leaves interference for some scales/observables which would invalidate the collapse. So it is really a question about how one views these difficult/impossible to access interferences. "Meaningless" according to Omnes. "Proof that QM cannot describe the occurrence of actual facts, which lie outside the theory" according to Bub.

 

[Demystifier]

Wigner must know not to use unitary evolution whenever something crosses the "macroscopic" threshold, but then we are faced with the ambiguity of where this threshold is.

That's very interesting. To make this ambiguity even more interesting, I would suggest to reformulate the Wigner friend type of paradoxes in terms of systems in which a "friend" is a mesoscopic system (say, a system made of 100 atoms) for which it is not intuitively obvious whether we should treat it as "classical" or "quantum".

 

[Demystifier]

That's very interesting. To make this ambiguity even more interesting, I would suggest to reformulate the Wigner friend type of paradoxes in terms of systems in which a "friend" is a mesoscopic system (say, a system made of 100 atoms) for which it is not intuitively obvious whether we should treat it as "classical" or "quantum".

I think I have solved it. The solution is inspired by Bohmian interpretation, but does not depend on it. To generalize the notion of a "friend", consider any degrees of freedom that get entangled with the spin. These degrees of freedom may constitute a macro, a meso or a micro system, my analysis will not depend on it. In particular, those degrees of freedom may be the position of the particle with spin itself. The entanglement of those degrees of freedom with the spin takes the form
$$|\psi_{\uparrow}\rangle | {\uparrow}\rangle + |\psi_{\downarrow}\rangle | {\downarrow}\rangle$$
(for simplicity I suppress the overall normalization of the state). The crucial thing to consider are wave functions in the position space $\psi_{\uparrow}(\vec{x})=\langle \vec{x}|\psi_{\uparrow}\rangle$, $\psi_{\downarrow}(\vec{x})=\langle \vec{x}|\psi_{\downarrow}\rangle$. If those wave functions have a negligible overlap in the sense of Eq. (4) in my "Bohmian mechanics for instrumentalists", then Wigner has to treat the system as if the system effectively collapsed. Otherwise, he has not.

Why does Wigner has to treat it as if the system effectively collapsed? Because all Hamiltonian interactions are local, so the measurement made by Wigner is a local measurement. Hence he cannot measure a non-local observable. On the other hand, an observable of which the superposition above is an eigenstate is a non-local observable when the two wave functions have a negligible overlap in the position space. Hence the Wigner's measurement in this thought experiment cannot really be performed.

 

[DarMM]

Why does Wigner has to treat it as if the system effectively collapsed? Because all Hamiltonian interactions are local, so the measurement made by Wigner is a local measurement. Hence he cannot measure a non-local observable. On the other hand, an observable of which the superposition above is an eigenstate is a non-local observable when the two wave functions have a negligible overlap in the position space. Hence the Wigner's measurement in this thought experiment cannot really be performed

You arrive at a conclusion very similar to Omnès. When the overlaps are virtually negligable the observable required is physically impossible to realize. The only difference between the Bohmian and Copenhagen case is that you say "effective collapse". For the Bohmian the overlap is still there but has no influence on the statistics of any observable. For Copenhagen this however is true collapse as the wavefunction is nothing but statistics for realizable observables. The overlap is only present when the wavefunction is considered as a state on an algebra of self-adjoint operators strictly larger than the actual observable algebra, on the true algebra it's not present.

 

[charters]

Why does Wigner has to treat it as if the system effectively collapsed? Because all Hamiltonian interactions are local, so the measurement made by Wigner is a local measurement. Hence he cannot measure a non-local observable. On the other hand, an observable of which the superposition above is an eigenstate is a non-local observable when the two wave functions have a negligible overlap in the position space. Hence the Wigner's measurement in this thought experiment cannot really be performed

But Wigner's procedure is assumed to successfully bring all the relevant degrees of freedom back into local contact before measuring the |+> or |-> observable, which restores coherence and overcomes this problem.

The idea of Wigner's Friend and the un-making of Friend's measurement is equivalent to what Susskind and Bousso describe as case 3 in figure 9 here: https://arxiv.org/abs/1105.3796. The only difference is in WF the number of degrees of freedom we would need to control is much larger.

 

[Demystifier]

But Wigner's procedure is assumed to successfully bring all the relevant degrees of freedom back into local contact before measuring the |+> or |-> observable, which restores coherence and overcomes this problem.

Fine, but then the two seemingly incompatible observables are measured at different times, which can make them compatible. For a simple example consider this. A particle momentum is measured first, then a confining potential is turned on so that the particle enters a hole narrow in the position space, after which the particle position is measured. The result of the position measurement can be predicted with arbitrary precision (because the width of the hole can be arbitrarily small) despite the fact that the momentum has been measured at an earlier time.

 

[charters]

Fine, but then the two seemingly incompatible observables are measured at different times, which can make them compatible. For a simple example consider this. A particle momentum is measured first, then a confining potential is turned on that so that the particle enters a hole narrow in the position space, after which the particle position is measured. The result of the position measurement can be predicted with arbitrary precision (because the width of the hole can be arbitrarily small) despite the fact that the momentum has been measured at an earlier time.

The issue is not compatibility, its reversibility. If Friend's measurement indeed collapses/reduces the state, so that one of the branches disappears at that time, Wigner's subsequent reversal protocol doesn't restore |+> every time, even though unitary QM dictates it should have done so.

 

[Demystifier]

The issue is not compatibility, its reversibility. If Friend's measurement indeed collapses/reduces the state, so that one of the branches disappears at that time, Wigner's subsequent reversal protocol doesn't restore |+> every time, even though unitary QM dictates it should have done so.

But we already concluded that if an irreversible collapse has occured, then the paradox is resolved. The issue raised later was what if it is not clear whether the collapse has occurred or not. If you take a look at the first post on this thread, you will see that it is precisely the compatibility what is at stake here.

 

[charters]

But we already concluded that if an irreversible collapse has occured, then the paradox is resolved. The issue raised later was what if it is not clear whether the collapse has occurred or not. If you take a look at the first post on this thread, you will see that it is precisely the compatibility what is at stake here.

The issue in #1 is that it seems at first glance like Friend can predict with certainty two non-commuting observables.

In #51, you are concluding Wigner can't make his |+> or |-> measurement at all, which I think is ultimately a question of whether unitary reversibility is possible for large systems.

But maybe I lost the thread of the conversation and am missing the connection.

 

[Demystifier]

In #51, you are concluding Wigner can't make his |+> or |-> measurement at all, which I think is ultimately a question of whether unitary reversibility is possible for large systems.

But maybe I lost the thread of the conversation and am missing the connection.

For the smooth connection see #50.

 

[DarMM]

The issue is not compatibility, its reversibility. If Friend's measurement indeed collapses/reduces the state, so that one of the branches disappears at that time, Wigner's subsequent reversal protocol doesn't restore |+> every time, even though unitary QM dictates it should have done so.

I thin`k partially the issue comes down to what exactly is contained in unitary QM. As mentioned above calculations by Omnes and others cast doubt on these unitaries or the appropriate interference observables even existing. In that case the only problem is considering all abstract self-adjoint operators and unitaries to be part of the theory and there's no contradiction between collapse and the actual physical unitaries and observables.

Berthold-Georg Englert also has the example that in general if we evolve from $\left[0,t\right)$ via the unitary $E^{-iHt}$ then reversal over time $\left[t,2t\right)$ would require the unitary $e^{-iH^{'}t}$ where because the spectrum of $H$ is unbounded above the spectrum of $H^{'}$ has to be unbounded below. Only when the Hamiltonian is bounded as for spin degrees of freedom is $H^{'}$ realisable.

Or we could evolve from $\left[t,2t\right)$ with the same Hamiltonian but perform conjugation of the state first, then evolve and apply conjugation again. This conjugation operation seems to be impossible outside of simple degree of freedom like spin.

Finally Itamar Pitowsky has given a purely kinematic conjecture about the increasing rarity of entangled states that are operationally distinct from product states as the size of the system gets larger¹. Even well before objects on our scale the rays are most likely vanishingly rare and thus to obtain results demonstrating interference and entanglement would require increasingly difficult fine tuning of the macroscopic ket. Since there are general results in quantum measurement theory showing that experimental errors grow quicker than this, the ket can't be brought to such a state.

So we have four independent lines of reasoning where well before our scale measurements are truly irreversible for the actual algebra of observables and unitaries.

¹ The conjecture has been proven for a large class of entanglement witnesses, i.e. observables whose eigenvalues lie in some range $\left[a,b\right]$ with $a < -1, b > 1$, but whose eigenvalues on seperable states is bound to $\left[-1,1\right]$. This includes those witnesses usually used to give estimations in Quantum Information.

 

[Morbert]

But the boolean algebra of a un-made/unitarily reversed event can't be appropriate, right?

Sorry, I forgot to respond to this even though I said I would.

I got some interesting (and hopefully correct) results. For simplicity, I'll represent Wigner's friend, his device, and his lab all with $F$, and Wigner's own lab including himself with $W$. I'll also ignore the coin toss (which I think is not relevant for this new question).

CH let's us use unitary evolution or collapse, depending what properties we want to discuss. A fully unitary evolution of the whole system (particle $\psi$, friend $F$, and Wigner $W$) is
\begin{eqnarray*}
U(t_0,t_1)|\psi\rangle|F_\Omega\rangle|W_\Omega\rangle &=& |1_\mathcal{X}\rangle|W_\Omega\rangle\\
U(t_1,t_2)|1_\mathcal{X}\rangle|W_\Omega\rangle &=& |1_\mathcal{X}\rangle|W_{1}\rangle
\end{eqnarray*}
In the histories formalism, this unitary evolution implies the family of histories containing only one history with nonzero probability:
$$[\psi,F_\Omega,W_\Omega]_{t_0}\odot[1_\mathcal{X},W_\Omega]_{t_1}\odot[1_\mathcal{X},W_1]_{t_2}$$
We have a history with probability $1$ that contains Wigner's measurement outcome as the property $[W_1]$.

If, on the other hand, we insert a collapse for Wigner's friend's measurement, we get evolution that looks like (e.g. for a measurement outcome $F_\uparrow$)
\begin{eqnarray*}
U(t_0,t_1)|\psi\rangle|F_\Omega\rangle|W_\Omega\rangle &=& |1_\mathcal{X}\rangle|W_\Omega\rangle\\
|F_\uparrow\rangle\langle F_\uparrow|1_\mathcal{X}\rangle|W_\Omega\rangle &=& \frac{1}{\sqrt{2}}|\uparrow\rangle|F_\uparrow\rangle|W_\Omega\rangle\\
U(t_1,t_2)\frac{1}{\sqrt{2}}|\uparrow\rangle|F_\uparrow\rangle|W_\Omega\rangle &=& \frac{1}{2}|\uparrow\rangle|F_\uparrow\rangle(|W_{0}\rangle-|W_{1}\rangle) + \frac{1}{2}|\downarrow\rangle|F_\downarrow\rangle(|W_{0}\rangle+|W_{1}\rangle)\end{eqnarray*}
This implies the possible histories
\begin{eqnarray*}
[\psi,F_\Omega,W_\Omega]_{t_0}&\odot&[\uparrow,F_\uparrow,W_\Omega]_{t_1}&\odot&[\uparrow,F_\uparrow,I_W]_{t_2}\\
[\psi,F_\Omega,W_\Omega]_{t_0}&\odot&[\uparrow,F_\uparrow,W_\Omega]_{t_1}&\odot&[\downarrow,F_\downarrow,I_W]_{t_2}\\
[\psi,F_\Omega,W_\Omega]_{t_0}&\odot&[\downarrow,F_\downarrow,W_\Omega]_{t_1}&\odot&[\uparrow,F_\uparrow,I_W]_{t_2}\\
[\psi,F_\Omega,W_\Omega]_{t_0}&\odot&[\downarrow,F_\downarrow,W_\Omega]_{t_1}&\odot&[\downarrow,F_\downarrow,I_W]_{t_2}
\end{eqnarray*}
But this family is inconsistent! The 1st + 3rd family don't decohere. Nor do the 2nd + 4th. This is due to the dynamics $U(t_1,t_2)$ in our evolution. If Wigner left the system alone, they would decohere. But they don't.

This means we cannot reason about the correlations between the property $F_\uparrow$ (i.e. the friend recording $\uparrow$) after Wigner's measurement, with respect to the property before Wigner's measurement. But we need this kind of reasoning to make statements like "Wigner's friend still remembers his measurement, even after Wigner's measurement". We cannot identify a record of Wigner's friend's measurement at any time after Wigner's measurement. Erasure! I suspect (but haven't proved) that this is true for any of Wigner's friend's experiences since they should all fail to commute with Wigner's measurement.

I also looked at a special case where the particle is prepared with the property $\uparrow$. I ended up with the consistent family
\begin{eqnarray*}
[\uparrow,F_\Omega,W_\Omega]_{t_0}&\odot&[\uparrow,F_\uparrow,W_\Omega]_{t_1}&\odot&[\uparrow,F_\uparrow,I_W]_{t_2}\\
[\uparrow,F_\Omega,W_\Omega]_{t_0}&\odot&[\uparrow,F_\uparrow,W_\Omega]_{t_1}&\odot&[\downarrow,F_\downarrow,I_W]_{t_2}
\end{eqnarray*}
with a probability of $0.5$ for each. This implies $P([F_\uparrow]_{t_1} \vert [F_\uparrow]_{t_2}) = P([F_\uparrow]_{t_1}\vert [F_\downarrow]_{t_2}) = 0.5$. I.e. Zero correlation. So even though we can now construct histories referencing the friend's memory both before and after Wigner's measurement, there is no correlation with past memories. I.e. Still no record.

[edit] - What's also weird is it implies Wigner's friend is still alive! Wigner's precise interactions, with its exotic dynamics, precisely tampers with his friend's memories, but keeps him healthy.

All of the above comes with the caveat that we are hardly talking about a realistic scenario, as discussed by DarMM. So none of the above should imply any of it is realizeable.

[edit]- I also put this out there for criticism. There might be all sorts of issues with my reasoning.

[edit 2] - PS my conclusion that Wigner's friend is alive hinges on Wigner perfoming a non-destructive measurement of $\mathcal{X}$.