Question regarding the Many-Worlds interpretation

[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.

 

[mfb]

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.

Right, we cannot distinguish between interpretations experimentally, all interpretations give the same observations. That's why they are called interpretations.

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

"But this is an issue, Copenhagen has no way to generate multiple branches!"
(Don't take this seriously, that's how some argument here look like to me).


Okay, seriously, I don't think further discussion with me here will help anyone.

 

[S.Daedalus]

Right, we cannot distinguish between interpretations experimentally, all interpretations give the same observations. That's why they are called interpretations.

But that's the point: the MWI you propose does not account for our observations.

 

[tom.stoer]

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?

mfb, the argument against "is not required" is not Copenhagen but experiment. You constantly ignore the fact that there are experimentally observed statistical frequencies which require an interpretation. You ignore the fact that we can use tr(Pρ) to calculate something formally which approximates these observed statistical frequencies.

The simple questions to MWI are
Why does this work in so many cases?
What replaces the Born rule probabilities and explains the statistical frequencies of 90% - 10% in my original question?

The answer "there are no probabilities" may eliminate the concept of probability from an interpretation but it does not eliminate the entity tr(Pρ) from the formalism. So if you want to interpret the formalism and its application to the real world you should be able to explain why tr(Pρ) does work FAPP even so it is not required.

In order to make this claim against your position work I do rely on Copenhagen but on "shut up and calculate"; I am asking why this works, so please explain.

 

[stevendaryl]

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

I asked this question earlier in this thread, and got no answer: How does getting rid of the other branches change anything, such as the predictiveness of the Born rule, on THIS branch?

 

[S.Daedalus]

I asked this question earlier in this thread, and got no answer: How does getting rid of the other branches change anything, such as the predictiveness of the Born rule, on THIS branch?

Because you end up with a state in some subspace, over which Gleason's theorem provides a measure, which gives you the Born rule. Also, because it introduces alternatives in the first place: this thing happens rather than that one, making an appeal to probability coherent.

 

[stevendaryl]

Because you end up with a state in some subspace, over which Gleason's theorem provides a measure, which gives you the Born rule. Also, because it introduces alternatives in the first place: this thing happens rather than that one, making an appeal to probability coherent.

I'm saying that the nonexistence of other "branches" is not something that is observable in this branch (well, not in practice--to observe the effects of other branches would require detecting interference among branches, and if the branches involve macroscopically different states, then this is impossible). So whatever rule of thumb you are using to extract predictive content from QM doesn't actually require collapse. You can do the same procedure whether or not collapse happens, and you'll get the same results.

 

[S.Daedalus]

I'm saying that the nonexistence of other "branches" is not something that is observable in this branch (well, not in practice--to observe the effects of other branches would require detecting interference among branches, and if the branches involve macroscopically different states, then this is impossible). So whatever rule of thumb you are using to extract predictive content from QM doesn't actually require collapse. You can do the same procedure whether or not collapse happens, and you'll get the same results.

As I said, the applicability of Gleason's theorem to extract the Born probabilites depends on having a state that is really in one of the subspaces, rather than a superposition. This is where the collapse comes in; without it, Gleason's theorem simply doesn't talk about the probabilities. There are no observable differences, no, but important conceptual ones that mean you must reason differently if assuming different ontologies. Assuming a collapse, probabilities follow; without it, I don't see how.

 

[stevendaryl]

As I said, the applicability of Gleason's theorem to extract the Born probabilites depends on having a state that is really in one of the subspaces, rather than a superposition. This is where the collapse comes in; without it, Gleason's theorem simply doesn't talk about the probabilities. There are no observable differences, no, but important conceptual ones that mean you must reason differently if assuming different ontologies. Assuming a collapse, probabilities follow; without it, I don't see how.

There is something fishy about this argument. There is no observable difference between situation A and situation B, but the fact that there is a mathematical technique that works for situation A, but not for situation B is an argument that A must be the case?

The statement of Gleason's theorem does not mention the collapse hypothesis. It's about a measure on subspaces of a Hilbert space.

 

[stevendaryl]

There is something fishy about this argument. There is no observable difference between situation A and situation B, but the fact that there is a mathematical technique that works for situation A, but not for situation B is an argument that A must be the case?

This business about collapse destroying all the branches but one reminds me of a philosophical argument about Star Trek's teleporter.

Suppose that teleporters are invented some day, and the way they work is this:

1. At the transmitting end, a laser, or X-ray, or some kind of beam scans every atom in your body and records its state.
2. As a side-effect, it blasts your body into its component atoms.
3. At the receiving end, a matter assembler takes the information and builds a new body with the same atomic states. (I'm disregarding quantum mechanics here.)

For all intents and purposes, the traveler leaves the transmitter, and is transported at the speed of light to the receiver. But now let's change things so that step 2 doesn't happen. The original "you" is NOT destroyed in the process. Would you still consider this a way of traveling at the speed of light? From the point of view of the original "you", what happens is that you enter a booth, are scanned, and then walk out of the booth in the same location you started in. Except you are poorer by the cost of the teleportation fees. You'd feel ripped off. But for some reason, you wouldn't be comforted by the offer to have your body torn into its component atoms, reducing the situation to the previous case.

 

[S.Daedalus]

There is something fishy about this argument. There is no observable difference between situation A and situation B, but the fact that there is a mathematical technique that works for situation A, but not for situation B is an argument that A must be the case?

No. There's a mathematical argument that explains why things look the way they do in situation A, but not in situation B; so from the point of view of explaining why things look a certain way, A gives the better explanation.

The statement of Gleason's theorem does not mention the collapse hypothesis. It's about a measure on subspaces of a Hilbert space.

No, but it mentions subspaces on Hilbert space. And the general state won't be in any of the relevant subspaces. That's why you need the collapse.