Question regarding the Many-Worlds interpretation

[bhobba]

That's not what I have quoted or referred to. My reply was specifically about your claim that it makes no sense to have a special basis in the process of observation.

I never claimed that - in fact I don't even know what you mean by that. To be clear my claim is that decoherence solves the preferred basis problem as stated on page 113 of the reference I gave before by Schlosshauer. He gives 3 issues the measurement problem must solve:

1. The preferred basis problem
2. The problem of non observability of interference
3. The problem of why we have outcomes at all.

The statement he makes is my position:
'it is reasonable to conclude decoherence is capable of solving the first two, whereas the third problem is linked to matters of interpretation'

And that is exactly it - the first two have had considerable work done that indicates decoherence will likely solve them - in fact for a number of models given in the book it does. To solve the third one you need further assumptions and the detail of those assumptions is peculiar to each interpretation.

Thanks
Bill

 

[bhobba]

I've not been missing your point. I'm intimately familiar with the arguments used in decoherence, and I still disagree. The way you lay it out the argument depends highly on the construction of the density matrix to encode quantum ensembles. And this construction is only motivated if you assume that upon observations quantum probabilities mix with classical (ensemble) probabilities. It doesn't matter then how you construct improper ensembles or if they're sensible constructs because the real ensembles are already problematic.

This is going around in circles.

One more time - then that's the end of it for this thread for me. However based on past experience it will be rehashed.

Decoherence adherents, unless they are being disingenuous like the paper cited before, do not claim it solves the measurement problem. What they claim is its non issue because its observationally the same as a proper mixture and gives the appearance of wavefunction collapse.

Thanks
Bill

 

[stevendaryl]

This is why I specified proper mixture: improper mixtures that arise via tracing out a part of the system look mathematically indistinguishable from 'true' mixtures that arise from uncertainty about what the actual state is,

The MWI interpretation basically amounts to the claim that all mixtures are "improper" in that sense.

but can't be given an ignorance interpretation (this error is also at the root of Art Hobson's recent 'resolution' of the measurement problem which we've discussed here).

I have not followed that thread, but as I understand it, MWI absolutely depends on there being no observational difference between "proper" and "improper" mixtures.

 

[S.Daedalus]

Decoherence adherents, unless they are being disingenuous like the paper cited before, do not claim it solves the measurement problem. What they claim is its non issue because its observationally the same as a proper mixture and gives the appearance of wavefunction collapse.

I just don't see in what sense that's true; it's trivial to distinguish a proper from an improper mixture. Take the state
$$|\Phi^+\rangle=\frac{1}{\sqrt{2}}(|00\rangle + |11\rangle).$$
Locally, both Alice and Bob describe it by the mixture
$$\rho_A=\rho_B=\frac{1}{2}(|0\rangle\langle 0| + |1\rangle\langle 1|),$$
and all of their observations will be in line with this assignment. But if they now believe that therefore, their respective states are in an actual mixture of $|0\rangle$ and $|1\rangle$, they must also believe that the global state corresponds to
$$\rho_{AB}=\rho_A\otimes\rho_B=\frac{1}{4}(|00\rangle\langle 00| + |01\rangle\langle 01| + |10\rangle\langle 10| + |11\rangle\langle 11|),$$
simply because that is the state the system would be in if it were the case that each local state were a proper mixture, one for instance generated by producing the states $|0\rangle$ or $|1\rangle$ equiprobably at random. But of course, this state is observationally very different from $|\Phi^+\rangle$: for instance, both Alice and Bob would expect their measurements to be entirely uncorrelated, but in fact, they will be perfectly correlated. This amounts to a falsification of the belief that their states can be described by a proper mixture, i.e. that they can be given an ignorance interpretation. The states can't be identified, even though the local observations are equivalent.

Alternatively, you can just measure $|\Phi^+\rangle \langle\Phi^+|$: clearly, while $|\Phi^+\rangle$ is an eigenstate, and thus, the measurement will return +1 determinately, $\rho_{AB}$ is not, and the outcome will be random; the assumption of being able to give an ignorance interpretation to their states leads Alice and Bob to make false predictions.

So in what sense could you consider these states equivalent?

 

[bhobba]

That's not what I have quoted or referred to. My reply was specifically about your claim that it makes no sense to have a special basis in the process of observation.

I have had a look at the responses here and there has been a misunderstanding of contexts.

The original quote you gave was in reference to the assumption of non-contextuality in the proof of Gleason's theorem which is the measure is basis independent. That didn't make sense in that context and I assumed it meant that decoherence didn't single out a specific basis.

Thanks
Bill

 

[stevendaryl]

The MWI interpretation basically amounts to the claim that all mixtures are "improper" in that sense.

I have not followed that thread, but as I understand it, MWI absolutely depends on there being no observational difference between "proper" and "improper" mixtures.

So I took a look at one of your comments in that thread:

Mathematically, this is the same object that one would use to describe a system that is prepared either in the state |M1⟩ or |M2⟩ with a respective probability of |c1|2 or |c2|2. However---and this is where the argument goes wrong, I believe---in case this object is arrived at by tracing out the degrees of freedom of another subsystem, one can't interpret it in the way that the system is in fact in either of the states |M1⟩ or |M2⟩, and we just don't know which.

Is that supposed to be a criticism of MWI? MWI essentially claims that it's incorrect to attribute QM probabilities to ignorance. It's incorrect to attribute mixed states to ignorance. So it's no criticism of MWI that it has a conclusion that is different from the conclusion of a theory that attributes mixed states to ignorance--that's the whole point of MWI. The real issue isn't whether the mixed states are due to ignorance (they are, in some interpretations, and they are not, in other interpretations). The issue is whether and how both interpretations are consistent with what we observe.

 

[bhobba]

I just don't see in what sense that's true; it's trivial to distinguish a proper from an improper mixture.

What exactly about the answer based of Decoherent Histories I gave in the thread you linked to didn't you understand?

Added Later:

You seemed to understand the issue in that thread:
'Now as I said, you can augment the scheme so as to avoid this problem---by, say, going the modal route, or by just working within the framework of consistent histories as Griffiths does; but then you are adding an extra interpretation to the quantum formalism and not, as Hobbes claims to do, resolving the problem 'from within' (something which by the way runs headlong into multiple insolubility theorems of the measurement problem from within quantum mechanics formulated over the years, starting with Fine (1970)). And of course, all of these interpretations do have their own problems (contradictory inferences in consistent histories, Kochen-Specker contradictions/inconsistent value state assignments in modal theories, etc.).'

The answer is still the same, and I have never shied away from it, but for some reason it seems to be bought up all the time, you need further assumptions. MWI is one route, Decoherent Histories is another.

Thanks
Bill

 

[S.Daedalus]

Is that supposed to be a criticism of MWI?

No, that was a criticism of Hobson's approach. It's also relevant in the context of this thread because people keep attempting to appeal to Gleason's theorem in order to recover the Born probabilities in the MWI, but this fails for a similar reason, namely that proper and improper mixtures are not the same thing.

In a way, getting a proper from an improper mixture is what the measurement problem is all about. Collapse interpretations solve it by fiat: someone snips their fingers, and out comes the desired proper mixture. Many people find this dissatisfying, and with good reason. But then proposing a solution that ends up depending on the very same sleight of hand is no progress at all.

 

[S.Daedalus]

What exactly about the answer based of Decoherent Histories I gave in the thread you linked to didn't you understand?

Well, you didn't really give me something to understand, you merely gave a reference. And furthermore, many worlds and consistent histories are two distinct interpretations, with different problems; that a problem can be solved in one, doesn't mean it can be solved in the other. Besides, from what I know about it, I don't see how consistent histories permits the identification of improper and proper measurements; after all, it still must account for Bell tests somehow. But at any rate, if you think that you can show how, in the scenario I outlined above, Alice and Bob can't simply meet up and compare their measurement records in order to find out that their states couldn't have been proper mixtures, I'm all ears.

Response to the bit you added later: yes, I do think that the problem of probability is (at the very least) less severe in consistent histories, but that doesn't mean that it solves the problems of the MWI; you can't fix the MWI by just switching to a different interpretation. It's within the MWI framework that we need to solve the problem here; you can add all manner of things in order to get the right probabilities (though some people wouldn't even believe that), but the question is whether you can (as is often claimed) resolve it within the MWI itself.

 

[stevendaryl]

I just don't see in what sense that's true; it's trivial to distinguish a proper from an improper mixture. Take the state
$$|\Phi^+\rangle=\frac{1}{\sqrt{2}}(|00\rangle + |11\rangle).$$
Locally, both Alice and Bob describe it by the mixture
$$\rho_A=\rho_B=\frac{1}{2}(|0\rangle\langle 0| + |1\rangle\langle 1|),$$
and all of their observations will be in line with this assignment. But if they now believe that therefore, their respective states are in an actual mixture of $|0\rangle$ and $|1\rangle$, they must also believe that the global state corresponds to
$$\rho_{AB}=\rho_A\otimes\rho_B=\frac{1}{4}(|00\rangle\langle 00| + |01\rangle\langle 01| + |10\rangle\langle 10| + |11\rangle\langle 11|),$$
simply because that is the state the system would be in if it were the case that each local state were a proper mixture, one for instance generated by producing the states $|0\rangle$ or $|1\rangle$ equiprobably at random. But of course, this state is observationally very different from $|\Phi^+\rangle$: for instance, both Alice and Bob would expect their measurements to be entirely uncorrelated, but in fact, they will be perfectly correlated. This amounts to a falsification of the belief that their states can be described by a proper mixture, i.e. that they can be given an ignorance interpretation. The states can't be identified, even though the local observations are equivalent.

Alternatively, you can just measure $|\Phi^+\rangle \langle\Phi^+|$: clearly, while $|\Phi^+\rangle$ is an eigenstate, and thus, the measurement will return +1 determinately, $\rho_{AB}$ is not, and the outcome will be random; the assumption of being able to give an ignorance interpretation to their states leads Alice and Bob to make false predictions.

So in what sense could you consider these states equivalent?

I think you miss the point of the use of decoherence in saying that proper and improper mixed states are observationally indistinguishable. There is a mathematical difference between the two, because the improper mixed state contains "interference terms" that are absent in the proper mixed state. But in order to observe these interference effects, you have to perform a measurement that has different outcomes (or different probabilities of outcomes) if the interference terms are present. Basically, what that amounts to is performing a measurement that "unmixes" the states. But when the subsystems involve many, many states (an observer, or the environment) this is a practical impossibility on the order of putting a broken pane of glass back together. Mixing is irreversible in practice for the same reason that classical physics of 10^23 particles is irreversible in practice, even both are reversible in theory.

I don't think your mathematical demonstration is correct. I don't think you combine mixed states that way.

 

[stevendaryl]

No, that was a criticism of Hobson's approach. It's also relevant in the context of this thread because people keep attempting to appeal to Gleason's theorem in order to recover the Born probabilities in the MWI, but this fails for a similar reason, namely that proper and improper mixtures are not the same thing.

In a way, getting a proper from an improper mixture is what the measurement problem is all about. Collapse interpretations solve it by fiat: someone snips their fingers, and out comes the desired proper mixture. Many people find this dissatisfying, and with good reason. But then proposing a solution that ends up depending on the very same sleight of hand is no progress at all.

That's absolutely not true. The progress is that you do away with a nonphysical collapse hypothesis. I agree that there are conceptual difficulties with MWI, but what the use of "improper" mixtures shows is that there is really no evidence that any particular measurement collapses the wave function. So there is no evidence that macroscopic objects (even measurement devices and observers) can't be treated quantum mechanically.

I certainly agree that there is still mystery involved in the interpretation of quantum mechanics, measurements and probabilities and all that. But I don't consider the use of mixed states to be still a mystery.

 

[bhobba]

Well, you didn't really give me something to understand, you merely gave a reference.

I beg to differ. As I said in that thread there is a key assumption in your argument:

Basically you are assuming it possesses those properties SIMULTANEOUSLY globally - which the situation doesn't require. The complete analysis in given in the reference and hinges on the concept of framework used in that interpretation - basically one is free to choose frameworks as long as they are consistent - and a framework exists where it has both those properties locally - but not globally.

You even said it in that thread - assuming an epistemic interpretation that you left out in what you posted here.

Thanks
Bill

 

[S.Daedalus]

I think you miss the point of the use of decoherence in saying that proper and improper mixed states are observationally indistinguishable. There is a mathematical difference between the two, because the improper mixed state contains "interference terms" that are absent in the proper mixed state. But in order to observe these interference effects, you have to perform a measurement that has different outcomes (or different probabilities of outcomes) if the interference terms are present.

No. That's the point of Alice and Bob's story: they only make the local measurements, which don't detect interference terms and look just like the states actually were in a mixed state. But upon comparing their measurement records, they will notice that they always got the same results---i.e. that they are perfectly correlated---while the prediction on the belief of proper mixedness is that they would not observe any correlation at all.

That's not to say that for all practical purposes, you can consider decohered states to be effectively mixed, since as you say the relevant measurements are pretty much impossible to perform; but this is not just a FAPP question: if you want to appeal to Gleason in order to get probabilities in the MWI, then you require that the identification can be made exactly; in any other case, Gleason just doesn't talk about probability in the MWI.

I don't think your mathematical demonstration is correct. I don't think you combine mixed states that way.

Of course you do.

 

[S.Daedalus]

I beg to differ. As I said in that thread there is a key assumption in your argument:

Even that quote just directs me to the reference. I repeat: what is it that prohibits Alice and Bob from just writing down their results and comparing them?

 

[stevendaryl]

No. That's the point of Alice and Bob's story: they only make the local measurements, which don't detect interference terms and look just like the states actually were in a mixed state. But upon comparing their measurement records, they will notice that they always got the same results---i.e. that they are perfectly correlated---while the prediction on the belief of proper mixedness is that they would not observe any correlation at all.

That's not to say that for all practical purposes, you can consider decohered states to be effectively mixed, since as you say the relevant measurements are pretty much impossible to perform; but this is not just a FAPP question: if you want to appeal to Gleason in order to get probabilities in the MWI, then you require that the identification can be made exactly; in any other case, Gleason just doesn't talk about probability in the MWI.

Of course you do.

The article specifically says "If A and B are two distinct and independent systems then $\rho_{AB}=\rho_{A}\otimes\rho_{B}$ which is a product state."

They don't elaborate on what "independent systems" means, but in fact, that rule is only valid if you ignore entanglement. It's absolutely incorrect in the case we're talking about.

 

[S.Daedalus]

They don't elaborate on what "independent systems" means, but in fact, that rule is only valid if you ignore entanglement. It's absolutely incorrect in the case we're talking about.

Yes, that's the point I'm trying to get across: attaching an ignorance interpretation to their local states is exactly ignoring the entanglement between them, and erroneously stipulating that they can treat the system as being in a definite, but unknown, state---which is simply not the case if you have entanglement. It's exactly the error made in the argument that proper and improper mixtures can be treated the same!

 

[kith]

Seems that I am the only one where "the penny did not drop".

I'm in the same boat. I find it difficult to get to the core of the issue and even to ask the right questions.

What I get is the assertion that probabilities either don't make sense (but I don't get how replacing them by amplitudes is sufficient) or if we insist on having them, they follow from Gleason's theorem (but I don't get if and what additional assumptions are made in this case).

 

[kith]

Locally, that's true, but once Alice and Bob get together and compare their measurement results, or a measurement is carried out on both parts of the system, you get results that falsify the idea that the parts of the system are in some definite state, and we just don't know which---correlations that we can't account for with such a model in the first case, and interference results in the second. These are perfectly valid observations, so I don't see how it's true that the two states are 'observationally equivalent'.

I think if you want to argue that improper and proper mixtures are the same in the framework of the MWI, you have to look at it from the other viewpoint: every mixture is improper. For seemingly proper mixtures, it is just to complicated to track the correlations between the system and the large and uncontrollable environment .

 

[S.Daedalus]

I think if you want to argue that improper and proper mixtures are the same in the framework of the MWI, you have to look at it from the other viewpoint: every mixture is improper. For seemingly proper mixtures, it is just to complicated to track the correlations between the system and the large and uncontrollable environment .

Yes, I believe that's a valid point of view. But of course, the issue of proper mixtures only arose because of the attempt to appeal to Gleason's theorem in order to get the Born rule in the MWI (I know, we're getting deeper and deeper down the rabbit hole here), for which (I have argued) you need proper mixtures (i.e. states that really are in a given subspace---otherwise, the measure on subspaces Gleason gives just has nothing to say).

 

[kith]

Yes, I believe that's a valid point of view. But of course, the issue of proper mixtures only arose because of the attempt to appeal to Gleason's theorem in order to get the Born rule in the MWI (I know, we're getting deeper and deeper down the rabbit hole here), for which (I have argued) you need proper mixtures (i.e. states that really are in a given subspace---otherwise, the measure on subspaces Gleason gives just has nothing to say).

I think the main point of mfb was that we shouldn't talk about probabilities at all in the MWI. In this case, it is meaningless to talk about the Born rule or it's derivation. If we insist to talk about probabilities we have to make additional assumptions (?) which allow us to derive the Born rule via Gleason's theorem.

I am much more interested in discussig if and how we get and verify predictions from QM using the MWI without talking about probabilities. I think it is crucial to understand this viewpoint first and then look at how it is connected to the probabilistic picture.

 

[bhobba]

Even that quote just directs me to the reference. I repeat: what is it that prohibits Alice and Bob from just writing down their results and comparing them?

Nothing - but how does that imply the proper mixed state you wrote down?

Basically your supposed proper mixed state is wrong. The proper mixed state depends on who does the observing and it only contains two terms.

Thanks
Bill

 

[stevendaryl]

Yes, that's the point I'm trying to get across: attaching an ignorance interpretation to their local states is exactly ignoring the entanglement between them, and erroneously stipulating that they can treat the system as being in a definite, but unknown, state---which is simply not the case if you have entanglement. It's exactly the error made in the argument that proper and improper mixtures can be treated the same!

I think you're wrong. Using $\rho_{AB} = \rho_A \otimes \rho_B$ is the same sort of assumption as using $P(X \wedge Y) = P(X) \times P(Y)$ to compute joint probabilities. It's correct under the assumption that $X$ and $Y$ are independent random variables, but not in general.

It's not the fact that they are "improper" mixtures that prevents you from combining density matrices that way; it's the fact that you are treating systems as independent that have a common history. Probabilities due to ignorance are NOT independent for systems that share a common history.

 

[S.Daedalus]

I think the main point of mfb was that we shouldn't talk about probabilities at all in the MWI. In this case, it is meaningless to talk about the Born rule or it's derivation. If we insist to talk about probabilities we have to make additional assumptions (?) which allow us to derive the Born rule via Gleason's theorem.

I am much more interested in discussig if and how we get and verify predictions from QM using the MWI without talking about probabilities. I think it is crucial to understand this viewpoint first and then look at how it is connected to the probabilistic picture.

I think this is what mfb meant with his 'hypothesis testing', but I'm afraid the point is lost on me---as I said, I can certainly form the hypothesis that relative frequencies are asymptotically distributed according to Born, test it, and become convinced it's right; I can also form the hypothesis that there are green apples, test it, and become convinced of it. Both merely means that the MWI is consistent with that hypothesis, but in neither case does it then follow from it, so it only posits something which is not explained within the MWI.

Nothing - but how does that imply the proper mixed state you wrote down?

The proper mixed state is implied by Alice and Bob's believe that they can attach an ignorance interpretation to their local states. The fact that they will observe perfect correlations once they compare their measurements falsifies this belief.

I think you're wrong. Using $\rho_{AB} = \rho_A \otimes \rho_B$ is the same sort of assumption as using $P(X \wedge Y) = P(X) \times P(Y)$ to compute joint probabilities. It's correct under the assumption that $X$ and $Y$ are independent random variables, but not in general.

It's not the fact that they are "improper" mixtures that prevents you from combining density matrices that way; it's the fact that you are treating systems as independent that have a common history. Probabilities due to ignorance are NOT independent for systems that share a common history.

No, it's the fact that Alice and Bob apply an ignorance interpretation to their states that makes me write down the state in that way---that's just what an ignorance interpretation means: they belief that their state is actually in either of the states $|0\rangle$ or $|1\rangle$; from this alone, it follows that they must hold the global state to be the mixture I wrote down.

I'm really not sure where the disconnect lies. To me, 'improper mixtures aren't proper mixtures' is nothing but a 40-some year old cut-and-dried textbook result, preached to every student of QM who finds out that if he traces out the measured system, what's left looks like a statistical mixture of distinct apparatus states, and then thinks to have solved the measurement problem. The argument is given in many different ways by different authors---first, perhaps, by d'Espagnat (though it was probably known earlier), or in the textbook by Hughes, and so on. Quick googling brought up this by Mittelstaedt, and many other statements like it. I don't really think there's any controversy around this issue.

 

[bhobba]

It's not the fact that they are "improper" mixtures that prevents you from combining density matrices that way; it's the fact that you are treating systems as independent that have a common history. Probabilities due to ignorance are NOT independent for systems that share a common history.

I also think he is misunderstanding what decoherence says. Whoever does the observation does the decoherence - it is at that point it becomes an improper mixed state and it is exactly the same as the proper one - mathematically that is.

Thanks
Bill

 

[stevendaryl]

No, it's the fact that Alice and Bob apply an ignorance interpretation to their states that makes me write down the state in that way.

But what you wrote is NOT correct, under the ignorance interpretation of mixed states.

As it says in the article:

$\rho_{A} = tr_A \rho_{AB}$
$\rho_{B} = tr_B \rho_{AB}$

From these two definitions, it does not follow that
$\rho_{AB} = \rho_A \otimes \rho_B$
except in special cases.

You can't, in general, combine subystem density matrices that way, regardless of whether the mixtures arose from "ignorance".

 

[Jazzdude]

Decoherence adherents, unless they are being disingenuous like the paper cited before, do not claim it solves the measurement problem. What they claim is its non issue because its observationally the same as a proper mixture and gives the appearance of wavefunction collapse.

I understand very well what you are claiming. But even that "proper and improper mixtures are observationally indistinguishable" requires the measurement postulate, or something equivalent. Because otherwise ensembles cannot be expressed in terms of density matrices.

Prove me wrong by deriving the density matrix formalism for ensembles without referring in any way to any form of the measurement postulate.

Cheers,

Jazz

 

[Jazzdude]

I never claimed that - in fact I don't even know what you mean by that.

Then please let's discuss what you don't understand. Just saying that it's not true won't get us any farther.

To be clear my claim is that decoherence solves the preferred basis problem as stated on page 113 of the reference I gave before by Schlosshauer. He gives 3 issues the measurement problem must solve:

1. The preferred basis problem
2. The problem of non observability of interference
3. The problem of why we have outcomes at all.

The statement he makes is my position:
'it is reasonable to conclude decoherence is capable of solving the first two, whereas the third problem is linked to matters of interpretation'

Yes, I know Schlosshauer's publications. And what I'm saying is that 1) is very questionable for reasons I gave earlier, 2) is a valid conclusion unless we gain a possibly more complete understanding of observation that invalidates is and 3) is absolutely out of reach of the decoherence framework because it doesn't even look at single outcomes.

But you add 4) that decoherence gives a behavioristic explanation for the world we see by stating that ensembles and reduced states are indistinguishable by experiments. That takes what we know experimentally about observation (namely that it only results in probabilities and we can therefore encode ensembles more densely by putting them in a form compatible with the measurement rule) to explain observation. That is circular.

Cheers,

Jazz

 

[bhobba]

Prove me wrong by deriving the density matrix formalism for ensembles without referring in any way to any form of the measurement postulate.

How can I prove you wrong for something I am not claiming. The claim is - it don't matter - not that the measurement postulate isn't still there.

Even with proper mixed states you have the measurement postulate - its still there - its just that it conforms to our intuition of having the value prior to observation and you don't have this collapse because its revealing what's already there. It simply means you can interpret in a more reasonable manner that APPEARS to solve the measurement problem.

Thanks
Bill

 

[Jazzdude]

How can I prove you wrong for something I am not claiming. The claim is - it don't matter - not that the measurement postulate isn't still there.
Even with proper mixed states you have the measurement postulate - its still there - its just that it conforms to our intuition of having the value prior to observation and you don't have this collapse because its revealing what's already there. It simply means you can interpret in a more reasonable manner that APPEARS to solve the measurement problem.

So you're saying, in a setting where you have decoherence AND the measurement postulate is valid, you get something that explains how a decohered system looks like a mixture, and that explains all practical aspects of measurement. Why use decoherence at all? This follows from only the measurement postulate already.

Cheers,

Jazz

 

[bhobba]

But you add 4) that decoherence gives a behavioristic explanation for the world we see by stating that ensembles and reduced states are indistinguishable by experiments.

Sigh.

I say it gives the APPEARANCE that the world behaves in way that conforms more readily to our intuition and hence gives the APPEARANCE of solving the problem.

Thanks
Bill