# A Someone observing Wigner and his friend

#### A. Neumaier

Summary
A query
Wigner has a mixed state for his friend's experimental device because he has maximal possible knowledge (a pure state) for the Lab as a whole. The friend does not track the lab as a whole and thus he can have maximal knowledge (a pure state) for the device.
A human asking Wigner and his friend about the states they assign has certain and complete information about both Wigner's pure state of the lab and his friend's pure state of the device. How do these certainties show up (as certainties) in the pure state of ''lab + Wigner + his friend''? Since the pure state is supposed to be a state of maximal knowledge these certainties should be deducible deterministically from this state.

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#### DarMM

Gold Member
Summary: A query

A human asking Wigner and his friend about the states they assign has certain and complete information about both Wigner's pure state of the lab and his friend's pure state of the device
So an observer outside of Wigner? Wigner's boss, Goheen I guess. The President of Princeton at the time

When does Goheen learn these states in this thought experiment? Before or after the friend performs a measurement?

#### A. Neumaier

Wigner's boss, Goheen I guess. [...]When does Goheen learn these states in this thought experiment? Before or after the friend performs a measurement?
After, but before he told his answer to Wigner.

#### charters

Maybe relevant is https://arxiv.org/abs/1901.11274. In particular in Step (a), where we see Friend can tell Wigner she has a pure state of the device as long as she doesn't tell him which pure state it is. Friend could also pass this note to Boss, and Wigner could (more easily) pass a note to Boss saying he has a pure state for the whole lab.

#### A. Neumaier

Maybe relevant is https://arxiv.org/abs/1901.11274. In particular in Step (a), where we see Friend can tell Wigner she has a pure state of the device as long as she doesn't tell him which pure state it is. Friend could also pass this note to Boss, and Wigner could (more easily) pass a note to Boss saying he has a pure state for the whole lab.
I'd like the Friend tell Wigner his assumed pure state before the measurement, and his Boss the assumed pure state after his measurement. And Wigner to tell the Boss his assumed pure state, before he gets to know the friend's measurement result, which may be assumed to be 0 (for a binary system)

#### charters

and his Boss the assumed pure state after his measurement
If Friend "breaks the seal" of her Lab like this, she becomes entangled with Boss. So at this point Wigner can't assign a pure state to the Lab, only to Lab+Boss. So either Wigner understands the experimental protocol and can't answer, or he has ignorance about the experiment, and just tells Boss something factually incorrect.

#### A. Neumaier

If Friend "breaks the seal" of her Lab like this, she becomes entangled with Boss.
But the dynamics of Lab+Boss entangles the states anyway. Surely Friend can talk to Boss if he can talk to Wigner.

#### charters

But the dynamics of Lab+Boss entangles the states anyway. Surely Friend can talk to Boss if he can talk to Wigner.
It depends what sort of "talking" you mean. Friend can pass a note to either W or B saying something like "I have now measured the qubit along the z axis and observed a definite outcome" and F/Lab will remain unentangled with W or B as long as the note is identical regardless of which outcome F has seen. But if F tells B something like "I saw |0>" now there is entanglement between the Lab and the receiver of the note. So, if B receives the latter note, there is no pure state of F and her lab anymore for W to describe.

#### A. Neumaier

It depends what sort of "talking" you mean. Friend can pass a note to either W or B saying something like "I have now measured the qubit along the z axis and observed a definite outcome" and F/Lab will remain unentangled with W or B as long as the note is identical regardless of which outcome F has seen. But if F tells B something like "I saw |0>" now there is entanglement between the Lab and the receiver of the note. So, if B receives the latter note, there is no pure state of F and her lab anymore for W to describe.
First F tells W and B his assumed pure state of the system before the measurement. Then W tells B his assumed pure state of the lab. Then F tells B about the outcome of the measurement. Then B tells me his pure state of the lab+W. I cannot see any problem with that.

#### charters

First F tells W and B his assumed pure state of the system before the measurement.
The overall state is

So F will currently say |0+1>

Then W tells B his assumed pure state of the lab.
W will tell B the lab is |0+1>⊗|F-ready>, which everyone already knew from the initial conditions

Then F tells B about the outcome of the measurement
So, first the state has to evolve to

Then, F tells B their result and we go to

Then B tells me his pure state of the lab+W
This will be

OR

Whereas W will say there is still a global superposition of the qubit, F and B.

#### A. Neumaier

The overall state is
So F will currently say |0+1>
--
W will tell B the lab is |0+1>⊗|F-ready>, which everyone already knew from the initial conditions
--
So, first the state has to evolve to
Then, F tells B their result and we go to
--
This will be
OR
Whereas W will say there is still a global superposition of the qubit, F and B.
the friend's measurement result, which may be assumed to be 0 (for a binary system)
then the or is gone...

A human asking Wigner and his friend about the states they assign has certain and complete information about both Wigner's pure state of the lab and his friend's pure state of the device. How do these certainties show up (as certainties) in the pure state of ''lab + Wigner + his friend''? Since the pure state is supposed to be a state of maximal knowledge these certainties should be deducible deterministically from this state.
How do I extract from the final state of B the pure state of W, which is part of the knowledge of B about ''lab + Wigner + his friend''?

#### charters

"Ready" just means the object in that ket is in a state that indicates no outcome of the qubit measurement (eg equipment dial is in the neutral state, the human's brain has not arranged neurons into a memory).

W is pure (or very close to pure if the experimental design is imperfect) according to B because B knows the initial conditions of your experiment, which defines W as such, and which requires W to remain isolated from both F and B throughout.

#### A. Neumaier

"Ready" just means the object in that ket is in a state that indicates no outcome of the qubit measurement (eg equipment dial is in the neutral state, the human's brain has not arranged neurons into a memory).
This looks very fishy to me. Whar is the state space used by B in your analysis? - assume a perfect design.

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#### charters

I am not sure what you are asking or objecting to. This is a very standard way to describe Wigner's Friend/any measurement in Dirac notation.

#### A. Neumaier

I am not sure what you are asking or objecting to. This is a very standard way to describe Wigner's Friend/any measurement in Dirac notation.
I don't know yet what precisely I am objecting to. To me, your state looks at present incomplete, so I want to understand what you wrote well enough to be able to spell out in detail what I am objecting to - like you did when trying to understand my thermal interpretation.

The last time I had looked at a formal analysis of Wigner's friend was over 30 years ago, so I don't know the standard assumptions and notation of current expositions. Moreover, my setting is a modification probably not in the literature. Thus I want to fully understand the formal assumptions in your analysis of my setting.

I know and accept all standard math of quantum mechanics, including (for the purposes of this discussion) the informal rules for its Bayesian interpretation. I introduced my query for a specific purpose - to find out the formal details needed to reason precisely about the Bayesian interpretation.

Any quantum analysis is based on a Hilbert space. I am asking what is the Hilbert space used in your analysis by B to describe system+friend+Wigner, and what is the meaning of the basis assumed?

Given your answer to this question, I may either accept it, or - more likely - I may wish to modify it to accommodate all the information given in my query of post #1.

#### charters

I introduced my query for a specific purpose - to find out the formal details needed to reason precisely about the Bayesian interpretation.
Maybe that's the problem as I am describing it from a unitary/Everettian perspective. I didn't see any reference to Bayesianism above.

I am asking what is the Hilbert space used in your analysis by B to describe system+friend+Wigner, and what is the meaning of the basis assumed?
The qubit, Friend, Wigner, and Boss all have their own Hilbert spaces, and the global state is a tensor product of each Hilbert space. The Hilbert spaces of the humans are highly coarse grained of course.

In https://arxiv.org/abs/quant-ph/0703160, Zurek describes the idea of a "Von Neumann" chain - eq (1) and (4) being key, and what I was using above. Wigner's friend is just a VN chain with 2 "apparatus/observer" links. You are simply increasing this to 3 links, which doesn't introduce any new complications beyond the 2 observer case.

#### A. Neumaier

I didn't see any reference to Bayesianism above.
But my query had Bayesian input: ''states they assign'' and ''complete information'':
''A human asking Wigner and his friend about the states they assign has certain and complete information''
I am describing it from a unitary/Everettian perspective.
In the Everettian perspective, all states of subsystems coexist. How then can the particular ''state that Wigner assigns'' be encoded?

#### A. Neumaier

The qubit, Friend, Wigner, and Boss all have their own Hilbert spaces, and the global state is a tensor product of each Hilbert space. The Hilbert spaces of the humans are highly coarse grained of course.
Which Hilbert space? coarse-grained to what? I need the basis used, and its meaning.

#### DarMM

Gold Member
To be honest there isn't much difference formalism wise between an Everettian view and Bayesian views like Neo-Copenhagen and QBism for this thought experiment since they both assume unitarity on all scales.

After, but before he told his answer to Wigner.
The friend for instance is using either $|\uparrow\rangle$ or $|\downarrow\rangle$ after the measurement. Wigner is using:
$$\frac{1}{\sqrt{2}}\left(|\uparrow, D_{\uparrow}, L_1\rangle + |\downarrow, D_{\downarrow}, L_2\rangle\right)$$
with $D_{\uparrow}$ denoting the state of the friends device and $L_1$ being a state of the lab. That is I use a "collapsed" state, but the superobserver does not.

So let's say Boss learns Wigner's state. He then goes to learn the friends state. If he does so without contacting Wigner again he essentially becomes part of the friends lab and thus Wigner's state is now:
$$\frac{1}{\sqrt{2}}\left(|\uparrow, D_{\uparrow}, L_1 , B_{\uparrow}\rangle + |\downarrow, D_{\downarrow}, L_2 , B_{\downarrow}\rangle\right)$$
where $B_{\uparrow}$ represents Boss knowing that the friends outcome was $\uparrow$.

However as such the Boss doesn't add much here, because the friend himself once he is sealed off from Wigner could deduce that Wigner is going to use the superposed state:
$$\frac{1}{\sqrt{2}}\left(|\uparrow, D_{\uparrow}, L_1\rangle + |\downarrow, D_{\downarrow}, L_2\rangle\right)$$
So he would know his own state of $|\uparrow\rangle$ and Wigner's state with superposition.

However these can't be combined to form a tighter state, which I think is what you are after. For somebody in Wigner's position has access to super-observables related to the lab's atomic structure for which the superposed state is required. However somebody in the experimental position to measure these super-observables by necessity cannot know which measurement outcome occurred for the friend.

#### charters

In the Everettian perspective, all states of subsystems coexist. How then can the particular ''state that Wigner assigns'' be encoded?
We can use a relative state. Any observer's "personal" state is their own ket and each ket they are tensored to, aka the state taken to be reduced to only their branch/world

Which Hilbert space? coarse-grained to what? I need the basis used, and its meaning
I don't really understand what sort of statement you consider an acceptable answer to this, you may need to ask more precise questions. The conventional Hilbert space treatment of these macro objects is just to treat them as basic 2 or 3 discrete level systems, which stand in for eigenstates of the decoherence basis (which is roughly a coarse grained position basis here).

If the Zurek paper did not help try https://arxiv.org/abs/1811.09062 or maybe the old Von Neumann paper Zurek cites.

#### A. Neumaier

We can use a relative state. Any observer's "personal" state is their own ket and each ket they are tensored to, aka the state taken to be reduced to only their branch/world
Then the observer's "personal" state is not a property of the universe (though all properties of the whole observer should be) but an additional mental input in addition to the Everettian state of the universe!? Thus an observer picks the branch s/he lives in, by making up his mind that operates independent of the universe?? So we are masters of our fate if we choose our personal state to be one with a favorable future???

I don't really understand what sort of statement you consider an acceptable answer to this, you may need to ask more precise questions. The conventional Hilbert space treatment of these macro objects is just to treat them as basic 2 or 3 discrete level systems, which stand in for eigenstates of the decoherence basis (which is roughly a coarse grained position basis here).
This is an acceptable answer and makes sense of your calculations. But to say it is a coarse-grained description is misleading; it is just a severe (and physically probably not acceptable) truncation of the Hilbert space. In contrast, a coarse-grained description in the sense of statistical mechanics would involve density operators or semiclassical observables.

I consider such a few-level description an irrelevant caricature of the real situation.

"Someone observing Wigner and his friend"

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