# A Wigner's Friend and Incompatibility

#### Demystifier

2018 Award
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

Gold Member
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

2018 Award
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.

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#### 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

2018 Award
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 occured 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 occured 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

2018 Award
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

Gold Member
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 lets 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.

 - 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 realiseable.

- 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 $\mathcal{X}$.

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"Wigner's Friend and Incompatibility"

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