In what sense does MWI fail to predict the Born Rule?

[akvadrako]

I hope this answers your question.

It's clear enough now; thanks for the clarifications.

 

[bhobba]

The usual criticism is that for actual finite measurements the coefficients in front of maverick worlds

I have never even heard of them. But did a search and sure Everett talks about them. I got a paper on it:
https://arxiv.org/pdf/1511.08881.pdf

I simply do not get the issue - it says:
Everett defined maverick branches of the state vector as those on which the usual Born probability rule fails to hold – these branches exhibit highly improbable behaviors, including possibly the breakdown of decoherence or even the absence of an emergent semi-classical reality.

I am still in the dark. A world happens after decoherence - how can it loose decoherence - beats me. I think it's to do with his view on the proof of the Born Rule - but Wallace is a more modern approach that doesn't seem to have this issue - if an issue it is.

Zurek and Wallace do not, they have a different use of the Born rule as you say. This is why they attempt to derive it.

That's probably the answer - I have only really studied Wallace.

Thanks
Bill

 

[DarMM]

I am still in the dark. A world happens after decoherence - how can it loose decoherence - beats me. I think it's to do with his view on the proof of the Born Rule - but Wallace is a more modern approach that doesn't seem to have this issue - if an issue it is.

I think what might be meant is stability, that some of the "worlds" will interfere with each other, so there is no long lived classical physics. Although I'd say something like those branches simply aren't worlds, as you mentioned. Everett permitted any arbitrary partitioning of the Hilbert Space, so he allows basis where decoherence doesn't occur.

Maverick Worlds in modern MWI are the stable quasi-classical branches (i.e. decohered branches) along whose history experimental frequencies don't hold to the Born rule, non-decoherent basis aren't considered worlds. Hsu is sticking to Everett's original use of "world", which is in essence, any 1D subspace.

That's probably the answer - I have only really studied Wallace.

Wallace assumes the existence of (highly) stable quasi-classical branches to begin with and hence there are no "coherent worlds". His MWI does have Maverick worlds in the modern sense, but it is always more rational to act is if your world will stay or become non-Maverick.

Zurek similarly already has quasi-classical branches as well and there are maverick worlds but, as his method reduces to a form of branch counting, there is simply less of them.

 

[DarMM]

If anybody wants the short version of either proof, the idea as such.

Zurek is basically saying, if quantum states are already associated with probabilities in some way and the world post-measurement is known to evolve into a decoherent form to near perfect accuracy, then environmental noncontextuality (environment has no affect on probabilities) allows us to demonstrate that terms with equal magnitude coefficients are equiprobable. This provides us all we need to prove Born's rule in general via branch counting, as the general case can always be reduced to the equal magnitude case by using the environment to split branches with larger coefficients into multiple branches with equal coefficients. That is in $\sqrt{\frac{1}{3}}|\uparrow\rangle + \sqrt{\frac{2}{3}}|\downarrow\rangle$ the environment can be used to show that there are two down worlds.

Wallace says, if we live in a multiverse with highly decohered parallel branches to begin with and agents in those branches have a certain control over the environment (erasure axiom), then under a certain definition of rationality*, it is always more rational to act as if the Born Rule were true.

*Rationality here has been criticised, but essentially the agent doesn't care about superpositions or branching in and of themselves, i.e. two worlds where you won the lottery are as valuable as one world where you did. Also the agent values more experiments whose average payout among his post-measurement selves is highest. Agents who value things like "best highest payout world" or "best worst outcome world" are excluded from the definition of rational.

 

[stevendaryl]

Wallace assumes the existence of (highly) stable quasi-classical branches to begin with and hence there are no "coherent worlds". His MWI does have Maverick worlds in the modern sense, but it is always more rational to act is if your world will stay or become non-Maverick.

I don't see the latter. If you imagine a world in which relative frequencies for events turn out to not be those predicted by QM, then in that world, nobody would have the slightest reason to develop QM in the first place. So rational agents certainly would not have reason to think the world would soon become non-Maverick.

 

[DarMM]

Yes, I should point out, the proof implicitly assumes they know QM in full, i.e. they are aware of their Maverick status.

 

[Jehannum]

In the Transactional Interpretation the Born Rule arises quite naturally.

 

[Derek P]

In the Transactional Interpretation the Born Rule arises quite naturally.

If you can call messages from the future "natural". But anyway, the ingredients for calculating a density matrix may all be there but the mechanism for selecting one interaction is, correct me if I'm wrong, a random choice "agreed" by the two participants. So it's a bit irrelevant to MWI, which is deterministic and does not have any such choice. Is that fair?

 

[Jehannum]

If you can call messages from the future "natural". But anyway, the ingredients for calculating a density matrix may all be there but the mechanism for selecting one interaction is, correct me if I'm wrong, a random choice "agreed" by the two participants. So it's a bit irrelevant to MWI, which is deterministic and does not have any such choice. Is that fair?

Yes, it's fair. I was actually replying to this much earlier post, which I should have quoted:

In what sense does any approach to QM derive the Born Rule? - as opposed to taking it as an assumption.

Perhaps that's a good topic for another thread.

And yes, I suppose the advanced waves do take a little getting used to.

 

[Derek P]

Zurek is basically saying, if quantum states are already associated with probabilities in some way and the world post-measurement is known to evolve into a decoherent form to near perfect accuracy, then environmental noncontextuality (environment has no affect on probabilities) allows us to demonstrate that terms with equal magnitude coefficients are equiprobable. This provides us all we need to prove Born's rule in general via branch counting, as the general case can always be reduced to the equal magnitude case by using the environment to split branches with larger coefficients into multiple branches with equal coefficients.

Okay. So we need a set of axioms to ensure environmental noncontextuality. I need to go over what you've posted but what's the short version of how Zurek does this? I am thinking that it might depend on some assumptions about how particles interact - locally, reversibly etc - to prove the existence of an ensemble from which the actual environment provides an unpredictable sample?

 

[Derek P]

And yes, I suppose the advanced waves do take a little getting used to.

Backwards causality always does!

 

[Derek P]

Yes, I should point out, the proof implicitly assumes they know QM in full, i.e. they are aware of their Maverick status.

Does it? How can the best strategy depend on the agent's knowledge? I thought the best strategy was to use the numbers that we know are given by the Born Rule whether it seems to fit with the agent's experience or not and whether the agent knows how to calculate them or not. They might not even know what the numbers are if, when the Mavericity has abated, they pick a strategy "at random". The best ones after would still be the ones that reflected the Born Rule even if it was a mystery to the agent as to why picking a strategy out of a hat has proved so successful. Or am I missing the point completely?

 

[Michael Price]

If anybody wants the short version of either proof, the idea as such.

Zurek is basically saying, if quantum states are already associated with probabilities in some way and the world post-measurement is known to evolve into a decoherent form to near perfect accuracy, then environmental noncontextuality (environment has no affect on probabilities) allows us to demonstrate that terms with equal magnitude coefficients are equiprobable. This provides us all we need to prove Born's rule in general via branch counting, as the general case can always be reduced to the equal magnitude case by using the environment to split branches with larger coefficients into multiple branches with equal coefficients. That is in $\sqrt{\frac{1}{3}}|\uparrow\rangle + \sqrt{\frac{2}{3}}|\downarrow\rangle$ the environment can be used to show that there are two down worlds.

...

Can you provide a link or ref to Zurek's paper where he does this? It sounds like the right approach to me, and merits further detailed examination.

 

[ftr]

I have indicated about this issue before, but this time I will ask a direct question.

In standard QM Born rule follows on axiom of the wavefunction( squaring), so shouldn't the derivation be for Schrodinger equation first.

Secondly, in physics we have equations then we interpret its part as corresponding to some elements in reality( which themselves are interpretations of measurement), so the interpretation of QM seems dubious. It is like people in Faraday's days spending all their time interpreting the ontology of the lines of force, while the correct way would have been to find the "more correct" equations/relations that correspond to reality, shouldn't it.

 

[Derek P]

I have indicated about this issue before, but this time I will ask a direct question.

In standard QM Born rule follows on axiom of the wavefunction( squaring), so shouldn't the derivation be for Schrodinger equation first.

Secondly, in physics we have equations then we interpret its part as corresponding to some elements in reality( which themselves are interpretations of measurement), so the interpretation of QM seems dubious. It is like people in Faraday's days spending all their time interpreting the ontology of the lines of force, while the correct way would have been to find the "more correct" equations/relations that correspond to reality, shouldn't it.

Well, I put this in an A level thread with a request (largely respected) to keep the maths in check. So the vector state and Hilbert space formalism can be taken for granted. Indeed, they have not been queried. And in MWI they are a "given". Zurek's proof of the Born Rule is simple once he establishes that the Schmidt decomposition is degenerate. The difficulty is in establishing axioms to prove that the environment does have the necessary characteristics. It seems clear to me that any kind of proof is going to involve the way individual interactions within the environment occur. Which is an eye-opener.

 

[Michael Price]

Well, I put this in an A level thread with a request (largely respected) to keep the maths in check. So the vector state and Hilbert space formalism can be taken for granted. Indeed, they have not been queried. And in MWI they are a "given". Zurek's proof of the Born Rule is simple once he establishes that the Schmidt decomposition is degenerate. The difficulty is in establishing axioms to prove that the environment does have the necessary characteristics. It seems clear to me that any kind of proof is going to involve the way individual interactions within the environment occur. Which is an eye-opener.

I believe Zurek's derivation will require the following property of the environment (=rest of the universe), namely that you can decompose it into an orthonormal basis dim(N*N) :
$|env\rangle=\frac{1}{N} Σ ^{N^2}_{i=1}|env^N_i\rangle$
which is why I'd like the explicit ref so I can check.

 

[A. Neumaier]

I believe Zurek's derivation will require the following property of the environment (=rest of the universe), namely that you can decompose it into an orthonormal basis dim(N*N) :
$|env\rangle=\frac{1}{N} Σ ^{N^2}_{i=1}|env^N_i\rangle$
which is why I'd like the explicit ref so I can check.

The environment contains particles, hence is necessarily represented by an infinite dimensional Hilbert space.

 

[Derek P]

The environment contains particles, hence is necessarily represented by an infinite dimensional Hilbert space.

I am sure there are some standard theorems, not invented specially for proving the Born Rule, which say whether and when continuous variables can be represented to any desired accuracy by discrete values. It is a very plausible conjecture given the linearity of QM, at least to my non-mathematical mind. So a finite dimensional Hilbert space should be just fine unless the state space representation introduces unnecessary restrictions.

 

[Derek P]

I believe Zurek's derivation will require the following property of the environment (=rest of the universe), namely that you can decompose it into an orthonormal basis dim(N*N) :
$|env\rangle=\frac{1}{N} Σ ^{N^2}_{i=1}|env^N_i\rangle$
which is why I'd like the explicit ref so I can check.

I'm being a bit dense but where does the N*N come from?

 

[Michael Price]

The environment contains particles, hence is necessarily represented by an infinite dimensional Hilbert space.

The Fock space is infinite dimensional, although the Bekenstein bound would cap that dimensionality, according to the entropy. But these are red herrings, as far as In understand it.

 

[Michael Price]

I'm being a bit dense but where does the N*N come from?

N is whatever you choose it to be. If someone gives me the Zurek red I'll double check and report back.

 

[A. Neumaier]

I am sure there are some standard theorems, not invented specially for proving the Born Rule, which say whether and when continuous variables can be represented to any desired accuracy by discrete values.

To have a valid claim that you are sure, you must be able to point to the sources. They don't exist. See the discussion here.

 

[Michael Price]

I believe Zurek's derivation will require the following property of the environment (=rest of the universe), namely that you can decompose it into an orthonormal basis dim(N*N) :
$|env\rangle=\frac{1}{N} Σ ^{N^2}_{i=1}|env^N_i\rangle$
which is why I'd like the explicit ref so I can check.

Just to answer my own question, the above is what Carroll and Sebens use in "Many Worlds, the Born Rule, and Self-Locating Uncertainty". And they credit Zurek for the insight, so I am happy that Zurek uses the same property of the environment vector.
So the question is answered. MWI does predict the Born rule.

 

[DarMM]

Just to answer my own question, the above is what Carroll and Sebens use in "Many Worlds, the Born Rule, and Self-Locating Uncertainty". And they credit Zurek for the insight, so I am happy that Zurek uses the same property of the environment vector.
So the question is answered. MWI does predict the Born rule.

Zurek reference is here:
https://arxiv.org/abs/quant-ph/0405161

However Zurek himself acknowledges that the derivation contained in the paper is circular, relying on a well-defined branching structure that has decohered already. Something that can only be shown using the Born rule.

Hence, I do not how you can claim that MWI does predict the Born rule.

 

[Michael Price]

Zurek reference is here:
https://arxiv.org/abs/quant-ph/0405161

However Zurek himself acknowledges that the derivation contained in the paper is circular, relying on a well-defined branching structure that has decohered already. Something that can only be shown using the Born rule.

Hence, I do not how you can claim that MWI does predict the Born rule.

Zurek might be being a bit over-cautious there. Decoherence is a property of many body systems becoming progressively more entangled as time passes. The branching occurs in an irreversible fashion as a result of the decoherence, even without weighting the branches. All the Born rule is doing is supplying a weighting to the already-defined branches. Anyway, Carroll and Sebens don't seem to share Zurek's reservations, stating it works for the full range of classical to quantum.
https://arxiv.org/abs/1405.7577

 

[Derek P]

Zurek might be being a bit over-cautious there. Decoherence is a property of many body systems becoming progressively more entangled as time passes. The branching occurs in an irreversible fashion as a result of the decoherence, even without weighting the branches. All the Born rule is doing is supplying a weighting to the already-defined branches. Anyway, Carroll and Sebens don't seem to share Zurek's reservations, stating it works for the full range of classical to quantum.
https://arxiv.org/abs/1405.7577

It occurred to me a while back that if the environment provides enough branches then ordinary statistics kick in and things like the Central Limit Theorem apply to bundles of branches.. But I can see that you may have to axiomatize the conditions to avoid pathological distributions.

 

[Michael Price]

It occurred to me a while back that if the environment provides enough branches then ordinary statistics kick in and things like the Central Limit Theorem apply to bundles of branches.. But I can see that you may have to axiomatize the conditions to avoid pathological distributions.

I don't think we have to worry about such things. The derivation works due to the mere presence of the environment in the background - but the observer-observed pair don't have to interact, in any way, with the environment to get this result. The environment is not supplying the decoherence or branches - although if you do interact with the environment (as we would in real life) then more branching and decoherence occurs.

 

[Derek P]

I don't think we have to worry about such things. The derivation works due to the mere presence of the environment in the background - but the observer-observed pair don't have to interact, in any way, with the environment to get this result. The environment is not supplying the decoherence or branches - although if you do interact with the environment (as we would in real life) then more branching and decoherence occurs.

Hmm, I think that is where people will disagree. You can extend the state with dummy environmental states but that won't give you any of the dynamics of world separation. (Obviously, since a beam splitter does not give you separation of worlds despite an entire universe minus one photon, in the background.) I think you need to allow the interaction and show that the Schmidt terms are degenerate. But what would I know?

 

[Michael Price]

Hmm, I think that is where people will disagree. You can extend the state with dummy environmental states but that won't give you any of the dynamics of world separation. (Obviously, since a beam splitter does not give you separation of worlds despite an entire universe minus one photon, in the background.) I think you need to allow the interaction and show that the Schmidt terms are degenerate. But what would I know?

The environment states are not dummy states, and the dynamics is not affected by the presence or absence of the Born rule. The dynamics is given by the Schrödinger equation or equivalent EOM. Beam splitting does not split decohered worlds because it is not an irreversible event. The Born Rule doesn't need the splitting to be decohered and permanent.

 

[Derek P]

The environment states are not dummy states, and the dynamics is not affected by the presence or absence of the Born rule. The dynamics is given by the Schrödinger equation or equivalent EOM. Beam splitting does not split decohered worlds because it is not an irreversible event. The Born Rule doesn't need the splitting to be decohered and permanent.

Fairt enough. I was thinking in terms of MWI world splitting.

 

[DarMM]

Zurek might be being a bit over-cautious there. Decoherence is a property of many body systems becoming progressively more entangled as time passes. The branching occurs in an irreversible fashion as a result of the decoherence, even without weighting the branches. All the Born rule is doing is supplying a weighting to the already-defined branches.

And it is the tracing formula, in essence attaining the marginal probabilities for the system, that allows you to show that decoherence occurs. There is currently no derivation of decoherence without the Born rule present, that is decoherence can only be shown to occur if you weight the branches, otherwise it doesn't. This is the point Zurek concedes.

 

[Michael Price]

And it is the tracing formula, in essence attaining the marginal probabilities for the system, that allows you to show that decoherence occurs. There is currently no derivation of decoherence without the Born rule present, that is decoherence can only be shown to occur if you weight the branches, otherwise it doesn't. This is the point Zurek concedes.

No, you have misread Zurek. Zurek is explicit (page 25/6) that his derivation avoids using decoherence precisely because that would be circular. His derivation of the Born rule is fully quantum, being based on entanglement. Only once the Born rule is deduced can the "decoherence toolbox" (his phrase) be employed, if needed.

 

[DarMM]

Earlier Post: https://www.physicsforums.com/threa...dict-the-born-rule.946467/page-9#post-5994645

Throughout his papers Zurek proves his theorem from axioms A1-A3 that I listed earlier combined with a fourth axiom. The fourth axiom may be any of three I listed as B1-B3.

The paper I linked you to has the "best" version of Zurek's derivation, as it adopts axiom B3 as the fourth axiom, the only one with robust experimental confirmation. However this axiom means the proof no longer takes place within a Many-Worlds framework, but in the "Existential Interpretation", Zurek's own interpretation.

See
Zurek, W. (2010). Quantum Jumps, Born’s Rule, and Objective Reality. In: S. Saunders et al, ed., Many Worlds? Everett, Quantum Theory, and Reality, 1st ed. Oxford University Press, pp. 409-432.

 

[DarMM]

Also see the following papers for a discussion of how Zurek is assuming structure only known to be present after decoherence:

Howard Barnum. No-signalling-based version of zurek’s derivation of quantum probabilities:
A note on “environment-assisted invariance, entanglement, and probabilities in
quantum physics”. arXiv quant-ph/0312150, 2003.

C. M. Caves. Notes on Zurek’s derivation of the quantum probability rule.
info.phys.unm.edu/ caves/reports/ZurekBornderivation.ps

My example breakdown of Zurek's proof is based on the latter.

 

[Michael Price]

Earlier Post: https://www.physicsforums.com/threa...dict-the-born-rule.946467/page-9#post-5994645

Throughout his papers Zurek proves his theorem from axioms A1-A3 that I listed earlier combined with a fourth axiom. The fourth axiom may be any of three I listed as B1-B3.

The paper I linked you to has the "best" version of Zurek's derivation, as it adopts axiom B3 as the fourth axiom, the only one with robust experimental confirmation. However this axiom means the proof no longer takes place within a Many-Worlds framework, but in the "Existential Interpretation", Zurek's own interpretation.

See
Zurek, W. (2010). Quantum Jumps, Born’s Rule, and Objective Reality. In: S. Saunders et al, ed., Many Worlds? Everett, Quantum Theory, and Reality, 1st ed. Oxford University Press, pp. 409-432.

Well, Carroll and Sebens reference the Zurek paper you gave and I read, so it would it seem the definitive one. The derivation requires only one piece of calculation beyond elementary Hilbert space algebra, which I have already given. Namely:
$|env\rangle=\frac{1}{\sqrt{N}}∑^N_{i=1}|env^N_i\rangle$
N is is chosen to produce orthonormal states in the system-environment decomposition.
The environment can simply be the rest of the universe. No properties, including decoherence, are required of it, except that it live in a Hilbert space of exceedingly high dimension, perhaps even infinite.
The Caves notes wouldn't open on my tablet.