# Question regarding the Many-Worlds interpretation

• tom.stoer

#### tom.stoer

Let's prepare and experiment with individual photons moving along the z-axis and polarization in the xy-plane such that a detector registers
- polarization along the x-axis with 90% and
- polarization along the y-axis with 10%

According to the MWI for each registered photon there's a branching such that in every branch one experimental result (either x- or y-polarization) is realized. Let's repeat the experiment N times with N individual photons which results in 2N branches in total.

Now my problem is that according to the branching the observers expect a result "y-polarization" with 50% probability, whereas according to the experimental setup they expect a result "y-polarization" only with 10% probability. Of course we know that in practice the second calculation is correct, such that I as an observer will always live in a branch where the result set "xyxxyxxxx..." agrees with the 90%-10% probabilities.

How does the MWI resolve this contradiction?
How does the MWI forces me to find myself in one of the most probable branches?

The short answer is that it doesn't. That's one of the fundamental problems of Everett style interpretations: The count of observed events does not emerge from the dynamics itself but has to be added in form of an additional postulate. You can either directly postulate an observed frequency proportional to the squared amplitude magnitude, or you can postulate something more subtle and then derive the probabilities. This is what Deutsch/Wallace based on decision theory do, or Zurek based on stability assumptions and the ad-hoc existence of a probability measure.

I think that the arguments to derive the Born rule from MWI are deeply flawed. Not even logically, but they implicitly admit that emergence is not enough to deal with the measurement problem and undermine the very basis of the idea. This makes the concept of many worlds much harder to accept, because it doesn't take you the whole way and in the end you still have to make additional assumptions just like in any other proposed solution of the measurement problem.

Cheers,

Jazz

1 person
Of course we know that in practice the second calculation is correct, such that I as an observer will always live in a branch where the result set "xyxxyxxxx..." agrees with the 90%-10% probabilities.
The MWI doesn't agree. You will not always live in such a branch. There's no mechanism which makes a certain branch which agrees with the probabilities more "real" than a branch where you always get y. It is only much more likely to end up in such a branch.

But you are right that the emergence of the probabilities in the first place is a problem. As Jazzdude has pointed out, there isn't a universally accepted derivation of the Born rule.

I think that the arguments to derive the Born rule from MWI are deeply flawed. Not even logically, but they implicitly admit that emergence is not enough to deal with the measurement problem and undermine the very basis of the idea. This makes the concept of many worlds much harder to accept, because it doesn't take you the whole way and in the end you still have to make additional assumptions just like in any other proposed solution of the measurement problem.

In thinking about the Born rule and MWI, I have wondered what it would even mean to derive the Born rule for MWI. In some "possible worlds", relative frequencies for outcomes of experiments will be well-described by the Born rule, and in other possible worlds, they won't. The best you can do is to come up with a measure on the possible worlds such that the Born rule is valid for "most" worlds (according to that measure), but the only real measure we have is the Born rule itself. So that's kind of circular, but I don't know how it could be otherwise.

This is a philosophical problem that I think is common to all "ensemble" theories of probability--the theory isn't really falsifiable. Just about any outcome for any experiment is consistent with the ensemble theory.

Now my problem is that according to the branching the observers expect a result "y-polarization" with 50% probability
This is a common fallacy. Where did the number 50% come from? Who said the branches have to be equally probable? It's like saying I have a 50/50 chance of winning a lottery: either I win or I don't.

Consider a simple example: you send a random photon through a polarizing beam splitter. There are 2 branches, in one branch (branch 1) the photon goes left, in another (branch 2) it goes right. They both exist simultaneously but you have a 50/50 chance to be in one branch or the other. Now consider another beam splitter at 45 degrees in the right output path. Now we have 3 branches (1, 2a, 2b), branch 1 remains as it was, while branch 2 splits into 2a and 2b when the photon hits second polarizer. Clearly if you have a 50% chance to get into branch 2 and then another 50/50 chance to get into either 2a or 2b, then you have an overall 25% chance to end up in branch 2a. In other words, each of the branches 2a and 2b is twice "thinner" compared to branch 1.

There are derivations of Born rule form symmetry considerations. Some people claim the problem is solved, while other people say it is based on circular argument. Ensuing discussion usually descends into the abyss of philosophy:)

@Delta Kilo: I think that does not address the problem.

@All: I don't see how additional assumptions or postulates can resolve this contradiction. We know from QM and experiment that most observers will observe "x", but counting branches and observers only 50% will observe "x". So the problem is that in the majority of branches sequences "xyxxyyxyxxyyxyx" with "50% x - 50% y" are observed. Therefore most branches are incompatible with our observations.

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I don't see how additional assumptions or postulates can resolve this contradiction.

I fully agree. The only "natural" way to come up with a probability would involve branch counting. The counter argument of the Everettians contains two main arguments: 1) That is an ad-hoc assumption about the attribution of "reality" to the branches. 2) Branch counting is impossible in general, as the number of branches depends on a number of things and is highly subjective. So a probability metric depending on branch counting could not deliver objective probabilities.

To me 2) is also saying that the whole world equals branch idea is not applicable. Either we can talk about different discrete realities or we cannot. In the latter case MWI has a much more fundamental problem than the emergence of the Born rule.

Their answer is however that we simply have to postulate a probability measure for the branches that also gives the branches a weight that equals out all possible issues with branch identification, so fixing both problems at the same time. The Born rule would do that, if you believe them, so it must be the correct way to measure subjective probabilities of finding yourself inside a branch.

Here's a little thought experiment that illustrates my issues with the Born rule in MWI: Consider we have two copies of a universe, both containing the same history of branches but with different squared magnitudes. In one universe the history is very likely, in the other the history is very unlikely as dictated by the Born rule. However the final states of both histories in the final branch are identical, only their amplitude differs. Consequently, the relative evolution and information contained in both branches is identical. Now if the Born rule would really follow from the amplitudes only, then in one branch the observer would say "Now that outcome I observe was very likely" whereas in the other branch he would say "That was rather unlikely!". But they cannot say different things, because their branches are dynamically identical and not correlated to the history of the amplitudes. This is a direct consequence of the linearity of the evolution.

You can clearly see, I'm not an Everettian. But I would be if it delivered what it promised: A theory of structural emergence from just unitary global evolution.

Cheers,

Jazz

Now my problem is that according to the branching the observers expect a result "y-polarization" with 50% probability
I'm asking again where did the number 50% come from. Specifically starting from branch counting how did you arrive at the probability of being in particular branch. It appears that at some point in doing so you assume equal probability for the branches. This assumption is completely unfounded.

In reality, there is always a lot of stuff going on. Consider a photomultiplier. A photon hitting a detector triggers an avalanche of electrons. Each time a collision happens, there is a "branching" (MWI) / "wavefunction collapse" (Copenhagen), whichever you prefer. By the time the signal the appears at the output, the world has split / wavefunctions collapsed many times over. So instead of 2 separate branches (photon X/photon Y), we in fact have a gazillion of branches, differing between each other in minute details, like the number and positions of all the electrons knocked out at each stage. Roughly 0.9 gazillion of branches (roughly) will have the photon in state X and another 0.1 gazillion in state Y. If you are in the game of counting branches you should count all these.

So the problem is that in the majority of branches sequences "xyxxyyxyxxyyxyx" with "50% x - 50% y" are observed. Therefore most branches are incompatible with our observations.
These outcomes are possible but have a very low probability of occurring. Which just means that the majority of branches have very small measure (which is another way of saying exactly the same thing).

Jazzdude: would you not solve that by interpreting worlds as divergent rather than splitted? like alastairwilson.org does?

I don't see how additional assumptions or postulates can resolve this contradiction. We know from QM and experiment that most observers will observe "x", but counting branches and observers only 50% will observe "x". So the problem is that in the majority of branches sequences "xyxxyyxyxxyyxyx" with "50% x - 50% y" are observed. Therefore most branches are incompatible with our observations.

The branches aren't incompatible with observation, as the outcome in any single branch is a physically possible outcome. The problem is that the ratios of numbers of branches of a given x:y ratio doesn't lead to a prediction of the probabilities that matches observation; and from that I can only conclude that either those ratios have no physical significance or (as Delta Kilo suggests) we aren't counting them right.

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In reality, there is always a lot of stuff going on. Consider a photomultiplier. A photon hitting a detector triggers an avalanche of electrons. Each time a collision happens, there is a "branching" (MWI) / "wavefunction collapse" (Copenhagen), whichever you prefer. By the time the signal the appears at the output, the world has split / wavefunctions collapsed many times over. So instead of 2 separate branches (photon X/photon Y), we in fact have a gazillion of branches, differing between each other in minute details, like the number and positions of all the electrons knocked out at each stage. Roughly 0.9 gazillion of branches (roughly) will have the photon in state X and another 0.1 gazillion in state Y. If you are in the game of counting branches you should count all these.

That's one way of explaining the observed macroscopic probabiities, given that we only get to investigate a single path through the tree. However, I find it less than completely satisfying (full disclosure: it is unlikely that anything will ever induce me to say anything nice about MWI) as in principle I ought to be able to separate the first interaction from the cascade by an almost arbitrary distance. Say I prepare a particle in the spin-up state then pass it through a Stern-Gerlach device oriented at 45 degrees that will route it off towards either of two very distant galaxies. Because the SG device is at 45 degrees not 90, the probability that the particle will be detected at one galaxy a million years later is different than the probability of its detection at the other - but how is this difference encoded in the branching that MWI says happens at the SG device? It's somewhat tempting to look for internal state in the particle itself, such that there are more "goes left" branches than "goes right" branches from the initial interaction with the SG apparatus.

Jazzdude: would you not solve that by interpreting worlds as divergent rather than splitted? like alastairwilson.org does?

No, that doesn't really change anything. At some point of the divergence process you must be able to separate the worlds by dynamic independence and you must be able to count them. Otherwise the whole concept is flawed.

... and from that I can only conclude that either those ratios have no physical significance or (as Delta Kilo suggests) we aren't counting them right.

You forget the most obvious option: MWI does not work out.

Cheers,

Jazz

You forget the most obvious option: MWI does not work out.

Cheers,

Jazz

But where is the error in Delta's solution?

Delta Kilo, the idea is very simple. The simple branches are
1) "x" and "y"
2) "xx", "xy", "xy", "yy"
...

The problem is that most observers exist in branches with arbitary low probability calculated using the "90% - 10% - rule". In the above mentioned example the branch "xx" has a probability of 81% but 3/4 of all observers exist in branches with only 19% in total. Changing probabilities (preparing different superpositions) does affect the QM probabilities but does not affect the branching.

So repeating and correcting myself the problem is that the ratio of the number of branches is incompatible with our observations.

Your idea that counting branches means that one should count all these branches and that the majority of branches have very small measure is correct, of course, but I hope you agree that MWI should then provide a means to define branch counting, a probability measure, a derivation of the QM probabilities, or at least a means to assign compatible measures (instead of simply claiming that the branches with small measure have small QM probability w/o being able to define or derive these statements). It seems that you want to hide these problems by introducing rather complex branch structures and ending up with a result of roughly 0.9 gazillion of branches having the photon in state X and another 0.1 gazillion in state Y. I would agree to this result, iff you are able to present a derivation of this result.

I think I agree with Jazz, especially that
I'm not an Everettian. But I would be if it delivered what it promised: A theory of structural emergence from just unitary global evolution.

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Let's prepare and experiment with individual photons moving along the z-axis and polarization in the xy-plane such that a detector registers
- polarization along the x-axis with 90% and
- polarization along the y-axis with 10%

So in this experiment 1000 photons hit 1000 detectors. X-axis polarization is detected by 900 detectors. Y-axis polarization is detected by 100 detectors.

When a detector detects one of the thousand photons, for some reason it becomes unable to detect the 999 other photons. Probably this has something to do with "branching".

Of course in reality the number of photons and detectors is much larger than 1000.

The above is either the standard many worlds interpretation or my wrong idea about the standard many worlds interpretation.

@jartsa: there is only one detector in each branch.
According to the MWI for each registered photon there's a branching such that in every branch one experimental result (either x- or y-polarization) is realized. Let's repeat the experiment N times with N individual photons which results in 2N branches in total.

@jartsa: there is only one detector in each branch.

Ok, we can think that way, but why not think this way:

A quantum wave can be thought as superposition of many identical quantum waves.
A quantum wave of a detector can be thought as superposition of many identical quantum waves of a detector.
A detector can be thought as superposition of many identical detectors.

In the experiment the thicker of the two resulting branches contains 10 times more detectors than the other branch.

So instead of 2 separate branches (photon X/photon Y), we in fact have a gazillion of branches, differing between each other in minute details, like the number and positions of all the electrons knocked out at each stage. Roughly 0.9 gazillion of branches (roughly) will have the photon in state X and another 0.1 gazillion in state Y. If you are in the game of counting branches you should count all these.

That is quite a strong assertion. And it is fundamentally incompatible with the linearity of the state evolution in quantum theory. The weights of the initial superposition cannot have influence on the number of states in either category. Do the math and see for yourself.

Cheers,

Jazz

the ratio of the number of branches is incompatible with our observations.
the ratio of the number of branches is meaningless unless we can say something definitive about their relative probabilities.

I hope you agree that MWI should then provide a means to define branch counting, a probability measure, a derivation of the QM probabilities, or at least a means to assign compatible measures.
Compatible measures are assigned according to the Born rule, just like in any other interpretation. As for the rest, yes, it would be very nice to have, but then it would make MW a theory instead of interpretation.

It seems that you want to hide these problems by introducing rather complex branch structures ...
I didn't mean to suggest any particular branch counting scheme that works. On the contrary I was just trying to show that branch counting is meaningless unless a) you can say something about relative measures of the branches (eg from symmetry considerations) and b) there is an unambiguous way to count the branches.

But where is the error in Delta's solution?

There are many possible errors. First of all, his fine grained counting does not work out because of linearity. That's a well known no-go for MWI and easy to prove. Next, the assumptions underlying MWI can be blatantly wrong. That would include the ontology of the quantum state on a universal level or the unitarity of the evolution. Finally the mechanism of entangling an observer with the observation outcome may not be the dominant mechanism. It may be fragile (think of size of the system that an observer represents. Why could your left eye not entangle with a different result than your right eye?) or work only on a greater time scale than a different mechanism that is really responsible for "measurement", and so never become relevant. The ways in which Everett can go wrong are plenty. The statement "If you are a realist you have to be an Everettian" is also wrong. I am a realist, and I'm not an Everettian. And I can give very good reasons for that.

Cheers,

Jazz

the ratio of the number of branches is meaningless unless we can say something definitive about their relative probabilities.

Where would these relative probabilities come from in a framework where everything should emerge? The basic idea of MWI is that you can explain collapse and subjective randomness by looking at a duplicated observer. Imagine you have a xerox machine for people. Make a copy of yourself. Both wake up with your mind and have no good reason for finding themselves in the body the woke up with. Of course an ensemble of such experiments would yield equal outcome counts for both options.

The only way to change the probability weights of the outcome is to split the different paths asymmetrically. Like you did in your first example, we just keep subdividing one branch and leave the other alone. Or we kill one of the individuals in one outcome.

In MWI each branch contains a functional copy of the observer. The relative amplitude of the copy does not affect his consciousness or anything in his branch, because of the linearity of the evolution. So we also have to count copies, without looking at amplitudes.

Unfortunately the asymmetric branching that could give us different probability weights for each branch cannot possibly depend on the original weights of the superposition, again because of linearity. So any "natural" emergence of probabilities, using the same mechanism that MWI itself claims solves the subjective collapse problem, does not result in the Born rule.

Compatible measures are assigned according to the Born rule, just like in any other interpretation. As for the rest, yes, it would be very nice to have, but then it would make MW a theory instead of interpretation.

For the reasons I just gave, this is not really an option. Either you give a mechanism for emergence ,or you don't. Half-way is not a solution. And just for the record, Everett DID present MWI as a theory, not an interpretation. And that's exactly what we need, a theory of observation in quantum theory.

Cheers,

Jazz

That is quite a strong assertion. And it is fundamentally incompatible with the linearity of the state evolution in quantum theory. The weights of the initial superposition cannot have influence on the number of states in either category.

Well, yes and no. I certainly understand the implications of linearity. But branching is an emergent phenomenon and the keyword here is FAPP (For All Practical Purposes only). Branches diverge when decoherence reduces interference terms to 0. However these terms can never become precisely equal to 0 (that would contradict the linearity) but only FAPP. This is all very well at macro level where the division into branches is obvious and clear-cut, but as you go down considering more and more minute details of the interaction it becomes less and less clear. At some point it becomes a judgment call whether particular branches have diverged irreversibly or there is still possibility of interference. The usual way to deal with this situation it is to introduce a cut-off, then move it up and down and see how it affects things. Obviously, once the cut-off is in place, linearity goes out of the window.

Again I'm not proposing this as a definite method for obtaining probabilities from counting branches but I hope something along these lines might work eventually.

Delta Kilo, the idea is very simple. The simple branches are
1) "x" and "y"
2) "xx", "xy", "xy", "yy"
...
It is not that simple. Imagine a modified setup: if the measurement is "y" (and only then), you send another photon in a different setup through a polarizer, where you get a and b with the same rate. Now you get:

1) "x", "ya" and "yb"
2) "xx", "xya" and "xyb", "yax" and "ybx", "yaya" and "yayb", "ybya" and "ybyb" (9 results: 1 xx, 4 xy/yx, 4 yy)

Does that change the "probabilities" (what does that even mean in a deterministic interpretation?) of x and y? That would mean you could influence the probabilities of x and y after you got the measurement! Simple counting does not work.
And the way to avoid that is the squared amplitude as measure for the branches. There are

branches which see "wrong" (very "unlikely") results. That is equivalent to probabilistic interpretations, where we get "wrong" (very "unlikely") results with a small probability. You always have them, or the probability to get them, in every interpretation.

There are branches which see "wrong" (very "unlikely") results. That is equivalent to probabilistic interpretations, where we get "wrong" (very "unlikely") results with a small probability. You always have them, or the probability to get them, in every interpretation.

Exactly. That's why I think it's misguided to try to prove the correctness of the Born measure. You can't prove that relative frequencies will obey the Born rule for probabilities, because there are possible worlds in which they don't. You might be able to show that the set of possible worlds that DON'T obey the Born rule has a very small probability, but then you need a notion of probability for sets of possible worlds. How do prove that that probability is correct? You can't rely on relative frequency for THAT.

It seems to me that there is no such thing as proof of the correctness of probabilistic predictions. The best you can hope for is to have an unproved conjecture about the probabilities for events, and then demonstrate empirically that relative frequencies are observed to agree with these predictions. There is, in addition, a self-consistency constraint, which is that if you have a notion of measure for sets of possible worlds, in addition to probabilities for events, then you need to prove that the set of worlds where relative frequencies don't approach the predicted probabilities is of measure zero. (Or at least, very unlikely, according to your measure.) But there is no noncircular way to prove the correctness of your measure, it seems to me.

What some people have claimed is that the self-consistency constraint is so severe, in the case of quantum mechanics, that only the Born measure satisfies it. I'm not sure about those proofs, but I've never seen an alternative measure that has any kind of plausibility when applied to quantum mechanics.

The basic idea of MWI is that you can explain collapse and subjective randomness by looking at a duplicated observer.
I thought the basic idea was to apply SE to Shroedinger's Cat and take the absurdly-sounding result "superposition of dead and alive" for a face value. We then ask "what does it mean for a cat to be in superposition"? As it turns out that, the wavefunction of the cat + the box + the researcher who saw the cat + the lab + the whole campus etc etc quickly evolves into a superposition of two separate non-interacting and internally consistent solutions. Each solution describes a researcher who sees a cat in a definitive state and the rest of the world (eyewitness reports, recording devices) being entirely consistent with it. This pretty much answers the question "what is it like to be in superposition": you only ever see 1 definite outcome, which is consistent with our everyday experience. The next step is to accept the conclusions and to assume that both solutions indeed exist in superposition. The rest just follows.

Either you give a mechanism for emergence ,or you don't. Half-way is not a solution. And just for the record, Everett DID present MWI as a theory, not an interpretation. And that's exactly what we need, a theory of observation in quantum theory.
Well, as far as I know, MWI is generally considered an interpretation of QM and not a theory in its own right. MWI does not include anything beyond standard QM. Any prediction of MWI can be made using any other interpretation of QM. The recipe is simple: you just move the boundary between micro and macro all the way up and treat everything (including the apparatus, observer and the environment) quantum-mechanically. So if Born rule is somehow derived in the context of MWI, it would automatically apply to all other interpretations, which would be a good thing for everybody but would not prove anything special for MWI.

Well, yes and no. I certainly understand the implications of linearity. But branching is an emergent phenomenon and the keyword here is FAPP (For All Practical Purposes only). Branches diverge when decoherence reduces interference terms to 0. However these terms can never become precisely equal to 0 (that would contradict the linearity) but only FAPP.

This is quite certainly not accurate. Linearity implies continuity, and if you can arbitrarily close to 0 then you can also reach zero. If anything it would violate the orthogonality of states, but not even that is true. The constraint you refer to is the time independence of the evolution generator in a finite dimensional hilbert space. But it already disappears if you allow for infinite dimensions.

This is all very well at macro level where the division into branches is obvious and clear-cut, but as you go down considering more and more minute details of the interaction it becomes less and less clear. At some point it becomes a judgment call whether particular branches have diverged irreversibly or there is still possibility of interference. The usual way to deal with this situation it is to introduce a cut-off, then move it up and down and see how it affects things. Obviously, once the cut-off is in place, linearity goes out of the window.

A slightly stronger version of the no-go theorem following from linearity also excludes a solution like that.

Cheers,

Jazz

Exactly. That's why I think it's misguided to try to prove the correctness of the Born measure. You can't prove that relative frequencies will obey the Born rule for probabilities, because there are possible worlds in which they don't. You might be able to show that the set of possible worlds that DON'T obey the Born rule has a very small probability, but then you need a notion of probability for sets of possible worlds. How do prove that that probability is correct? You can't rely on relative frequency for THAT.

That problem only appears if you try to count events along a single line of history. That's not what is usually done. We have the alternative of counting over the branching ensemble of a single event. For that the born rule must be accurate, because all options are realized independently. So the scenario you describe is not an issue.

Cheers,

Jazz

I thought the basic idea was to apply SE to Shroedinger's Cat and take the absurdly-sounding result "superposition of dead and alive" for a face value. We then ask "what does it mean for a cat to be in superposition"?
...
The next step is to accept the conclusions and to assume that both solutions indeed exist in superposition.

This is all correct and not in disagreement with what I wrote. Still, the primary explanation of the collapse and subjective randomness is based on the notion of finding "copies" of the observer in different branches. So my explanation is according to the standard understanding of MWI.

The rest just follows.

It would be interesting to understand exactly, what you mean with "the rest".

The recipe is simple: you just move the boundary between micro and macro all the way up and treat everything (including the apparatus, observer and the environment) quantum-mechanically. So if Born rule is somehow derived in the context of MWI, it would automatically apply to all other interpretations, which would be a good thing for everybody but would not prove anything special for MWI.

No, other interpretations only mix badly in terms of ontology with MWI. In fact, most interpretations are fundamentally incompatible regarding their understanding of the meaning of the quantum state and measurement. Even if you could derive the Born rule in an Everett context that would mean nothing for Bohmian mechanics or transactional QT. I admit that it would give MWI a better position in comparison with the other interpretations and would make it preferable. I would even go so far to say it would make MWI a valid theory of quantum measurement and render "interpretations" more or less irrelevant. But we're far from that.

Cheers,

Jazz

We have the alternative of counting over the branching ensemble of a single event.
How do you define a "single event" in the real world, where decoherence happens everywhere all the time, and for all those events parallel?

How do you define a "single event" in the real world, where decoherence happens everywhere all the time, and for all those events parallel?

You don't have to do that in a real world. Instead you can fabricate your own hypothetical setup and describe it mathematically. A single event would then be any reasonably isolated process with temporally asymptotic stable input and output states. If the Born rule emerges, that emergence must be reducible to some such system and can be studied there.

Cheers,

Jazz

the ratio of the number of branches is meaningless unless we can say something definitive about their relative probabilities.

...

On the contrary I was just trying to show that branch counting is meaningless unless a) you can say something about relative measures of the branches (eg from symmetry considerations) and b) there is an unambiguous way to count the branches.
The problem is that as long as you are not able to say anything about branch counting, measures etc., you have gained nothing.

Let's come back to my experiment. It is clear that changing the superposition of the polarization and their Born probabilities must somhow affect the branching. What has the MWI to say about that?

That problem only appears if you try to count events along a single line of history. That's not what is usually done. We have the alternative of counting over the branching ensemble of a single event. For that the born rule must be accurate, because all options are realized independently. So the scenario you describe is not an issue.

Cheers,

Jazz

I don't think it matters how you do the counting. The fact is that the only information we have is about what happens along a single line of history. That doesn't logically tell us anything about the Born rule unless we assume that our particular history is "typical" in some sense. But the notion of "typical" is theory-dependent.

Simple counting does not work.
I fully agree. I started with a very simple counter example, but of course there are more sophisticated examples.

And the way to avoid that is the squared amplitude as measure for the branches.
What would that mean for the branch structure in my example?

I don't think it matters how you do the counting. The fact is that the only information we have is about what happens along a single line of history. That doesn't logically tell us anything about the Born rule unless we assume that our particular history is "typical" in some sense. But the notion of "typical" is theory-dependent.

You don't have to make any such assumption if you can show that the Born rule holds for the ensemble of observations following from a single event. Because that already implies that we are in a "typical history" with overwhelming likelihood if enough such events are concatenated.

Cheers,

Jazz