A Realization of a Basic Wigner's Friend Type Experiment

[DarMM]

But there's no logical inference by which you can get specific F outcomes just from the Ws knowing their ok/fail outcomes

Well the whole set up of the Frauchiger-Renner paper essentially attempts to.

Depending on the paper the contradiction is either presented as:
$$w = ok \Rightarrow F=tails\\
\bar{w} = \overline{ok} \Rightarrow F=heads$$
or as:
$$w = ok \Rightarrow F=tails \Rightarrow \bar{w} = \overline{fail}\\
|\psi\rangle \Rightarrow P(\overline{ok}) \neq 0$$

I agree that using modal logic in QM doesn't seem to make much sense. Thus I'm not really sure what FR says about any interpretation.

 

[charters]

But these inferences are not valid in MWI/unitary QM. The Fs always take both paths and both paths get intertwined during recombination, such that |ok> and |fail> outcomes both receive amplitude from both F paths. Bub is still acting like F is a Copenhagen observer when treating the problem in MWI, which isn't fair.

 

[DarMM]

But these inferences are not valid in MWI/unitary QM. The Fs always take both paths and both paths get intertwined during recombination, such that |ok> and |fail> outcomes both receive amplitude from both F paths. Bub is still acting like F is a Copenhagen observer when treating the problem in MWI, which isn't fair.

I'm not arguing for Bub ( I need to think about it more). I'm just saying that's what FR do.

 

[DarMM]

Yes, agreed. Or you accept you have a "single user" theory or (as you mention) you deny superobservers somehow.

Earlier we characterised Copenhagen explanations of Wigner's friend and its extensions as either

1. Have transcendent elements of reality fill the role of storing some trace of the preparation
2. Single user theory
3. No superobservers

Where would you place the Consistent Histories explanation? FR treats Consistent histories in their paper,as does this paper:
https://arxiv.org/abs/1907.10095
It clearly breaks assumption C but it doesn't seem to obviously do so by the methods above, more so just restricting reasoning in general. FR explain it a bit differently to Losada et al.

 

[charters]

Where would you place the Consistent Histories explanation?

I think CH will deny superobservers; the idea of a histories partition assumes you can absolutely neglect the risk of "recoherence" in the classical limit. So after F measures, W has no hope of pulling of the necessary measurement.

 

[DrChinese]

I don't believe this paper has been referenced yet in this thread, as it just recently came out:

https://arxiv.org/abs/1907.05607
Testing the reality of Wigner's friend's experience
Kok-Wei Bong, Aníbal Utreras-Alarcón, Farzad Ghafari, Yeong-Cherng Liang, Nora Tischler, Eric G. Cavalcanti, Geoff J. Pryde, Howard M. Wiseman

Does quantum theory apply to observers? A resurgence of interest in the long-standing Wigner's friend paradox has shed new light on this fundamental question. Brukner introduced a scenario with two separated but entangled friends. Here, building on that work, we rigorously prove that if quantum evolution is controllable on the scale of an observer, then one of the following three assumptions must be false: "freedom of choice", "locality", or "observer-independent facts" (i.e. that every observed event exists absolutely, not relatively). We show that although the violation of Bell-type inequalities in such scenarios is not in general sufficient to demonstrate the contradiction between those assumptions, new inequalities can be derived, in a theory-independent manner, which are violated by quantum correlations. We demonstrate this in a proof-of-principle experiment where a photon's path is deemed an observer. We discuss how this new theorem places strictly stronger constraints on quantum reality than Bell's theorem.

This discusses the subject both on the theoretical and experimental levels, concluding that nature violates "Local Friendliness" (their version of 3 assumptions mentioned in Brukner's paper).

 

[DarMM]

I don't believe this paper has been referenced yet in this thread, as it just recently came out:

I did read it, I hope I understood it correctly. The locality and freedom of choice are well known, but the Objective Facts one is the most interesting. Dropping it means acknowledging a new form of complimentarity.

An interesting thing about Consistent histories as a formalism is that it has shown us that there are diachronic instances of complimentarity that go beyond the usual cases, in that one can have sequences of operators at a set of times $t_1,\dots,t_n$ that are all part of the same context/Boolean frame at each time, but the histories overall are not.

No Objective facts seems to expand complimentarity in a different direction, between the descriptions of an external and encapsulated observer. One cannot discuss Wigner and the Friend's experiences within the same Boolean context.

Of course to get around this you might reject superobservers.

I know some forms of Copenhagen (e.g. QBism and Consistent Histories) drop the Objective facts aspect, others reject superobservers (Neo-Copenhagen, e.g. Peres). Bohmian mechanics clearly drops locality. Many-Worlds drops Objective facts as well. Superdeterminism drops freedom of choice.

@RUTA what's your take?

 

[kimbyd]

Of course to get around this you might reject superobservers.

Rejecting superobservers is not a viable option, as observers can freely observe one another performing experiments. If there are observers, then there are by necessity super-observers.

You'd have to reject things in the other direction: that observers who can realistically exist in a superposition of states must necessarily be so simple that they can't really be considered to be observers. But then we're back to what I was saying earlier: you can then just prove that it is unnecessary to have an observer cause collapse by increasing the interactions of the pseudo-observer with its environment. By proving that collapse can occur with even these pseudo-observers, it makes nonsense of any such observer-based theories (e.g. you need a conscious observer to cause collapse).

 

[DarMM]

Rejecting superobservers is not a viable option, as observers can freely observe one another performing experiments. If there are observers, then there are by necessity super-observers

A superobserver isn't just an observer who can view another performing an experiment, it's an observer who is completely isolated from the first observer and has the ability to perform arbitrarily powerful observations on them down to the atomic level. Roland Omnes has a discussion of how realistic they are in chapter 7 of his "The Interpretation of Quantum Mechanics".

It'a related to the old question of whether all self-adjoint operators are part of the actual observable algebra.

 

[Morbert]

I think CH will deny superobservers; the idea of a histories partition assumes you can absolutely neglect the risk of "recoherence" in the classical limit. So after F measures, W has no hope of pulling of the necessary measurement.

That's definitely the case on a practical level, though I think the formalism is robust enough to handle them. E.g. If we have a Wigner's friend scenario characterised by an initial state $|\Psi\rangle = |\psi\rangle_p|\Omega\rangle_F|\Omega\rangle_W$ that evolves unitarily into a final state $(|0\rangle_p|0\rangle_F+|1\rangle_p|1\rangle_F)|+\rangle_W$ (ignoring normalisation), CH would let us describe Wigner's friend's measurement with the partition
$$[\Psi]\otimes\{[0]_F,[1]_F\}$$
or Wigner's measurement with the partition
$$[\Psi]\otimes\{[+]_W,[-]_W\}$$
CH even let's us go wild and build partitions like
$$[\Psi]\otimes\{[0]_p,[1]_p\}\otimes\{[0]_F,[1]_F\}$$
which contains propositions about the properties of the particle before Wigner's friend's measurement. Absolute crazytown! but partitions like
$$[\Psi]\otimes\{[0]_F,[1]_F\}\otimes\{[+]_W,[-]_W\}$$
are forbidden. Hence we cannot infer anything about what Wigner will measure based on any description of his friend's measurement. So e.g. if Wigner's friend says "I measured $[0]_p$ therefore Wigner has a 50-50 chance of measuring $[F]_+$", they would be incorrectly applying QM. As an aside, if the particle is prepared with the property $[0]_p$ then partitions like
$$[0]_p\otimes\{[0]_F,[1]_F\}\otimes\{[+]_W,[-]_W\}$$
which contains propositions about both measurements are permitted (provided Wigner's measurement happens after his friend's). In either case, CH would also agree re/ Wigner's friend's lab/brain having no record of Wigner's friend's measurement.

[edit]- PS In case it wasn't clear I was considering Wigner's measurement to be characterised by the isometry $J(|0\rangle_p|0\rangle_F+|1\rangle_p|1\rangle_F) = (|0\rangle_p|0\rangle_F+|1\rangle_p|1\rangle_F)|+\rangle_W$ which I think is in line with the discussion you were having.

 

[PeterDonis]

if quantum evolution is controllable on the scale of an observer

I think this is the critical premise of the argument, and in fact of the entire discussion about "superobservers" and such scenarios. Basically, a "superobserver" is an entity that can perform arbitrary unitary transformations on anything whatever, including "observers" like people. But this capability is equivalent to the capability to undo decoherence. And if there are entities that have the capability of undoing decoherence, then our entire conceptual framework surrounding "measurements" and "results" goes out the window. No measurement result could ever be considered to be permanently, irreversibly recorded, because a superobserver could always come along and erase it by the appropriate unitary operation that undid the decoherence.

the Objective Facts one is the most interesting. Dropping it means acknowledging a new form of complimentarity.

I think it's more drastic than that; dropping Objective Facts means what I described above, that there would be no such thing as "measurements" and "results" at all, because one could never consider anything to be permanently, irreversibly recorded.

 

[DarMM]

I think it's more drastic than that; dropping Objective Facts means what I described above, that there would be no such thing as "measurements" and "results" at all, because one could never consider anything to be permanently, irreversibly recorded.

This is my current understanding, I'm not confident on it.
In classical mechanics we would have the possibility of such superobervers, people who could rewind Hamiltonian or more generally Liouville evolution. The difference is in classical mechanics this reversal wouldn't leave incompatibility in the sense of the previous results $a,b$ and the superoberver results $c,d$ could still be considered to occur in a common sample space.

Also as far as I know Brukner and FR don't actually use unitary reversal, only Masanes's theorem does. FR and Brukner only use somebody with the power to measure to arbitrary strength. They still undo decoherence but not by reversal, just by an utterly destructive measurement.

 

[PeterDonis]

FR and Brukner only use somebody with the power to measure to arbitrary strength. They still undo decoherence but not by reversal, just by an utterly destructive measurement.

Yes, actually "reversal" is too narrow a term. The idea is that a superobserver can make a measurement on an observer that is the equivalent of measuring spin-x on a qubit for which the spin-z basis is the one that corresponds to a permanently recorded observation.

 

[DarMM]

That's definitely the case on a practical level, though I think the formalism is robust enough to handle them

I was playing around with CH regarding Wigner's friend myself and regarding:

but partitions like
$$[\Psi]\otimes\{[0]_F,[1]_F\}\otimes\{[+]_W,[-]_W\}$$
are forbidden

it seems that we can have the following consistent family:
$$\{[\Psi]\odot [0]_F \odot[+]_W, [\Psi]\odot [1]_F \odot[-]_W\}$$
and this one:
$$\{[\Psi]\odot [1]_F \odot[+]_W, [\Psi]\odot [0]_F \odot[-]_W\}$$
So indeed you can't form the partition as you said. The ones you can are pretty strange, i.e. that:
$$[\Psi]\odot [0]_F \odot[+]_W$$
and
$$[\Psi]\odot [1]_F \odot[+]_W$$
are not consistent and thus complimentary.

 

[DarMM]

Yes, actually "reversal" is too narrow a term. The idea is that a superobserver can make a measurement on an observer that is the equivalent of measuring spin-x on a qubit for which the spin-z basis is the one that corresponds to a permanently recorded observation.

Yes precisely and so we have bases complimentary to what we take to be macroscopic facts/results as you said.

Again in classical mechanics, in theory, one can imagine no result being permanent because somebody could reverse your local history. It's more this complimentarity that's odd.

 

[Morbert]

I was playing around with CH regarding Wigner's friend myself and regarding:

it seems that we can have the following consistent family:
$$\{[\Psi]\odot [0]_F \odot[+]_W, [\Psi]\odot [1]_F \odot[-]_W\}$$
and this one:
$$\{[\Psi]\odot [1]_F \odot[+]_W, [\Psi]\odot [0]_F \odot[-]_W\}$$
So indeed you can't form the partition as you said. The ones you can are pretty strange, i.e. that:

The stuff you can construct with CH can be pretty out there for sure.

One question: Typically a family of histories is a projective decomposition of the identity, such that the probabilities of the histories sum to one. For the families above, I compute a probability of 0.25 for each history, and two histories in each family mean their probabilities sum to 0.5? Although maybe there are more general accounts of families beyond PDIs I'm not aware of.

 

[vanhees71]

Isn't the summary of all these discussions that "superobservers" are simply "counterfactual fictitions"? To get sane again after all these debates, I recommend to read the classic

A. Peres, Quantum Theory: Concepts and Methods, Kluwer
Academic Publishers, New York, Boston, Dordrecht, London,
Moscow (2002).

It's a gem, i.e., a "no-nonsense book" about the foundations of QT.

 

[DarMM]

Isn't the summary of all these discussions that "superobservers" are simply "counterfactual fictitions"?

What do you mean by "counterfactual fictions"? Fictions I get, but I don't understand the counterfactual part.

A. Peres, Quantum Theory: Concepts and Methods, Kluwer
Academic Publishers, New York, Boston, Dordrecht, London,
Moscow (2002).

It's a gem, i.e., a "no-nonsense book" about the foundations of QT.

Having read it I think Peres is a very very good book, but on this point I think Omnes is better. Peres just states there are no superobservers as an axiom. Omnes provides reasons for thinking this via a first principles calculation in QM that bounds their size.

 

[vanhees71]

As I understand it Peres names ideas "counterfactual" things which are just thought about but not realizable in nature. The most simple example is to say, "I've prepared my electron's spin in the state $\sigma_z=1/2$" by running it through a Stern-Gerlach magnetic field and then ask "what if I'd have prepared it in the state $\sigma_x=-1/2$? Shouldn't also $s_x$ have a definite value, though I've not prepared it?" Of course that's fictitious and counterfactual: If you think about how the SG apparatus works in selecting spin states, it's clear you can prepare only either $s_z$ or $s_x$ (or the spin component in anyone direction), but then the other components are indetermined. Indeed, the SG apparatus works in entangling (almost maximally) the value of the spin component in direction of the large homogeneous part of the magnetic field, while those in the other direction are rapidly precessing such that they are complete indetermined even if you think in a semiclassical way, i.e., even in classical theory you couldn't determine the spin components not in direction of the field very precisely. QT tells us that there's no apparatus at all that could do this.

Which paper/book by Omnes are you talking about? He's one of the founders of the consistent-history interpretation, right? I've not read anything by him yet.

 

[DarMM]

One question: Typically a family of histories is a projective decomposition of the identity, such that the probabilities of the histories sum to one. For the families above, I compute a probability of 0.25

You're quite right. I'll have something longer up regarding consistent histories later today.

 

[DarMM]

As I understand it Peres names ideas "counterfactual" things which are just thought about but not realizable in nature.

even in classical theory you couldn't determine the spin components not in direction of the field very precisely. QT tells us that there's no apparatus at all that could do this.

Counterfactual refers to reasoning about the results of experiments unperformed. It's not directly the same as not realizable. Superobservers might not be realizable but the reasons have little to do with counterfactuals.

Which paper/book by Omnes are you talking about? He's one of the founders of the consistent-history interpretation, right? I've not read anything by him yet.

The Interpretation of Quantum Mechanics.

 

[vanhees71]

Counterfactual refers to reasoning about the results of experiments unperformed. It's not directly the same as not realizable. Superobservers might not be realizable but the reasons have little to do with counterfactuals.The Interpretation of Quantum Mechanics.

But isn't the upshot with "superobservers" usually that you claim they could "know" something due to measurements that are not changing the system, and then you think about a measurement (or rather a preparation) that's not realizable at all according to QM since it involves the simultaneous preparation in eigenstates of incompatible observables?

Isn't this the key mistake of the EPR argument, where you have a state of two particles, described by a two-particle wave function of the type
$$\Psi(\vec{r},\vec{P})=\psi(\vec{r}) \tilde{\phi}(\vec{P})$$
where $\psi$ is a sharply peaked relative-position wave function ($\vec{r}=\vec{x}_1-\vec{x}_2$) and $\tilde{\phi}$ a sharply peaked total-momentum wave function ($\vec{P}=\vec{p}_1+\vec{p}_2$). Note that these two observables are compatible and of course one can prepare such a state.

If you now measure $\vec{x}_1$ precisely, then also $\vec{x}_2$ is known precisely due to this preparation though, of course, nothing in measuring the position $\vec{x}_1$ disturbs the far distant particle at position $\vec{x}_2=\vec{x}_1-\vec{r}$. Then, of course neither $\vec{p}_1$ nor $\vec{p}_2$ can be known very precisely.

Now EPR argue that you could have measured $\vec{p}_1$ very precisely without disturbing particle 2 in any way (which is of course true), and then you know also $\vec{p}_2=\vec{P}-\vec{p}_1$ very well. The mistake by EPR now simply is to conclude that does you'd know both $\vec{x}_1$ and $\vec{p}_1$ precisely though that violates the HUP.

The mistake simply is that this is counterfactual: If you measure $\vec{p}_1$ precisely you cannot also determine $\vec{x}_1$ precisely and thus also not know $\vec{x}_2$ precisely, while you in fact know $\vec{p}_2$ precisely. So choosing which measurement (position or momentum) you perform on particle 1 also determines what's precisely known about particle 2 (either position or momentum, respectively).

As Peres puts it: "unperformed measurements have no result".

It's a simple exercise to calculate the corresponding Fourier transformations for position or momentum measurements on either particle, e.g.,
$$\tilde{Psi}(\vec{x}_1,\vec{x}_2)=\psi(\vec{x}_1-\vec{x}_2) \int_{\mathbb{R}^3} \mathrm{d}^3 P \exp[\mathrm{i} \vec{P}(\vec{x}_1+\vec{x}_2)/2]/\sqrt{(2 \pi)^3} \tilde{\phi}(\vec{P})=\psi(\vec{x}_1-\vec{x}_2) \phi[(\vec{x}_1+\vec{x}_2)/2].$$
Indeed since $\tilde{\phi}$ is sharply peaked in $\vec{P}$, $\phi$ is a wide distribution in $\vec{R}=(\vec{x}_1+\vec{x}_2)/2$. Looking at all particle pairs, the position distribution of particle 1 is given by
$$P_1(\vec{x}_1)=\int_{\mathbb{R}^3} \mathrm{d}^3 x_2 |\tilde{\Psi}(\vec{x}_1,\vec{x}_2)|^2$$
which of course is a wide distribution.

Yet determining $\vec{x}_2$ to be in a small region around $\vec{x}_{20}$, the corresponding probability distribution is
$$\tilde{P}_1(\vec{x}_1|\vec{x}_2 \simeq \vec{x}_{20}) \simeq \tilde{\Psi}(\vec{x}_1,\vec{x}_{20}),$$
which is sharply peaked in $\vec{x}_1$, because $\psi(\vec{x}_1-\vec{x}_{20})$ is sharply peaked.

The analogous arguments can be made with the momenta, using the momentum representation of $\Psi$.

What's well determined about particle 1 through measurements on particle 2 depends in this example (a) on the original preparation, and in this case the positions $\vec{x}_1$ and $\vec{x}_2$ are entangled in a specific sense though both alone are quite undetermined as well as the momenta $\vec{p}_1$ and $\vec{p}_2$ are entangled though also their individual values are quite indetermined, and on (b) what's measured (either $\vec{x}_2$ or $\vec{p}_2$). Though not disturbing particle 1 in any way by the measurement on particle 2, you cannot know both $\vec{x}_2$ and $\vec{p}_2$ by a feasible measurement on particle 1.

Ironically what EPR call "realistic" is in fact utmost inrealistic according to QT, because they envoked a "counterfactual argument". Of course in their time, they could well claim that QT is incomplete (answering the question in the title of this infamous article with "yes" based on the counterfactual argument), because at this time neither the Bell inequality as a consequence of local deterministic HV theories were known nor the corresponding experiments have been performed. With all these achievements in the last 85 years, of course, such an excuse is mute since the corresponding zillions of "Bell tests" have confirmed the predictions of QT, and the stronger-than-classically-possible correlations described by entanglement, can be described nevertheless with microcausal relativisitc QFT, i.e., there's indeed no hint at "spooky actions at a distance".

The EPR paper becomes the more disfactory when one takes into account that Einstein himself didn't like it, because his own real quibble with QT was precisely the "inseparability" issue, i.e., the possibility of long-ranged correlations described by entanglement.

 

[DarMM]

But isn't the upshot with "superobservers" usually that you claim they could "know" something due to measurements that are not changing the system, and then you think about a measurement (or rather a preparation) that's not realizable at all according to QM since it involves the simultaneous preparation in eigenstates of incompatible observables?

No, they perform a measurement that seems to be realizable according to QM the results of which seem to contradict the observer having obtained a fixed result.

 

[vanhees71]

Hm, that sounds also strange. Either the observer has obtained a fixed result or not. No superobserver can change it, because "fixed" means it's irreversible. I guess, it's hard to discuss without a concrete example.

 

[PeterDonis]

Either the observer has obtained a fixed result or not. No superobserver can change it, because "fixed" means it's irreversible.

If superobservers exist, the observer's observation of a result is not irreversible, because the superobserver has the ability to perform arbitrary unitary operations on the observer, including ones that reverse or destroy the observer's observation of the result. For example, if the observer observes, say, spin-z up, the superobserver could perform a unitary operation on the observer that took him from the "observed spin-z up" state to the state

$$
\frac{1}{\sqrt{2}} \left( | \text{observed spin-z up} \rangle + | \text{observed spin-z down} \rangle \right)
$$

which is a state in which the observer has not observed any definite result at all.

 

[DarMM]

Hm, that sounds also strange. Either the observer has obtained a fixed result or not. No superobserver can change it, because "fixed" means it's irreversible. I guess, it's hard to discuss without a concrete example.

The standard Wigner's friend scenario is an example, the canonical example.

Imagine a lab that consists of a microscopic spin-1/2 system, a device that can measure it and the rest of the lab environment.

Somebody inside this lab measures a particle in the state:
$$\frac{1}{\sqrt{2}}\left(|\uparrow\rangle + |\downarrow\rangle\right)$$
they observe $\uparrow$ let's say and thus their device shows $D_{\uparrow}$ let's say, just symbolic for however the $\uparrow$ outcome is displayed on the set up.

Wigner outside the lab models the lab as:
$$\frac{1}{\sqrt{2}}\left(|\uparrow\rangle + |\downarrow\rangle\right)\otimes|D_{0}\rangle\otimes |L_{0}\rangle$$
where $D_{0}$ represents the device in the "ready" state before any readings and $L_{0}$ is the initial state of the lab.

After under unitary evolution the entire lab evolves for Wigner as:
$$\frac{1}{\sqrt{2}}\left(|\uparrow\rangle \otimes |D_{\uparrow}\rangle \otimes |L_{\uparrow}\rangle + |\downarrow\rangle \otimes |D_{\downarrow}\rangle \otimes |L_{\downarrow}\rangle\right)$$
where $\{|D_{\downarrow}\rangle ,|D_{\uparrow}\rangle\}$ represents indicator states of the device and $\{|L_{\downarrow}\rangle ,|L_{\uparrow}\rangle\}$ are corresponding states of the lab.

Wigner can then measure the entire lab system in the basis:
$$\left\{\frac{1}{\sqrt{2}}\left(|\uparrow\rangle \otimes |D_{\uparrow}\rangle \otimes |L_{\uparrow}\rangle + |\downarrow\rangle \otimes |D_{\downarrow}\rangle \otimes |L_{\downarrow}\rangle\right),\\
\frac{1}{\sqrt{2}}\left(|\uparrow\rangle \otimes |D_{\uparrow}\rangle \otimes |L_{\uparrow}\rangle - |\downarrow\rangle \otimes |D_{\downarrow}\rangle \otimes |L_{\downarrow}\rangle\right)\right\}$$
corresponding to some observable $\mathcal{X}$. Assigning $\{E_{+},E_{-}\}$ to indicate the two outcomes for brevity, he will have
$$P(E_{+}) = 1$$
in seeming contradiction with the device having recorded some result.

 

[DarMM]

One question: Typically a family of histories is a projective decomposition of the identity, such that the probabilities of the histories sum to one. For the families above, I compute a probability of 0.25 for each history, and two histories in each family mean their probabilities sum to 0.5? Although maybe there are more general accounts of families beyond PDIs I'm not aware of.

Sorry a total error on my part. We have five families.

The first:
$$\left\{\left[ 0\right]\otimes\left[W_{0}\right] , \left[ 1\right]\otimes\left[W_{0}\right], \left[ 0\right]\otimes\left[W_{1}\right], \left[ 1\right]\otimes\left[W_{1}\right] \right\}$$
Only those with matching values have non-zero probability.

The second:
$$\left\{\left[ 0\right], \left[ 1\right] \right\}$$
both with 0.5

The third:
$$\left\{\left[W_{0}\right], \left[W_{1}\right] \right\}$$
both with 0.5

Obviously the second and third can be seen as a coarse-graining of the first.

The fourth:
$$\left\{\left[ 0\right]\otimes\left[W_{+}\right] , \left[ 1\right]\otimes\left[W_{+}\right], \left[ 0\right]\otimes\left[W_{-}\right], \left[ 1\right]\otimes\left[W_{-}\right] \right\}$$
with outcomes all having a probability of 0.25

The fifth:
$$\left\{\left[W_{+}\right], \left[W_{-}\right] \right\}$$
with the first outcome having $P\left(\left[W_{+}\right]\right) = 1$

Now the second family can be seen as a coarse graining of the fourth one, but the fifth is incompatible with the fourth.

The first and third are incompatible with both the fourth and fifth.

Or more succintly:
$$\mathcal{F}_{1} \equiv \mathcal{F}_{2} \equiv \mathcal{F}_{3}\\
\mathcal{F}_{2} \equiv \mathcal{F}_{4}\\
\mathcal{F}_{5}$$

 

[Morbert]

$$\left\{\left[ 0\right]\otimes\left[W_{0}\right] , \left[ 1\right]\otimes\left[W_{0}\right], \left[ 0\right]\otimes\left[W_{1}\right], \left[ 1\right]\otimes\left[W_{1}\right] \right\}$$
Only those with matching values have non-zero probability.

Yeah this is what some consistent historians call a "measurement scenario", whereby the property/event that is measured correlates with the event that does the measuring, such that (as you mentioned) $P(\left[ 0\right] \mid \left[W_{0}\right] ) = P(\left[ 1\right] \mid\left[W_{1}\right] ) = 1$ and $P(\left[ 0\right] \mid \left[W_{1}\right] ) = P(\left[ 1\right] \mid\left[W_{0}\right] ) = 0$

$$\left\{\left[ 0\right]\otimes\left[W_{+}\right] , \left[ 1\right]\otimes\left[W_{+}\right], \left[ 0\right]\otimes\left[W_{-}\right], \left[ 1\right]\otimes\left[W_{-}\right] \right\}$$

What's interesting about this family is it also highlights the how the probabilities/consistency of families can depend on the preparation as well as just the dynamics. E.g. if the particle is prepared in the state
$$\frac{1}{\sqrt{2}}\left(|0\rangle + |1\rangle\right)$$
as is usually the case, then the probabilities will all be 0.25, but the decoherence functional will (I think) report interference. E.g. $\mathcal{D}( \left[ 0\right]\otimes\left[W_{+}\right] , \left[ 1\right]\otimes\left[W_{+}\right]) = 0.25$ If, however, the particle is prepared in a state $|0\rangle$, then the interference all goes away and the family becomes consistent (with a probability of 0.5 for two and 0 for the others)

 

[kimbyd]

I think this is the critical premise of the argument, and in fact of the entire discussion about "superobservers" and such scenarios. Basically, a "superobserver" is an entity that can perform arbitrary unitary transformations on anything whatever, including "observers" like people. But this capability is equivalent to the capability to undo decoherence.

That's going pretty far beyond the requirements of the problem. It is not necessary to undo decoherence in general. It is only necessary for it to be possible for there to be an experimental apparatus that effectively does this in some situations. That's only non-viable if decoherence is both discontinuous and occurs identically for all possible observers.

Basically, it's a question of whether or not there is such a thing as objective wavefunction collapse. If there is, then the Wigner's Friend-type experiments cannot ever measure quantum effects after this collapse has occurred. If there is no such objective collapse, or if said collapse long after decoherence has made it impossible to measure, then we should see a regime where the quantum effects are visible, and the visibility of said effects should gradually disappear as the experimental apparatus is moved out of that regime.

 

[PeterDonis]

It is not necessary to undo decoherence in general. It is only necessary for it to be possible for there to be an experimental apparatus that effectively does this in some situations.

If the "some situations" includes situations involving conscious observers, or macroscopic devices that record measurement results, then the issue I raised still applies. Our common sense understanding of what it means to "observe" something or to "record a measurement result" would no longer apply in these situations.

Also, I'm not sure I see a realistic limitation on the situations. It seems highly unlikely to me that there could exist a superobserver that can only do the particular unitary operations on my brain that correspond to "undoing" (erasing, destroying, whatever) my observation of the results of certain particular QM experiments, but not anything else.

 

[vanhees71]

If superobservers exist, the observer's observation of a result is not irreversible, because the superobserver has the ability to perform arbitrary unitary operations on the observer, including ones that reverse or destroy the observer's observation of the result. For example, if the observer observes, say, spin-z up, the superobserver could perform a unitary operation on the observer that took him from the "observed spin-z up" state to the state

$$
\frac{1}{\sqrt{2}} \left( | \text{observed spin-z up} \rangle + | \text{observed spin-z down} \rangle \right)
$$

which is a state in which the observer has not observed any definite result at all.

Sure, you can always do another SG experiment, but then you change inevitably the state. Superobservers shouldn't be allowed to be outside the rules of QT, if you want to make an argument within QT. That's what always comes out, when attempts are made to beat QT: At one point some "godlike" entity was envoked in the argument which can do measurements violating the principles of QT.

 

[vanhees71]

The standard Wigner's friend scenario is an example, the canonical example.

Imagine a lab that consists of a microscopic spin-1/2 system, a device that can measure it and the rest of the lab environment.

Somebody inside this lab measures a particle in the state:
$$\frac{1}{\sqrt{2}}\left(|\uparrow\rangle + |\downarrow\rangle\right)$$
they observe $\uparrow$ let's say and thus their device shows $D_{\uparrow}$ let's say, just symbolic for however the $\uparrow$ outcome is displayed on the set up.

Wigner outside the lab models the lab as:
$$\frac{1}{\sqrt{2}}\left(|\uparrow\rangle + |\downarrow\rangle\right)\otimes|D_{0}\rangle\otimes |L_{0}\rangle$$
where $D_{0}$ represents the device in the "ready" state before any readings and $L_{0}$ is the initial state of the lab.

After under unitary evolution the entire lab evolves for Wigner as:
$$\frac{1}{\sqrt{2}}\left(|\uparrow\rangle \otimes |D_{\uparrow}\rangle \otimes |L_{\uparrow}\rangle + |\downarrow\rangle \otimes |D_{\downarrow}\rangle \otimes |L_{\downarrow}\rangle\right)$$
where $\{|D_{\downarrow}\rangle ,|D_{\uparrow}\rangle\}$ represents indicator states of the device and $\{|L_{\downarrow}\rangle ,|L_{\uparrow}\rangle\}$ are corresponding states of the lab.

Wigner can then measure the entire lab system in the basis:
$$\left\{\frac{1}{\sqrt{2}}\left(|\uparrow\rangle \otimes |D_{\uparrow}\rangle \otimes |L_{\uparrow}\rangle + |\downarrow\rangle \otimes |D_{\downarrow}\rangle \otimes |L_{\downarrow}\rangle\right),\\
\frac{1}{\sqrt{2}}\left(|\uparrow\rangle \otimes |D_{\uparrow}\rangle \otimes |L_{\uparrow}\rangle - |\downarrow\rangle \otimes |D_{\downarrow}\rangle \otimes |L_{\downarrow}\rangle\right)\right\}$$
corresponding to some observable $\mathcal{X}$. Assigning $\{E_{+},E_{-}\}$ to indicate the two outcomes for brevity, he will have
$$P(E_{+}) = 1$$
in seeming contradiction with the device having recorded some result.

But after the superobservers unitary transformation the state is given by that you wrote in the curly brackets, and that implies that the original spin is not in a state determined by the observer's measurement. In this case you applied the standard rules of QT, and that tells you that with the intervention of the friend the state changes from that prepared within the lab before.

 

[DarMM]

But after the superobservers unitary transformation the state is given by that you wrote in the curly brackets, and that implies that the original spin is not in a state determined by the observer's measurement. In this case you applied the standard rules of QT, and that tells you that with the intervention of the friend the state changes from that prepared within the lab before.

That unitary transformation is the time evolution representing the friend's measurement though, i.e. his measurement is complete at that stage. It's not a transformation the superobserver applies himself just his description of the measurement process. To the superobserver that is the state after the measurement has been performed. The superobserver has not seen any outcome at this point so he will not alter the state at that stage and he is still free to perform a measurement on the $\mathcal{X}$ observable.

 

[vanhees71]

But there's the measurement process within the lab. This should also be taken into account in the unitary time evolution by Wigner's friend, right?

 

[DarMM]

But there's the measurement process within the lab. This should also be taken into account in the unitary time evolution by Wigner's friend, right?

How would it be taken into account? In other words how is it not taken into account in the description above.

If the particle was originally in the state $|\uparrow\rangle$ then the state of the lab after the measurement is:
$$|\uparrow\rangle \otimes |D_{\uparrow}\rangle \otimes |L_{\uparrow}\rangle$$
If it was originally $|\downarrow\rangle$ then the state after measurement is:
$$|\downarrow\rangle \otimes |D_{\downarrow}\rangle \otimes |L_{\downarrow}\rangle$$

How would Wigner, who sits outside the lab, model the lab from an initial state of $\frac{1}{\sqrt{2}}\left(|\uparrow\rangle + |\downarrow\rangle\right)$ for the system but by
$$\frac{1}{\sqrt{2}}\left(|\uparrow\rangle \otimes |D_{\uparrow}\rangle \otimes |L_{\uparrow}\rangle + |\downarrow\rangle \otimes |D_{\downarrow}\rangle \otimes |L_{\downarrow}\rangle\right)$$

What different state do you think he should be using?