Question regarding the Many-Worlds interpretation

[stevendaryl]

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.

Which postulate are you calling the "measurement postulate"? There are basically four assumptions in the standard, collapse interpretation of quantum mechanics:

1. Observables correspond to Hermitian operators.
2. When you measure an observable, you get an eigenvalue of the corresponding operator.
3. The probability of getting eigenvalue o is the absolute square of the wave function projected onto the eigenstate corresponding to o.
4. Afterwards, the system is in the eigenstate corresponding to the eigenvalue obtained.

The use of density matrices certainly doesn't require assumption 4. It's not clear to me that it actually requires assumption 2, either.

 

[bhobba]

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.

I am in Australia where its 1.00 am and I stayed up to watch a big election we just had - its been decided so I will heading off to bed soon.

The reason you use decoherence is you can think the world is revealed by measurement. If it was an actual mixed state that would be the case.

I am 100% sure you know there are a number of interpretations that use decoherence in their foundations and they all do it for various reasons. I adhere to the decoherence ensemble interpretation as mentioned in the link previously. The advantage is it allows me to think it has the property prior to observation. Its a variation on the usual ensemble interpretation. The high priest of that interpretation Ballentine believes its a crock of rubbish because its not required. It purely a matter of what appeals.

Anyway off to bed.

Thanks
Bill

 

[Jazzdude]

I adhere to the decoherence ensemble interpretation as mentioned in the link previously. The advantage is it allows me to think it has the property prior to observation. Its a variation on the usual ensemble interpretation. The high priest of that interpretation Ballentine believes its a crock of rubbish because its not required. It purely a matter of what appeals.

You used the same argument in the context of an MWI discussion however. Any reference to the measurement postulate is not allowed here to draw any conclusions. That's the context I use for pointing out that your argument is not valid.

Good night,

Jazz

 

[stevendaryl]

I am in Australia where its 1.00 am and I stayed up to watch a big election we just had - its been decided so I will heading off to bed soon.

This is off-topic, but how did it go? The predictions were that Labor was going to lose big.

 

[S.Daedalus]

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".

Yes, you can't in general do that because doing so means ignoring the entanglement between the two states. But that is exactly what it means to attach an ignorance interpretation to their local states for Alice and Bob, as it means taking their respective systems to be in either the state |0\rangle or |1\rangle, while just not knowing which one applies. But if this is the correct description, there can be no entanglement.

In other words: if you're saying that Alice and Bob can validly apply an ignorance interpretation, you are saying that their combined state is in the mixture I wrote down. Consider coming at this from the other way, equiprobably creating the states |0\rangle and |1\rangle locally. Presumably, you'd then agree with me that the global state is as I wrote it down, no? But then an 'ignorance interpretation' just means that you can consider a state as being just so constituted; the two say exactly the same thing. That Alice and Bob can consider their local states to be either |0\rangle or |1\rangle is quite simply only true if the state \rho_{AB} as I have given it obtains.

 

[stevendaryl]

Yes, you can't in general do that because doing so means ignoring the entanglement between the two states.

It depends on what you mean by "entanglement". Do you consider classical cases where P(A \wedge B) \neq P(A) \times P(B) to be "entanglement"?

But that is exactly what it means to attach an ignorance interpretation to their local states for Alice and Bob, as it means taking their respective systems to be in either the state |0\rangle or |1\rangle, while just not knowing which one applies. But if this is the correct description, there can be no entanglement.

I don't agree with that. Bob believes that his subsystem is in state 0 or state 1, and he doesn't know which. Alice believes that her subsystem is in state 0 or state 1, but she doesn't know which. It doesn't follow that for the composite system, that all four states:

|00>, |01>, |10>, |11>

are possible.

 

[tom.stoer]

I think it is all written in the thread now. It would be pointless to repeat it.

perhaps it is ... hidden in 300 posts ...

 

[S.Daedalus]

I don't agree with that. Bob believes that his subsystem is in state 0 or state 1, and he doesn't know which. Alice believes that her subsystem is in state 0 or state 1, but she doesn't know which. It doesn't follow that for the composite system, that all four states:

|00>, |01>, |10>, |11>

are possible.

This is exactly what follows. Honestly, I don't think this is going to get any more productive; I'll just leave you with what Timpson says on the issue (I'm not entirely certain this link will stay valid; it's on page 253 of his book Quantum Information Theory and the Foundations of Quantum Mechanics, starting with 'When |\Psi\rangle_{12} is an entangled state...' and ending with 'Thus reduced states of an entangled system cannot be given an ignorance interpretation'). The book is from 2013, so I will simply accept it as accurately reflecting current knowledge.

 

[stevendaryl]

In other words: if you're saying that Alice and Bob can validly apply an ignorance interpretation, you are saying that their combined state is in the mixture I wrote down. Consider coming at this from the other way, equiprobably creating the states |0\rangle and |1\rangle locally. Presumably, you'd then agree with me that the global state is as I wrote it down, no? But then an 'ignorance interpretation' just means that you can consider a state as being just so constituted; the two say exactly the same thing. That Alice and Bob can consider their local states to be either |0\rangle or |1\rangle is quite simply only true if the state \rho_{AB} as I have given it obtains.

I don't agree. Let's take a classical example: I take two identical envelopes, and put $10 in one, and $20 in the other. I shuffle the envelopes and hand one to Alice and one to Bob. Then Alice would describe the state of her envelope as "It holds $10 with probability 1/2, and it holds $20, with probability 1/2". Bob would describe the state of his envelope the same way. But if they are trying to describe the composite state of the two envelopes, they can't just take the "product". In both cases, a mixed state arises from ignorance, but the two uncertainties are not independent. There is zero probability that both Alice and Bob have $20.

 

[stevendaryl]

This is exactly what follows.

That's not true about CLASSICAL probabilities, which are all due to ignorance (or can be interpreted that way).

 

[S.Daedalus]

That's not true about CLASSICAL probabilities, which are all due to ignorance (or can be interpreted that way).

Read the Timpson quote I provided: "The true state of the N-party system would then simply be the tensor product of each of these true states for subsystems, or a convex combination of these if there were further correlations between them (emphasis mine)." The part in italics takes care of the possibility of there being classical correlations present; while you're right that you could in principle thus account for the simple correlation measurements I've described, you could not violate a Bell inequality, for example, as there would still be no entanglement present.

 

[stevendaryl]

Read the Timpson quote I provided: "The true state of the N-party system would then simply be the tensor product of each of these true states for subsystems, or a convex combination of these if there were further correlations between them (emphasis mine)." The part in italics takes care of the possibility of there being classical correlations present; while you're right that you could in principle thus account for the simple correlation measurements I've described, you could not violate a Bell inequality, for example, as there would still be no entanglement present.

Definitely, the Bell inequality gives a limit to how far you can push an "ignorance" explanation.

 

[S.Daedalus]

Definitely, the Bell inequality gives a limit to how far you can push an "ignorance" explanation.

But then, that just means that there is no ignorance interpretation---or are you saying that there's one as long as you don't make Bell tests?

 

[mfb]

perhaps it is ... hidden in 300 posts ...

And you think the next 300 will be better?

But then, that just means that there is no ignorance interpretation---or are you saying that there's one as long as you don't make Bell tests?

Ignorance interpretations need something more than simple ignorance. The de-Brogle-Bohm is an example where probabilities just arise from our ignorance (of the particle positions), but it is nonlocal.

 

[Jazzdude]

Definitely, the Bell inequality gives a limit to how far you can push an "ignorance" explanation.

Bell hardly restricts ignorance, it merely shows that you cannot recover local realism using ignorance. In fact, I believe that ignorance of some sorts is what causes random measurement outcomes. And I don't mean something like Bohmian Mechanics.

Cheers,

Jazz

 

[mfb]

Bell hardly restricts ignorance, it merely shows that you cannot recover local realism using ignorance. In fact, I believe that ignorance of some sorts is what causes random measurement outcomes. And I don't mean something like Bohmian Mechanics.

Is this backed by some established interpretation, or just a feeling?
I think Bohmian mechanics is exactly the type of ignorance that could help (unlike the early ideas to explain it as ignorance in a classic theory) if you really want ignorance to explain quantum effects.

 

[lugita15]

mfb, if it's not too much trouble, could you respond to Tom's questions in post #254? I think that might clarify much of the confusion.

 

[Jazzdude]

Which postulate are you calling the "measurement postulate"? There are basically four assumptions in the standard, collapse interpretation of quantum mechanics:

1. Observables correspond to Hermitian operators.
2. When you measure an observable, you get an eigenvalue of the corresponding operator.
3. The probability of getting eigenvalue o is the absolute square of the wave function projected onto the eigenstate corresponding to o.
4. Afterwards, the system is in the eigenstate corresponding to the eigenvalue obtained.

The use of density matrices certainly doesn't require assumption 4. It's not clear to me that it actually requires assumption 2, either.

Let's see what we need to construct the density representation.

By definition, an ensemble is a list of states with the probability of finding them. Only one of the states is realized, but we do not know which one. So our fundamental ensemble is the set \{(|\psi_n\rangle,p_n):n\in\mathbb{N}\} where the |\psi_n\rangle are a family of (in general nonorthogonal) states and p_n are the associated probabilities. Now suppose that our Hilbert space is only 2 dimensional, that is we have a basis \{|0\rangle,|1\rangle\}.

A density operator in this Hilbert space is a hermitian 2x2 matrix. Clearly, this matrix encodes much less information than the (possibly very long) list of states in the ensemble. Since there are infinitely many possible realized (mutually non-orthogonal) states in this Hilbert space, there is no way of reducing the information down to a density matrix without throwing some of the information away.

The idea for reduction is that not all of that information is observable when we perform a measurement according to the measurement postulate. Let's pick an orthonormal measurement basis \{|a\rangle,|b\rangle\} and perform a measurement. Depending on which of the ensemble states is realized, we get an outcome |a\rangle with the measurement probability \langle\psi_n|a\rangle\langle a|\psi_n\rangle and the corresponding expression for |b\rangle. Since all single ensemble states are mapped to the same two measurement outcomes, we can collect the results and multiply both probabilities, the measurement outcome probability and the ensemble state probability. That means we have |a\rangle with probability \sum_n p_n \langle\psi_n|a\rangle\langle a|\psi_n\rangle and |b\rangle with probability \sum_n p_n \langle\psi_n|b\rangle\langle b|\psi_n\rangle. Both probabilities sum to 1 if the states are normalized and the ensemble proabilities sum to 1. That means \sum_n p_n \langle\psi_n|a\rangle\langle a|\psi_n\rangle+\sum_n p_n \langle\psi_n|b\rangle\langle b|\psi_n\rangle=1. We can rewrite this as \mathrm{tr}_{a,b}\sum_n|\psi_n\rangle p_n \langle\psi_n| = 1, which works in any basis, not just the indicated measurement basis. Let's call the matrix we trace over \rho. We can also read off the probability expressions, that for a projector P onto any subspace the associated probability of finding a measurement outcome in this subspace is \mathrm{tr}(\rho P). It's straight forward to find the expectation value of any operator A from here to be \langle A \rangle = \mathrm{tr}(\rho A). Other similar results follow.

So we have seen, that \rho encodes all possible measurement outcomes in any possible measurement basis, with potentially a lot less required information than encoded in the original ensemble description. But we had to use assumptions from the measurement postulate for this. Explicitly in your labeling I used 1) when I assumed that the measurement basis was orthogonal and again for the expectation value, where I also used 2). The remaining 3) and 4) were used in the measurement process of the single ensemble states.

Now you may know of a way to derive the same result without making these references. I would be curious to learn how. But unless such a way exists, the use of a density matrix requires the measurement postulate.

Cheers,

Jazz

 

[Jazzdude]

Is this backed by some established interpretation, or just a feeling?
I think Bohmian mechanics is exactly the type of ignorance that could help (unlike the early ideas to explain it as ignorance in a classic theory) if you really want ignorance to explain quantum effects.

Just to make sure I am understood correctly, I don't think ignorance is the way to any "classical" explanation of quantum theory. But I think we can at least get rid of the randomness. My statement is also founded on much more than a feeling. I am specifically referring to http://arxiv.org/abs/1205.0293, where the ignorance lies in the parts of the universe any kind of observer cannot interact with because of the locality of interactions.

Cheers,

Jazz

 

[mfb]

@lugita15: sure.

1) MWI is talking about branches and relies on decoherence to identify them, but is not able to count them or to derive a corresponding measure

It is a pointless attempt to count them. It is as meaningful as (correctly!) counting "I will win in the lottery XOR I will not win in the lottery" as 2 different results. What does that number of 2 tell us?
It has been shown that there is just one consistent, context-independent measure. What else do you want for a derivation?

2) My simple question regarding the "probability being in a certain branch" which I can identify via a result string seems to become meaningless

Without probabilities, there are no probabilities, indeed.

3) I still have the feeling that my concerns regarding the "missing link" between the experimentally inaccesable top-down perspective of the full Hilbert space with all its branches and the accessable bottom-up approach restricted to the branch I am observing right now have not been understood

I think that is right.

4) We have the above mentioned statistical frequencies, but I learn that MWI does not provide the corresponding probabilities - that there are no probabilities at all

MWI does not need probabilities.

5) It is often claimed that the Born rule can derived, but what does it mean if there are no probabilities?

There is no need for the Born rule.
If you want to add something like a "probability based on ignorance" (the interpretation itself does not need this at all), Gleason's theorem tells you you have no other choice than the Born rule.

 

[stevendaryl]

But then, that just means that there is no ignorance interpretation---or are you saying that there's one as long as you don't make Bell tests?

I think we're arguing at cross-purposes here. According to MWI, all mixed states are "improper" in the sense that they AREN'T simply due to ignorance. According to a "collapse" interpretation, if a wave function has already collapsed, but you haven't checked to see what state it collapsed to, then the system is in a "proper" mixed state. There is no practical way to decide between these two interpretations. The ignorance interpretation is ALWAYS wrong, according to MWI.

 

[stevendaryl]

MWI does not need probabilities.

There is no need for the Born rule.

I'm not sure what you mean by "need" here. The fact is that we find empirically that the Born rule accurately predicts relative frequencies for repeated, independent measurements. This is an empirical fact that I would think requires explanation. If you don't have the Born rule, then what kind of empirical support can there be for quantum mechanics? Well, I guess there is some: the eigenvalues for observables, maybe.

 

[mfb]

I'm not sure what you mean by "need" here. The fact is that we find empirically that the Born rule accurately predicts relative frequencies for repeated, independent measurements. This is an empirical fact that I would think requires explanation. If you don't have the Born rule, then what kind of empirical support can there be for quantum mechanics? Well, I guess there is some: the eigenvalues for observables, maybe.

You can do hypothesis testing, and the "right" hypotheses survive everywhere where the measurement results look like they follow the Born rule. Apparently we are in a branch where this was true in the past (within some variation).
Asking "why" is as meaningful as asking why we are on the 3rd planet around a specific main-sequence star and not on some other habitable planet, or asking why you are stevendaryl and not someone else.

 

[stevendaryl]

Bell hardly restricts ignorance, it merely shows that you cannot recover local realism using ignorance. In fact, I believe that ignorance of some sorts is what causes random measurement outcomes. And I don't mean something like Bohmian Mechanics.

Cheers,

Jazz

Yes, you're right.

 

[stevendaryl]

You can do hypothesis testing, and the "right" hypotheses survive everywhere where the measurement results look like they follow the Born rule. Apparently we are in a branch where this was true in the past (within some variation).
Asking "why" is as meaningful as asking why we are on the 3rd planet around a specific main-sequence star and not on some other habitable planet, or asking why you are stevendaryl and not someone else.

That's pretty unsatisfying. We could just as well throw out all of quantum mechanics, and say that every measurement returns some real number--which one is completely undetermined. We just happen to live in a possible world in which those real numbers seem to be predicted by the Born rule.

 

[mfb]

That's pretty unsatisfying. We could just as well throw out all of quantum mechanics, and say that every measurement returns some real number--which one is completely undetermined.

No, that would be completely different.

We just happen to live in a possible world in which those real numbers seem to be predicted by the Born rule.

We just happen to be in a likely world in probabilistic interpretations. That is exactly the same statement, as I don't care about branches with a small measure. They are just not interesting.

 

[stevendaryl]

No, that would be completely different.
We just happen to be in a likely world in probabilistic interpretations. That is exactly the same statement, as I don't care about branches with a small measure. They are just not interesting.

They are interesting if we happen to find ourselves in one of them.

 

[lugita15]

You can do hypothesis testing, and the "right" hypotheses survive everywhere where the measurement results look like they follow the Born rule. Apparently we are in a branch where this was true in the past (within some variation).

But just because it was true (with some variation) in the past, what causes it to continue to be likely to be true (with some variation) in the future? Or do you think we aren't justified in our confidence that the Born rule will continue to hold (with some variation) in the future?

 

[mfb]

They are interesting if we happen to find ourselves in one of them.

What does "if" mean? We do not.

But just because it was true (with some variation) in the past, what causes it to continue to be likely to be true (with some variation) in the future?

See my post about hypothesis testing.

 

[Quantumental]

mfb: are you claiming that you don't actually think that the multiverse in mwi obeys Born Rule, but that we happen to be on a branch where our past seems to confirm the Born Rule, but in reality this is just a illusion?

 

[stevendaryl]

What does "if" mean? We do not.

What do you mean, what does "if" mean? I didn't make up that word.

See my post about hypothesis testing.[/QUOTE]

I don't find what you said very satisfying. You claim we don't need probabilities, because the set of worlds where relative frequencies approach the Born predictions has measure 1 (or 1-ε). I don't see a big difference between using probability and using measure.

 

[mfb]

mfb: are you claiming that you don't actually think that the multiverse in mwi obeys Born Rule, but that we happen to be on a branch where our past seems to confirm the Born Rule, but in reality this is just a illusion?

What is an illusion?
We are in a branch where repeated experiments in the past gave results close to [the Born rule for probabilistic interpretations]. This is not an illusion, this is real.

What do you mean with "multiverse in MWI obeys Born rule"? How would a universe look like that does, and one that does not?

What do you mean, what does "if" mean? I didn't make up that word.

Sure, but you asked an if-question about something that is not true. "What happens if you stop beating your wife?"

 

[bhobba]

This is off-topic, but how did it go? The predictions were that Labor was going to lose big.

Yea - way off topic. They lost but not by as big an amount as everyone thought - was about halfway - the optimists thought about a 20 seat majority, the pessimists about a 40 or more majority and the Labor government decimated - it was about 30. If it was only something like 20 or less (ie 10 or less seats decided the outcome) Kevin may have been able to remain as leader of the opposition, but 30 was just too many so he is now just a lowly MP.

Still a bit tired - will have a read of the thread activity and do a post when I have digested it.

Thanks
Bill

 

[lugita15]

See my post about hypothesis testing.

How does that post address my past and future question? What is the reason that we should be confident that relative frequencies close to the Born rule will likely continue to hold in the future? Or should we not be confident about that?

 

[tom.stoer]

mfb, thanks for the answers in

1) MWI is talking about branches and relies on decoherence to identify them, but is not able to count them or to derive a corresponding measure

It is a pointless attempt to count them. It is as meaningful as (correctly!) counting "I will win in the lottery XOR I will not win in the lottery" as 2 different results. What does that number of 2 tell us?
It has been shown that there is just one consistent, context-independent measure. What else do you want for a derivation?

2) My simple question regarding the "probability being in a certain branch" which I can identify via a result string seems to become meaningless

Without probabilities, there are no probabilities, indeed.

3) I still have the feeling that my concerns regarding the "missing link" between the experimentally inaccesable top-down perspective of the full Hilbert space with all its branches and the accessable bottom-up approach restricted to the branch I am observing right now have not been understood

I think that is right.

4) We have the above mentioned statistical frequencies, but I learn that MWI does not provide the corresponding probabilities - that there are no probabilities at all

MWI does not need probabilities.

5) It is often claimed that the Born rule can derived, but what does it mean if there are no probabilities?

There is no need for the Born rule.
If you want to add something like a "probability based on ignorance" (the interpretation itself does not need this at all), Gleason's theorem tells you you have no other choice than the Born rule.

Summarizing your statements I get that there are no probabilities and therefore there is no Born rule in MWI.

But the way you get there is not satisfying for me. Regarding (4) and (5) you say that MWI does neither require probabilities nor Born's rule, but for me it's the other way round: It's not up to an interpretation to decide what is required or not, it's our empirical knowledge about nature which requires an explanation in terms of a formalism and an interpretation. If MWI does not provide this it is incomplete in terms of its explanatory capabilities and therefore no viable interpretative system.

Empricically we have statistical frequencies - therefore they have to be predicted by the formalism and therefore a corresponding postulate or theorem is required. Empirically we find that Born's rule does exactly this, and we know that it's the only valid probability measure on a Hilbert space (Gleason) - therefore these facts require an explanation or interpretation.

If MWI is not able to or not willing to interpret the meaning of Born's rule or Gleason's theorem, then MWI is incomplete in the sense that there are facts (experimental results, Gleason's theorem etc.) and there is some knowledge (these facts, the formalism and its successful applicability) which MWI does not explain.

Repeating myself: If MWI does not provide this it is incomplete in terms of its explanatory capabilities.

Regarding (3) you say that I am probably right. For me this would be the deepest concern simply b/c this is all what MWI is about
A) "copies of observers" with "one observer existing within a certain branch" and having its own bottom-up perspective w/o access to other branches
B) a formalism using the full Hilbert space i.e. a top-down perspective
So if the MWI cannot provide the link between its own notions and the formalism then this interpretation isn't getting us anywhere.

Thanks again for your time and your response - and sorry if this post sounds rather harsh - but unfortunately it seems that I still do not get it.

EDIT: It seems that your response is in-line with others, like Tegmark's "many words"; it basically says that MWI solves some interpretational problems and is internally consistent - provided that you stop asking certain questions being ill-posed w.r.t. to the "MWI paradigm"; for me it seems as if MWI is partially self-immunizing against any critique not compliant with the MWI paradigm or mindset; this seems to be one reason why so many circularity issues are raised against MWI.

 

[S.Daedalus]

I think we're arguing at cross-purposes here. According to MWI, all mixed states are "improper" in the sense that they AREN'T simply due to ignorance. According to a "collapse" interpretation, if a wave function has already collapsed, but you haven't checked to see what state it collapsed to, then the system is in a "proper" mixed state. There is no practical way to decide between these two interpretations. The ignorance interpretation is ALWAYS wrong, according to MWI.

Yes, I believe I said in my response above that I consider this to be a valid viewpoint. But it doesn't help you in getting the Born rule to work in the MWI via Gleason's theorem, since it only gives you a measure on subspaces, which only applies if you have a state that is in a subspace (one of the eigenspaces of the observable you're measuring), which it however in general won't be. The way to ensure that in collapse interpretations, you have a state in an appropriate subspace, and simply don't know which, is the collapse postulate, which makes thus the Gleason measure appropriate. But without it, and thus, especially in the case that all mixtures are improper, the theorem simply does no work at all.

As for what Jazzdude said regarding Bell inequalities, it may well be that there is something we are ignorant about that leads to the probabilities in quantum mechanics (although the PBR theorem seems to me to put very strict restrictions on that sort of thing), but I don't think this can be in the literal sense of an ignorance interpretation of an improper mixture; after all, applying this interpretation simply would lead to empirically wrong predictions. (Even disregarding Bell tests, you can do complete tomography using only local measurements, and reconstruct the full state, which will not in general correspond to a convex combination of product states---i.e. will in general be incompatible with an ignorance interpretation.)

You can do hypothesis testing, and the "right" hypotheses survive everywhere where the measurement results look like they follow the Born rule. Apparently we are in a branch where this was true in the past (within some variation).
Asking "why" is as meaningful as asking why we are on the 3rd planet around a specific main-sequence star and not on some other habitable planet, or asking why you are stevendaryl and not someone else.

I think this misses the point on why we do interpretation in general. Tom.stoer puts the finger on the issue:

But the way you get there is not satisfying for me. Regarding (4) and (5) you say that MWI does neither require probabilities nor Born's rule, but for me it's the other way round: It's not up to an interpretation to decide what is required or not, it's our empirical knowledge about nature which requires an explanation in terms of a formalism and an interpretation. If MWI does not provide this it is incomplete in terms of its explanatory capabilities and therefore no viable interpretative system.

Ultimately, we have empirical data, from which we build a mathematical formalism, and look to an interpretation to make sense of that formalism. If the interpretation can't do that---which seems to be what you're claiming: we observe that the Born rule holds, but in the MWI, it does so for no reason---, then it's just not a viable interpretation. There's a part of the formalism that simply finds no explanation. It's not analogous to contingently finding oneself on the third rock from the sun, but rather, to having a theory of stellar evolution that can't account for stellar fusion (except for adding the hypothesis 'stars undergo fusion' to the rest of the theory).

There's a difference between necessary and contingent features: being on this particular planet is contingent; that stars undergo fusion is not. Likewise, the Born rule does not seem to be contingent in quantum mechanics; and if you assume it to be so, then there ceases to be a reason to expect that it continues to hold.

 

[bhobba]

Yes, I believe I said in my response above that I consider this to be a valid viewpoint. But it doesn't help you in getting the Born rule to work in the MWI via Gleason's theorem, since it only gives you a measure on subspaces, which only applies if you have a state that is in a subspace (one of the eigenspaces of the observable you're measuring), which it however in general won't be.

I am not sure exactly what you mean, so my comment may be off the mark. But Gleason's theorem applies to any state, indeed it defines what a state is. It shows the only measure (with value 0 to 1) that can be defined on a Hilbert space that is basis independent is Tr(P|u><u|) where u is an element of the Hilbert space and P is a positive operator of unit trace. By definition P is the state of the system. A mixed state, improper or otherwise, is a positive operator of unit trace.

For the details check out:
http://kof.physto.se/theses/helena-master.pdf

Thanks
Bill

 

[mfb]

How does that post address my past and future question? What is the reason that we should be confident that relative frequencies close to the Born rule will likely continue to hold in the future? Or should we not be confident about that?

There is no "likely".
I wanted to stop repeating that several posts ago :(.

But the way you get there is not satisfying for me. Regarding (4) and (5) you say that MWI does neither require probabilities nor Born's rule, but for me it's the other way round: It's not up to an interpretation to decide what is required or not, it's our empirical knowledge about nature which requires an explanation in terms of a formalism and an interpretation. If MWI does not provide this it is incomplete in terms of its explanatory capabilities and therefore no viable interpretative system.

I don't see what would be missing here for MWI.
Do you think probabilistic interpretations are missing an explanation why we are not in the most probable world (more likely than ours by way more than billions orders of magnitude), but in a world that would have passed hypothesis tests in the past? If not, where is the difference?

Regarding (3) you say that I am probably right. For me this would be the deepest concern simply b/c this is all what MWI is about
A) "copies of observers" with "one observer existing within a certain branch" and having its own bottom-up perspective w/o access to other branches
B) a formalism using the full Hilbert space i.e. a top-down perspective
So if the MWI cannot provide the link between its own notions and the formalism then this interpretation isn't getting us anywhere.

You are probably right that your concerns are not understood. And as I don't understand them, I cannot even try to find out if those concerns are serious or not.

Ultimately, we have empirical data, from which we build a mathematical formalism, and look to an interpretation to make sense of that formalism. If the interpretation can't do that---which seems to be what you're claiming: we observe that the Born rule holds, but in the MWI, it does so for no reason

It does for the same reason as we were so "lucky" to see the Born rule in probabilistic interpretations.

 

[S.Daedalus]

I am not sure exactly what you mean, so my comment may be off the mark. But Gleason's theorem applies to any state, indeed it defines what a state is. It shows the only measure (with value 0 to 1) that can be defined on a Hilbert space that is basis independent is Tr(P|u><u|) where u is an element of the Hilbert space and P is a positive operator of unit trace. By definition P is the state of the system. A mixed state, improper or otherwise, is a positive operator of unit trace.

I agree with all of that, however, I still see the issue as I have laid out in this post. If you have a measure, say on the subsets of some set, then this can be used to give a probability to drawing something out of one of these subsets. Analogously, if you have a measure on the subspaces of a Hilbert space, then Gleason's theorem gives you the measure on that, and thus, if the state is in one of those subspaces, the probability that it is in a particular one. However, if you measure an observable \mathcal{O}, then in general it won't be the case that the state will be in one of the subspaces defined by the projectors onto the eigenstates of \mathcal{O}; rather, it will typically be superposed.

In the classical analogy, this would correspond to the case where something simply does not belong to one of the subsets you have a measure on in the first place, and thus, the measure tells you nothing about probability (in the example I gave, it's a marble that's neither red, green, blue, nor pink). The same is---or that's how it seems to me anyway---also true in the quantum case: the superposed state is not in one of the subspaces in which the system has some definite value for \mathcal{O}. But the measure given by Gleason is relevant only there. So the collapse theory proposes that upon measurement, the state jumps into one of the required subspaces; then, Gleason's theorem becomes applicable, and gives you the Born probabilities. The analogous process is missing in the MWI, since the state stays in superposition.

So Gleason is perfectly valid, it just talks about things that have no bearing on the situation. Measuring \mathcal{O}, a generic state may be expanded as |\psi\rangle=\sum_i\mu_i|o_i\rangle, where the |o_i\rangle are the eigenstates of \mathcal{O} to eigenvalues o_i. Gleason tells you that the measure of the subspace associated with the projector |o_i\rangle\langle o_i| is given by \mathrm{Tr}(|\psi\rangle \langle\psi|o_i\rangle \langle o_i|)=|\mu_i|^2. But the state |\psi\rangle is not in either of these subspaces; that would only be the case if it were reducible to the proper mixture \rho=\sum_i|\mu_i|^2|o_i\rangle\langle o_i|. So, Gleason gives you a measure on subspaces, which is of course only applicable to states in those subspaces.

For the details check out:
http://kof.physto.se/theses/helena-master.pdf

Thanks for that, I'd been looking for a reference that collects these things.

 

[tom.stoer]

I don't see what would be missing here for MWI.
Do you think probabilistic interpretations are missing an explanation why we are not in the most probable world (more likely than ours by way more than billions orders of magnitude), but in a world that would have passed hypothesis tests in the past? If not, where is the difference?

Sorry to say that, but this is exactly what I mean with

It seems that your response is in-line with others, like Tegmark's "many words"; it basically says that MWI solves some interpretational problems and is internally consistent - provided that you stop asking certain questions being ill-posed w.r.t. to the "MWI paradigm"; for me it seems as if MWI is partially self-immunizing against any critique not compliant with the MWI paradigm or mindset; this seems to be one reason why so many circularity issues are raised against MWI.

[you] don't see what would be missing here for MWI

, but this is not the point (neither are probabilistic interpretations). You (or MWI) decided not to see these issues (which are based on experimental results, not on interpretations) so you're conclusion is that there are no such issues in the context of MWI. That's circular reasoning.

There are facts (experimentally observed statistical frequencies, Gleason's theorem, success of Born's rule, ...) which MWI denies to interpret b/c they do not fit into the MWI.

You are probably right that your concerns are not understood. And as I don't understand them, I cannot even try to find out if those concerns are serious or not.

What is unclear in

this is all what MWI is about:
A) "copies of observers" with "one observer existing within a certain branch" and having its own bottom-up perspective w/o access to other branches
B) a formalism using the full Hilbert space i.e. a top-down perspective
So if the MWI cannot provide the link between its own notions and the formalism then this interpretation isn't getting us anywhere.

Many others do see and understand my point (3). You seem to miss it b/c it does not exist in the MWI context.

 

[S.Daedalus]

It does for the same reason as we were so "lucky" to see the Born rule in probabilistic interpretations.

There's no luck about it. Consider asking the question: "Why do we observe probabilities according to the Born rule?" to a) a collapse theorist, b) a many worldesian. On your view, the answers would be:

a) "The collapse ensures that the state is in some subspace defined by the observable we measure having a particular value; then, Gleason's theorem ensures us that the only possible probabilities are given by the Born rule."

b) "Stuff just happens that way."

So there's a clear difference here, I think. Furthermore, rationally, the evidence could never compel you to form a belief in the many worlds idea: as all we ever observe are relative frequencies, and relative frequencies are not predicted by the MWI, there is no actual evidence for the theory. The theory, without giving an account of the relative frequencies we observe, is simply empirically inadequate.

 

[S.Daedalus]

That is not a decision. I think this issue does not exist independent of the interpretation.
Can you explain why we live in a planet orbiting our sun, and not another star? If not, is this an issue?

Yes: that we live on this planet is a contingent feature of our theories; that the quantum probabilities are Born-distributed is either necessary, in which case the MWI must account for it, or contingent, in which case the theory isn't predictive, as there would be no reason to expect a Born distribution in the future. In either case, it's not a theory that should be accepted.

ETA: That post disappeared somewhere along the way...

 

[mfb]

I deleted my own post as it is really pointless to repeat the same arguments again. I try to avoid that. To answer your posts, new posts would not be better than my previous posts are.

 

[stevendaryl]

There's no luck about it. Consider asking the question: "Why do we observe probabilities according to the Born rule?" to a) a collapse theorist, b) a many worldesian. On your view, the answers would be:

a) "The collapse ensures that the state is in some subspace defined by the observable we measure having a particular value; then, Gleason's theorem ensures us that the only possible probabilities are given by the Born rule."

But we don't observe probabilities, we observe relative frequencies. They don't have to be the same as probabilities; it's possible to flip a coin 100 times and get 100 heads.

 

[S.Daedalus]

I deleted my own post as it is really pointless to repeat the same arguments again. I try to avoid that. To answer your posts, new posts would not be better than my previous posts are.

Well, you could try to engage with the arguments brought forth against your position, rather than just restating it.

But we don't observe probabilities, we observe relative frequencies. They don't have to be the same as probabilities; it's possible to flip a coin 100 times and get 100 heads.

Yes, that's a problem in the philosophy of probability. But as I said, the problem of the many worlds interpretation is that it doesn't even get that far.

 

[mfb]

Well, you could try to engage with the arguments brought forth against your position, rather than just restating it.

That is what I did.

 

[tom.stoer]

mfb, sorry to say that, but this is not fair. The are a couple of interested people here trying to understand (and challange ;-) your arguments, but in many cases we have to learn that "it is not required", "it is pointless", "it has been numerous times", ...

 

[mfb]

I just don't think new posts would add anything new, or make anything better. We need someone who can explain that better than me.

"is not required" comes from the attempts to apply Copenhagen to MWI. It is like asking "where are the additional branches in Copenhagen? Without them, I cannot see how Copenhagen could work!". What is the correct reply? "Copenhagen does not have or need multiple branches, it is pointless to ask how they enter the interpretation."
And the tenth time this question is asked would certainly annoy someone.

 

[S.Daedalus]

That is what I did.

Well, there's one question I haven't seen you answer, and would very much like to hear your views about: Why should one believe in the MWI, if it fails to predict the observed relative frequencies, but those frequencies are ultimately all the experimental evidence we have?

All I've heard you say in that direction is something about 'hypothesis testing'. And yes: you can form, test, and validate the hypothesis that the relative frequencies are Born-distributed. But if you do so, it's wholly independent of the MWI: it neither implies nor contradicts this hypothesis. So on these grounds, you foster belief in the hypothesis that the relative frequencies are Born-distributed, but your belief regarding the MWI is not affected at all. But then, the MWI would be wholly independent of observation.

This is compounded by the fact that the MWI was introduced in order to resolve the difficulties the standard way of explaining the relative frequencies offers, i.e. the apparent contradictions involved in the collapse. If the then resulting theory fails to lead any account at all, I just can't see how it's an improvement.

 

[S.Daedalus]

I just don't think new posts would add anything new, or make anything better. We need someone who can explain that better than me.

"is not required" comes from the attempts to apply Copenhagen to MWI. It is like asking "where are the additional branches in Copenhagen? Without them, I cannot see how Copenhagen could work!". What is the correct reply? "Copenhagen does not have or need multiple branches, it is pointless to ask how they enter the interpretation."
And the tenth time this question is asked would certainly annoy someone.

But 'where are the branches' has a clear-cut answer in Copenhagen: the collapse gets rid of them.