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

[DarMM]

I see now what you mean. I will need time to think about it, I'll get back to you after I cover Zurek's proof in more detail.

 

[A. Neumaier]

the Born rule doesn't by itself contradict the assumption that evolution is purely unitary.

Of course not.

The unitary evolution is postulated for a closed system, while the Born rule is claimed for an open, observed system. That makes all the difference, since they apply to disjoint regimes, and hence cannot contradict each other.

I don't necessarily think it contradicts unitary evolution either, there's just no proof it's a consequence of unitary evolution.

Under certain conditions, the Born rule for measuring a tiny subsystem by a large detector can be derived from statistical mechanics. (One doesn't need MWI, which only creates additional confusion.) We discussed this in the thread Collapse from unitarity.

 

[PeterDonis]

Is the journal peer reviewed ?

The Mentors have reviewed the paper and journal and it is an acceptable source for PF discussion.

 

[DarMM]

We start with a macroscopic system $\mathcal{S}$ in a state $|\psi\rangle$ and we have an apparatus that measures the basis $\{|\sigma_j\rangle\}$. There is also an environment $\mathcal{E}$.

The combined system-environment state is then assumed to evolve into Schmidt form after measurement:

$$\Psi_{\mathcal{S}\cdot\mathcal{E}} = \sum_i \alpha_i |\sigma_i\rangle |\eta_i\rangle$$

with $|\eta_i\rangle$ a orthonormal basis for the environment.

What we want to show is that the probability of measuring a particular outcome $j$ of the basis $|\sigma_j\rangle$ from the microstate $\psi$, denoted $p\left(j,\{\sigma_j\},\psi\right)$ is derivable from some property of the final state $\Psi_{\mathcal{S}\cdot\mathcal{E}}$. That is:

$$p\left(j,\{\sigma_j\},\psi\right) = F\left(\sigma_i,\eta_i,\Psi_{\mathcal{S}\cdot\mathcal{E}}\right)$$

and that this probability agrees with the Born rule.

Assumption 1, Environmental Noncontextuality, EN:
$$p\left(j,\{\sigma_j\},\psi\right) = F\left(\sigma_i,\Psi_{\mathcal{S}\cdot\mathcal{E}}\right)$$

The probabilities do not depend on the environmental states after measurement.

Envariance is really just a consequence of this assumption. For if some unitary on the microscopic system can be undone by the environment and probabilities do not depend on the environment, then that unitary can have no effect on the probabilities.

Assumption 2, Perfect correlation, PC:
The chance of observing an environmental state $\eta_i$, denoted $G\left(\eta_i,\Psi_{\mathcal{S}\cdot\mathcal{E}}\right)$, obeys:
$$G\left(\eta_i,\Psi_{\mathcal{S}\cdot\mathcal{E}}\right) = F\left(\sigma_i,\Psi_{\mathcal{S}\cdot\mathcal{E}}\right)$$

That is, in a Schmidt state the environment and the system are perfectly correlated.

Zurek's proof in the equal amplitude case then be seen more easily via looking at the state:

$$\psi = \sqrt{\frac{1}{2}}\left(|0\rangle + |1\rangle\right)$$

for which the post-measurement Schmidt state is:

$$\Psi_{\mathcal{S}\cdot\mathcal{E}} = \sqrt{\frac{1}{2}}\left(|00\rangle + |11\rangle\right)$$

So:

$$p\left(0,\{|0\rangle,|1\rangle\},\sqrt{\frac{1}{2}}\left(|0\rangle + |1\rangle\right)\right) = F\left(|0\rangle,\sqrt{\frac{1}{2}}\left(|00\rangle + |11\rangle\right)\right)$$

Swapping the microscopic states can be undone by swapping the environment states, so by assumption 1 it has no effect on the system, hence:

$$F\left(|0\rangle,\sqrt{\frac{1}{2}}\left(|00\rangle + |11\rangle\right)\right) = F\left(|0\rangle,\sqrt{\frac{1}{2}}\left(|10\rangle + |01\rangle\right)\right)$$

In this new state the chance to measure the system in $|0\rangle$ is the same as the environment in $|1\rangle$, hence:
$$F\left(|0\rangle,\sqrt{\frac{1}{2}}\left(|10\rangle + |01\rangle\right)\right) = G\left(|1\rangle,\sqrt{\frac{1}{2}}\left(|10\rangle + |01\rangle\right)\right)$$

We can then use the observation that swaps have no effect once again, this time on the environment:
$$G\left(|1\rangle,\sqrt{\frac{1}{2}}\left(|10\rangle + |01\rangle\right)\right) = G\left(|1\rangle,\sqrt{\frac{1}{2}}\left(|11\rangle + |00\rangle\right)\right)$$

And once more use the correlation between system and environment:
$$G\left(|1\rangle,\sqrt{\frac{1}{2}}\left(|11\rangle + |00\rangle\right)\right) = F\left(|1\rangle,\sqrt{\frac{1}{2}}\left(|11\rangle + |00\rangle\right)\right)$$

However this is just $p\left(1,\{|0\rangle,|1\rangle\},\sqrt{\frac{1}{2}}\left(|0\rangle + |1\rangle\right)\right)$, hence:

$$p\left(0,\{|0\rangle,|1\rangle\},\sqrt{\frac{1}{2}}\left(|0\rangle + |1\rangle\right)\right) = p\left(1,\{|0\rangle,|1\rangle\},\sqrt{\frac{1}{2}}\left(|0\rangle + |1\rangle\right)\right)$$

Note how this works. We use EN to swap environments, PC then let's us convert this to a probability for environment observations. The latter step is what allows us to alter it from a statement about $|0\rangle$ states to one about $|1\rangle$ states, we just have to use EN once more to reattach the system states in their original order.

So envariance alone only tells us we can swap environments in certain scenarios, but it will never allow us to convert a statement about one element of a basis to another, we need PC for that (which is B3 in my list above, you could also use B1 or B2).

Okay, onto problems next.

 

[DarMM]

I do agree that's another widespread misunderstanding, which is what prompted me to respond to your claim that WMI has to derive the Born rule.

Okay I have thought about this a bit more. Quite simply:

1. It has to derive the Born Rule as that is the empirical core of quantum mechanics. If unitary QM alone cannot obtain the Born rule, then it doesn't match experiment.
2. If you wish to alter Many-Worlds, so that it has a Born rule built-in, that is a different theory outside the scope of this thread, as it is no longer deriving the Born rule as a consequence of unitary evolution. That is not to say such theories will display an inconsistency between the Born rule and Unitary evolution.
   There is an interpretive difference between the Born rule as effective and the Born rule as primary.

I do think versions of MWI where the Born-rule is built in are quite different from just pure unitary QM where it is not. I think this can be demonstrated by trying to give a meaning to the assumed Born rule. You mentioned the coefficients as "measures of existence", such a concept is non-existent in pure unitary QM and hence its introduction changes the interpretation.

 

[stevendaryl]

I do think versions of MWI where the Born-rule is built in are quite different from just pure unitary QM where it is not. I think this can be demonstrated by trying to give a meaning to the assumed Born rule. You mentioned the coefficients as "measures of existence", such a concept is non-existent in pure unitary QM and hence its introduction changes the interpretation.

I would say that a pure wavefunction evolving as a deterministic function of time cannot possibly have a unique physical interpretation. Nothing physically follows from "There is a complex-valued function of 3N+1 dimensional space obeying such-and-such differential equation". You have to make psycho-physical assumptions (what is the relationship between the mathematical objects of the theory and our observations) before a theory can be said to have any empirical content whatsoever.

So if you're trying to derive the Born rule from pure unitary evolution, you have to make some interpretation assumptions to even make sense of the question: What is the probability of this or that happening? You have to at least identify what are the events that can either happen or not happen. Or it seems that way to me.

 

[akvadrako]

You mentioned the coefficients as "measures of existence", such a concept is non-existent in pure unitary QM and hence its introduction changes the interpretation.

It's not clear to me where our disagreement lies, but a theory without such a measure is incomplete, even disregarding the empirical connection. If have the state $\psi= 1|a\rangle + 0|b\rangle$, then to have a physical interpretation at all I need to be able to say something like $|a\rangle$ exists and $|b\rangle$ doesn't.

Anyone trying to derive the Born rule from just unitary QM must be assuming it has some measure. Otherwise it's hopeless from the start.

 

[DarMM]

It's not clear to me where our disagreement lies, but a theory without such a measure is incomplete, even disregarding the empirical connection. If have the state $\psi= 1|a\rangle + 0|b\rangle$, then to have a physical interpretation at all I need to be able to say something like $|a\rangle$ exists and $|b\rangle$ doesn't.

I think they'd all agree on this, as they would see it as a branch being absent/present, however this is an easy special case. What is the difference between:

$$\sqrt{\frac{1}{2}}|a\rangle + \sqrt{\frac{1}{2}}|b\rangle$$

and

$$\sqrt{\frac{1}{3}}|a\rangle + \sqrt{\frac{2}{3}}|b\rangle$$

within the "measure of existence" concept?

Different versions of Many Worlds would distinguish these two states in a different manner.

Anyone trying to derive the Born rule from just unitary QM must be assuming it has some measure. Otherwise it's hopeless from the start.

Wallace and Zurek are explicitly avoiding assuming any measure, that's the point of their derivations.

 

[akvadrako]

I think they'd all agree on this, as they would see it as a branch being absent/present, however this is an easy special case.

It's two different issues:

(1) Can you derive the Born rule with no additional assumptions? My claim is it's obviously not possible, otherwise you have no reason to disregard 0-weight terms.
(2) Do the different sets of assumptions agree?

What is the difference between:
$$\sqrt{\frac{1}{2}}|a\rangle + \sqrt{\frac{1}{2}}|b\rangle$$
and
$$\sqrt{\frac{1}{3}}|a\rangle + \sqrt{\frac{2}{3}}|b\rangle$$
within the "measure of existence" concept?

The straightforward version just says the measure of existence of a term is the absolute squared amplitude. It's not a priori the same as what an observer would experience.

 

[DarMM]

I would say that a pure wavefunction evolving as a deterministic function of time cannot possibly have a unique physical interpretation. Nothing physically follows from "There is a complex-valued function of 3N+1 dimensional space obeying such-and-such differential equation". You have to make psycho-physical assumptions (what is the relationship between the mathematical objects of the theory and our observations) before a theory can be said to have any empirical content whatsoever.

So if you're trying to derive the Born rule from pure unitary evolution, you have to make some interpretation assumptions to even make sense of the question: What is the probability of this or that happening? You have to at least identify what are the events that can either happen or not happen. Or it seems that way to me.

I would agree, I think in the back and forth of this something has become unclear, as I am not proposing just a complex wavefunction with no physical meaning is assumed in Many Worlds.

Neither Wallace and Zurek discuss a purely mathematical "unitary evolution only" QM, neither would any MW interpretation for the reasons you mention. They do give a physical meaning to the mathematics (I'm not going to try to explain the meaning they give, because certainly in Wallace's case, I find it confusing still).

What I should say is that the only part of QM they give a physical meaning to at the outset is the unitary evolution component.

An example might be better.

There are versions of MWI (such as Mandolesi's) where additional structures beyond just the unitary evolution are given a direct physical interpretation at the beginning. For example neither Wallace nor Zurek attribute a direct physical meaning to the magnitude of branch coefficients, Mandolesi gives them the direct meaning of "measure of resistance to interference". Other interpretations directly say the coefficient magnitudes are a "count" of how many copies of that state there are (how many worlds).

I am saying that the versions like Wallace and Zurek's, where the coefficient magnitudes have no direct physical meaning (aside from distinguishing states) are the ones that have to "derive" the connection with experiment known as the Born rule. Others do not.

 

[DarMM]

The straightforward version just says the measure of existence of a term is the absolute squared amplitude. It's not a priori the same as what an observer would experience.

What does that mean physically? That is, what is the interpretative difference between the two states I mentioned? Mandolesi would say in the second state the $|b\rangle$ branch is more resistant to interference, other versions would say there are more $|b\rangle$ branches/worlds in the second state than the first.

 

[DarMM]

I will say one thing however, I think there is an assumption that Many-Worlds is one interpretation and Wallace, Zurek, DeWitt etc are all talking about the same thing. Hence some of the confusion on this thread. They are not. They have completely different explanations for what the Born rule is (in some it is subjective, in others not), what a world is, etc

A read of Wallace's book and some MWI papers shows them directly disagreeing with each other and arguing over the physical meaning of terms. Even some versions that look very similar initially, e.g. Vaidmann and Carroll/Seben, have fundamental differences.

 

[akvadrako]

What does that mean physically? That is, what is the interpretative difference between the two states I mentioned? Mandolesi would say in the second state the $|b\rangle$ branch is more resistant to interference, other versions would say there are more $|b\rangle$ branches/worlds in the second state than the first.

That's a term from Vaidman and I don't see a clear physical meaning, though he does mention the signifigance of $|b\rangle$ being more resistant to interference. In terms of WMI, one can consider worlds as operators, so I would say it makes sense to treat the coefficients as how much it contributes to the total state, in the same sense as a cosmological measure.

I will say one thing however, I think there is an assumption that Many-Worlds is one interpretation and Wallace, Zurek, DeWitt etc are all talking about the same thing. Hence some of the confusion on this thread. They are not. They have completely different explanations for what the Born rule is (in some it is subjective, in others not), what a world is, etc

A read of Wallace's book and some MWI papers shows them directly disagreeing with each other and arguing over the physical meaning of terms. Even some versions that look very similar initially, e.g. Vaidmann and Carroll/Seben, have fundamental differences.

You can call them different interpretations, but they all agree that the physical (ontic) situation maps 1-to-1 onto unitary QM. That's what the whole Everett program is about. The extra structure given to it is just descriptive. There are different uses of the Born rule - one about subjective experience of agents and one about the objective coefficients of worlds, so that's a fair point to disagree about.

 

[akvadrako]

To expand on the zero-weight case for why a measure must be assumed, let's say we have this setup:

$|Y\rangle(-1|A\rangle) + |X\rangle(a|A\rangle + b|B\rangle)$

If you can manipulate this so the two parts in parentheses are together then you'll have:

$|?\rangle(-1|A\rangle + a|A\rangle + b|B\rangle)$

When $a = 1, b = 3$, the $|A\rangle$ branch weight becomes 0 and no longer exists, which is different then when $a = b = 2$. So even different non-zero branch weights must be given some different meaning.

 

[stevendaryl]

There is a notion of measure that is built-in to the definition of the Hilbert space of square-integrable functions. Wave functions are only defined up to a set of measure zero. That is, if you have two wave functions, $\psi_1(x)$ and $\psi_2(x)$, then they are equal, as elements of the Hilbert space, if

$\int |\psi_1(x) - \psi_2(x)|^2 dx = 0$

I actually read an article once that tried to bootstrap from: "Two functions are equal if they only differ on a set of measure zero" to the Born rule. The argument went this way:

Imagine that there is some measurement that has two results: $U$ or $D$. The amplitude for one result is $\alpha$ and the amplitude for the other is $\beta$.

Now, imagine repeating the measurement over and over again, and recording it in some persistent storage.

Then the wave function of the whole universe can afterwards be written as a superposition of possible worlds, each of which has a different history of Us and Ds.

Some of those worlds will have records of Us and Ds that agree with the Born rule, some will not. The claim made was that in some kind of limit as the number of measurements goes to infinity, the measure of the set of "maverick" worlds (where the relative frequency disagrees with Born) goes to zero.

To the extent that the limiting case actually makes sense, you can say that there are no worlds that disagree with the Born rule, because sets of measure zero are ignored in the definition of Hilbert space.

I don't remember the details of how this limit actually makes sense, so take it with a grain of salt...

 

[Michael Price]

There is a notion of measure that is built-in to the definition of the Hilbert space of square-integrable functions. Wave functions are only defined up to a set of measure zero. That is, if you have two wave functions, $\psi_1(x)$ and $\psi_2(x)$, then they are equal, as elements of the Hilbert space, if

$\int |\psi_1(x) - \psi_2(x)|^2 dx = 0$

I actually read an article once that tried to bootstrap from: "Two functions are equal if they only differ on a set of measure zero" to the Born rule. The argument went this way:

Imagine that there is some measurement that has two results: $U$ or $D$. The amplitude for one result is $\alpha$ and the amplitude for the other is $\beta$.

Now, imagine repeating the measurement over and over again, and recording it in some persistent storage.

Then the wave function of the whole universe can afterwards be written as a superposition of possible worlds, each of which has a different history of Us and Ds.

Some of those worlds will have records of Us and Ds that agree with the Born rule, some will not. The claim made was that in some kind of limit as the number of measurements goes to infinity, the measure of the set of "maverick" worlds (where the relative frequency disagrees with Born) goes to zero.

To the extent that the limiting case actually makes sense, you can say that there are no worlds that disagree with the Born rule, because sets of measure zero are ignored in the definition of Hilbert space.

I don't remember the details of how this limit actually makes sense, so take it with a grain of salt...

That sounds equivalent to Bryce Dewitt's argument from his 1970 article, Quantum Mechanics and Reality, although he expressed it slightly differently. But, yes, the measure (Hilbert space norm) of worlds where the Born Rule is violated vanishes - hence the Born Rule is obeyed. I've always found this argument quite persuasive and always been somewhat bemused at the complexity of other "derivations".

 

[stevendaryl]

That sounds equivalent to Bryce Dewitt's argument from his 1970 article, Quantum Mechanics and Reality, although he expressed it slightly differently. But, yes, the measure (Hilbert space norm) of worlds where the Born Rule is violated vanishes - hence the Born Rule is obeyed. I've always found this argument quite persuasive and always been somewhat bemused at the complexity of other "derivations".

Well, there are no possible worlds in which there have been infinitely many measurements---such a possible world would have to be infinitely old. So the situation that has measure zero (infinitely many measurements, whose results disagree with the Born rule) is impossible, anyway. So I would not say that it's clear-cut that this argument says anything about the actual universe (with only finitely many measurements).

 

[Michael Price]

Well, there are no possible worlds in which there have been infinitely many measurements---such a possible world would have to be infinitely old. So the situation that has measure zero (infinitely many measurements, whose results disagree with the Born rule) is impossible, anyway. So I would not say that it's clear-cut that this argument says anything about the actual universe (with only finitely many measurements).

And yet the worlds which obey the Born Rule are also infinitely old. Their norm tends to 1, whilst the norm of the worlds where the Born Rule is violated tends to 0. That has to be telling us something.

 

[A. Neumaier]

Wave functions are only defined up to a set of measure zero. That is, if you have two wave functions, $\psi_1(x)$ and $\psi_2(x)$, then they are equal, as elements of the Hilbert space, if

$\int |\psi_1(x) - \psi_2(x)|^2 dx = 0$

But wave functions convey physics only up to an x-independent phase. Your distance doesn't and hence is physically meaningless.

 

[A. Neumaier]

there are no possible worlds in which there have been infinitely many measurements---such a possible world would have to be infinitely old. So the situation that has measure zero (infinitely many measurements, whose results disagree with the Born rule) is impossible, anyway.

But finitely many measurements do not at all contribute to the mathematical ensemble mean which is always a mean over infinitely many cases - so according to the same argument as you employ here, measurements are completely unrelated to ensemble means! This shows that arguing with rigor about something that has only an approximate connection to reality can never prove anything about reality...

 

[bhobba]

That sounds equivalent to Bryce Dewitt's argument from his 1970 article, Quantum Mechanics and Reality, although he expressed it slightly differently. But, yes, the measure (Hilbert space norm) of worlds where the Born Rule is violated vanishes - hence the Born Rule is obeyed. I've always found this argument quite persuasive and always been somewhat bemused at the complexity of other "derivations".

How can the Born rule be violated in any world since its non-contextual? Wallace proves it, but it doesn't really make any sense for a theory with just the wave function to be contextual.

Thanks
Bill

 

[stevendaryl]

But wave functions convey physics only up to an x-independent phase. Your distance doesn't and hence is physically meaningless.

I think you're confused. The Hilbert space of square-integrable functions includes phase information, and as I said, is only defined modulo sets of measure 0. Then of functions in the Hilbert space, we further say that physical states are rays. But you can't work with Hilbert space by dealing with rays as the basis elements.

 

[stevendaryl]

How can the Born rule be violated in any world since its non-contextual?

I think we might be talking about different things. If I prepare an electron to be spin-up in the z-direction, and then measure its spin in the x-direction, the Born rule says that it's 50% likely to be spin-up or spin-down. If I perform a billion such measurements, then there is nonzero probability that the number of spin-up results will be 0 or a billion or whatever. There is no guarantee, for a finite number of trials, that the relative frequencies will be equal to (or close to) what is calculated using the Born rule.

I guess it's a matter of terminology as to whether you say that that's a violation of the Born rule, or is a violation of the expectation that relative frequencies are approximately equal to theoretical probabilities.

 

[A. Neumaier]

I think you're confused.

No. Any measure that can mean something physical necessary must satisfy that the distance between $\psi(x)$$and $\lambda\psi(x)$ is zero when $\lambda|=1$, since these correspond to identical state. Your measure doesn't satisfy this trivial requirement.,

 

[stevendaryl]

No. Any measure that can mean something physical necessary must satisfy that the distance between $\psi(x)$$and $\lambda\psi(x)$ is zero when $\lambda|=1$, since these correspond to identical state. Your measure doesn't satisfy this trivial requirement.,

I don't know what you're talking about. I'm not proposing a measure on possible worlds, I'm stating a fact about the Hilbert space of square-integrable functions.

 

[bhobba]

And yet the worlds which obey the Born Rule are also infinitely old. Their norm tends to 1, whilst the norm of the worlds where the Born Rule is violated tends to 0. That has to be telling us something.

Well in MW it follows the Schrodinger equation - which usually deals with norms of 1, - so exactly how does it tend to 1? I can see after decoherence a world may have a norm less that 1 so we need to normalize it to apply the Schrodinger equation - but after infinite time without that re-normalization it would tend to zero.

As I have mentioned Gell-Mann thinks his Decoherent-Histories is just MW with different semantics - he calls it a post Everton interpretation or something like that. In Decoherent Histories QM is a stochastic theory of histories but only one history exists - so you don't have this dilution issue of MW - nor this exponentially increasing number of worlds. This guy thinks they are isomorphic:
https://www.hedweb.com/manworld.htm#many

I would not go that far, but there is a strong connection such that the differences SEEM to disappear or be trivial on careful examination.

As I said I learned a lot about decoherent histories from reading Wallace - but have not found a really good book on Decohreret Histories - Griffiths Consistent Histories is a bit too basic for me. Good for those starting out - say after reading Susskind.

Thanks
Bill

 

[bhobba]

I think we might be talking about different things.

Yes in that sense its just the law of large numbers - but since we are taking probabilities it really comes with the territory.

Thanks
Bill

 

[A. Neumaier]

I don't know what you're talking about. I'm not proposing a measure on possible worlds, I'm stating a fact about the Hilbert space of square-integrable functions.

But this fact is irrelevant for physics, which only cares for states (rays in Hilbert space), so your subsequent conclusions about the Born rule are irrelevant for physics, too.

 

[DarMM]

To the extent that the limiting case actually makes sense, you can say that there are no worlds that disagree with the Born rule, because sets of measure zero are ignored in the definition of Hilbert space.

I don't remember the details of how this limit actually makes sense, so take it with a grain of salt...

I haven't found much formal treatment of DeWitt's limit in the literature. The usual criticism is that for actual finite measurements the coefficients in front of maverick worlds are still non-zero and without any meaning attached to the coefficients via Born's rule, there is no reason to discount them, they still dominate a pure count of branch terms. DeWitt himself says that one has to introduce some meaning to the branch terms, like a "width" of worlds etc to get his result, but it's never really explained what these sorts of ideas mean physically, i.e. what is the "width of a world"?

 

[DarMM]

You can call them different interpretations, but they all agree that the physical (ontic) situation maps 1-to-1 onto unitary QM. That's what the whole Everett program is about. The extra structure given to it is just descriptive. There are different uses of the Born rule - one about subjective experience of agents and one about the objective coefficients of worlds, so that's a fair point to disagree about.

Why do you think WMI cannot just assume the Born rule, interpreted as a measure of existence?

I hope this answers your question. Originally when you raised this I was discussing Wallace and Zurek assuming the results of decoherence, when they in fact need to derive them. Other versions of Many-Worlds might not need to do this, as they assume the Born Rule or a weaker version of it, etc. 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.
I think you took my statement as concerning all MWI versions rather than those of Wallace and Zurek.

For ones like Zurek's, if their derivation makes use of a structure (post-decoherence branching) that thus far has only been shown to arise via use of Born's rule they are in a bind, as Zurek himself has said, hence his Quantum Darwinism project.

 

[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.