# A Wigner's Friend and Incompatibility

#### DarMM

Gold Member
For clarity we should separate two issues. 1) Is Wigner's operation possible in practice or in principle? 2) If Wigner's operation is possible in principle, and if we at least pretend it is therefore possible in practice with sufficient tech, is it conceivable Friend is alive and apparently sentient afterwards? I was only saying yes to the latter. I think, at a minimum, this is straighforwardly true in the Masanes type "full unitary reversal" versions of the protocol.
Agreed, if it is possible in principal you have a form of outcome subjectivity. Crudely speaking in a Copenhagenish view QM is a probability calculus for impressions left in a Classical/Boolean background by Quantum/Non-Boolean systems. If we have a Masanes type superobserver in principal then it has to be acknowledged that this classical background is not in principle unique and there can be other classical backgrounds whose events cannot be logically combined with our own. I sketched a PR-box version of Masanes due to Bub here:

In the Deutsch version of Wigner's friend where the backgrounds later combine this leads to observers with incompatible memories.

Regarding (1) I do think there are strong arguments it's not possible in principle. It seems $\mathcal{X} \notin \mathcal{A}\left(\mathcal{O}\right)$ where $\mathcal{O}$ is the observable horizon.

#### charters

Regarding (1) I do think there are strong arguments it's not possible in principle. It seems X∉A(O)X∉A(O)\mathcal{X} \notin \mathcal{A}\left(\mathcal{O}\right) where OO\mathcal{O} is the observable horizon.
Is this the de Sitter horizon?

#### Morbert

In non-encapsulated QM two events $E,F$ that don't commute can't usually have $P(E) = P(F) = 1$. Is this then a special feature of histories with encapsulation?
Actually my previous answer might be completely wrong and your intuition correct. If Wigner's friend wants to compute a prediction that Wigner definitely records $1$, he needs a record of his entire lab including himself, which should be incompatible with his record of his own measurement. I don't think he can be aware of both the pure state of his lab and his measurement.

#### DarMM

Gold Member
Is this the de Sitter horizon?
I'm not fully conversant on all the horizons in modern cosmology, but if you mean the $t \rightarrow \infty$ limit of the cosmological horizon, i.e. roughly 63 billion light years under current models, then yes.

#### charters

I'm not fully conversant on all the horizons in modern cosmology, but if you mean the $t \rightarrow \infty$ limit of the cosmological horizon, i.e. roughly 63 billion light years under current models, then yes.
Right, so this is dependent on the existence of dark energy/accelerated expansion right? Because otherwise the horizon does not asymptote to a finite volume

#### DarMM

Gold Member
Right, so this is dependent on the existence of dark energy/accelerated expansion right? Because otherwise the horizon does not asymptote to a finite volume
Not entirely. One would need the accelerating expansion to be false and then wait a long time until $10^{10^{18}}$ particles could fit within the horizon without collapsing into a black hole. The device would be way beyond a google light years across which would seem to prevent it operating on the time scales required, e.g. the average timescale of thermal fluctuations of the first device.

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#### Demystifier

2018 Award
However if the friend does a measurement of $S_z$ he knows for a fact he will get whatever result he originally got.
That's true.

However he also knows Wigner will obtain the 1 outcome with certainty.
That's not true. To simplify the notation, let me define $|{\rm up}\rangle\equiv |L_{\uparrow_z}, D_{\uparrow_z}, \uparrow_z \rangle$ and similarly for $|{\rm down}\rangle$. If the friend obtained the result $|\uparrow_z \rangle$, then the friend knows that Wigner should correctly describe the system as $|{\rm up}\rangle$. Alternatively, if the friend obtained the result $|\downarrow_z \rangle$, then the friend knows that Wigner should correctly describe the system as $|{\rm down}\rangle$. In either case, the friend can not predict what will be the Wigner's result of measurement of $\chi$.

But how can Wigner know that he should not describe the system as a superposition $|{\rm up}\rangle+|{\rm down}\rangle$? He can know it by knowing how to use quantum mechanics. He knows that $|{\rm up}\rangle$ and $|{\rm down}\rangle$ are macroscopically distinct, and he knows that in this case a collapse rule should be applied. It can either be the true collapse (a'la von Neumann or GRW), or many-world effective collapse (because Wigner compares him with his friend in the same branch), or Bohmian effective collapse (one of the branches in the superposition is empty, which makes it effectively irrelevant), or a qbist effective collapse (Wigner knows that friend knows that the spin has a definite value). But some kind of a collapse rule must be applied, which renders QM consistent.

The source of an apparent inconsistency in your simple example is the same as in the (in)famous FR theorem, it results from a hidden assumption that somehow an effective collapse both did and didn't happen. Once one realizes that the effective collapse definitely happened, the inconsistency disappears.

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#### DarMM

Gold Member
Definitely collapse "all the way down" takes care of this problem.

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#### Morbert

To expand a bit on my last messages, and to lay out my understanding of events: Let's say Wigner prepares eveything in the state $\rho = [\psi_0,D_\mathrm{ready},F_\mathrm{ready},L_\mathrm{ready}]\otimes\frac{1}{2}([\mathrm{heads}]+[\mathrm{tails}])$ (Where I have included an additional description of the friend as $[F_\mathrm{ready}]$. If Wigner wants to reason about his friend's experiences, he can use the framework (omitting histories with $0$ probability)

[\uparrow_z,D_{\uparrow_z},F_{\uparrow_z},L_{\uparrow_z}]_{t_1}&\odot&[\mathrm{tails}]_{t_2}\\
[\downarrow_z,D_{\downarrow_z},F_{\downarrow_z},L_{\downarrow_z}]_{t_1}&\odot&[\mathrm{tails}]_{t_2}
\end{eqnarray*}

Or he can use an alternative framework to discuss the property he is about to measure

\begin{eqnarray*}
[\mathrm{tails}]_{t_2}&\otimes&[\mathcal{X_+}]_{t_4}
\end{eqnarray*}

Using the first framework, Wigner can reason that his friend measures, and therefore knows, the spin of the particle at time $t_1$, e.g. $[F_{\uparrow_z}]$. He can also reason that his friend, a competent physicist, knows that a future measurement at time $t_3$ will yield the same result. He can't however, conclude his friend knows $[\mathcal{X}_+]_{t_4}$, since that is only a property in the alternative framework. And vice verse: The framework with $[\mathcal{X}_+]$ makes no mention of the friend's knowledge of $[\mathcal{X}_+]$. In order for Wigner to reason that his friend knows his measurement outcome, and also knows what Wigner's measurement outcome will be, Wigner would have to construct a consistent framework with both of those propositions. I suspect this is impossible, since commutation relations forbid a record of both $[\mathcal{X}_+]$ and $[\uparrow]_z,[\downarrow_z]$ at the same time.

Some liberties I have taken: I haven't explicitly included any of the friend's doxastic properties. Only epistemic ones re/ previous measurements. There might be some loophole there that I'm not considering. I have also couched the whole conversation in Wigner's perspective. It might be the case that his friend can build his own corresponding frameworks, but that he will have to use some alternative $\rho$ to make his own predictions, and he will not be able to predict with certainty the property $[\mathcal{X}_+]_{t_4}$ either way.

#### Morbert

But how can Wigner know that he should not describe the system as a superposition $|{\rm up}\rangle+|{\rm down}\rangle$? He can know it by knowing how to use quantum mechanics. He knows that $|{\rm up}\rangle$ and $|{\rm down}\rangle$ are macroscopically distinct, and he knows that in this case a collapse rule should be applied.
Interesting. In the conventional WF thought experiment, it's usually supposed that Wigner is able to model his friend's lab with unitary evolution, right up to the point of measurement. If he should not do that, can he still know beforehand, with certainty, the result of his measurement outcome?

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

#### Demystifier

2018 Award
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

2018 Award
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.

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#### 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

2018 Award
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

2018 Award
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

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#### Demystifier

2018 Award
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

2018 Award
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

2018 Award
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

2018 Award
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.

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#### 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

Gold Member
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.

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#### DarMM

Gold Member
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?

#### DarMM

Gold Member
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

2018 Award
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".

"Wigner's Friend and Incompatibility"

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