Can the Born Rule Be Derived in the Many Worlds Interpretation?

[bhobba]

So the probability distribution is "entangled".

Standard proability theory does not exhibit entanglement - it fact QM's ability to do that is what distinguishes it from probability theory:
http://arxiv.org/pdf/0911.0695v1.pdf

Thanks
Bill

 

[stevendaryl]

I didn't say that reality is a certain way because of science. I only said that a scientific theory needs to be falsifiable. It's the idea that a good theory has to be more than that that's wishful thinking.

Well, in a certain sense, it's wishful thinking that it's possible to do science, at all. There is no reason for the universe to be predictable or comprehensible in any way. The idea behind science is that we optimistically hope that things are comprehensible. There is no reason for them to be.

You are making the distinction between science as a means of making predictions, and science as something more than that. You're calling the second "wishful thinking". But both are wishful thinking, in the sense that there is no necessary reason for nature to be humanly comprehensible, or predictable, at all.

 

[atyy]

(I think, the assumption of a spontaneous collapse, which is outside of the dynamics of quantum theory is inconsistent with Einstein causality as is a "cut" between a quantum and a classical world; everything is quantum, the appearance of a classical behavior is due to a coarse grained observation of macroscopic observables of objects of macroscopic scales and well-understood from quantum-many body theory). If you assume that it's Alice's measurement which causes Bob's positron to spontaneously get a determined spin-z component, you indeed violate Einstein causality, because no signal can travel faster with the speed of light to make Bob's spin determined although initially it was completely indetermined. Within the minimal interpretation, there is no problem, because you take the Born interpretation of states really seriously, i.e., before Alice's measurement the spin-z component of both the electron and the positron were (even maximally) undetermined, but due to the preparation in an entangled state, the correlations are already implemented when the electron-positron pair were prepared. Of course, such a thing is not describable with local deterministic hidden-variable theories, and as long as nobody finds a consistent non-local deterministic theory which is as successful as QT, I stick to (minimally interpreted) QT :-).

If everything is quantum and there is only unitary evolution, then there would be unitary evolution of the wave function of the universe. The "minimal interpretation" without collapse and without a classical quantum cut is not minimal - it is making a huge claim - that the unitary evolution of the wave function of the universe makes sense. If this were true, the minimal interpretation would have solved the problem that the many-worlds approach investigates.

 

[stevendaryl]

Standard provability theory does not exhibit entanglement - it fact QM's ability to do that is what distinguishes it from probability theory:
http://arxiv.org/pdf/0911.0695v1.pdf

Thanks
Bill

Maybe I'm using the wrong word, but it seems to me that in QM, when we say that two particles are entangled, what we mean is that the composite state is not expressible as a product of one-particle states. That concept has a direct analogy in classical probability theory: you have a probability distribution describing a composite system that cannot be expressed as a product of probability distributions of the component systems.

What I would say is different about quantum mechanics is not entanglement, but the fact that it is not possible to understand entanglement as lack of information about an unknown un-entangled state.

This is really the basis of Bell's inequality. We start with a joint probability distribution for Alice and Bob:

P(R_A, R_B | \alpha, \beta)

(The probability that Alice gets result R_A and Bob gets result R_B, given that Alice performs measurement \alpha and Bob performs measurement \beta).

There is a special class of joint probability distributions, the "factorable" ones, that can be written as follows:

P(R_A, R_B | \alpha, \beta) = P_A(R_A | \alpha) P_B(R_B | \beta)

I was using the word "entangled" to mean any joint probability distribution that cannot be factored that way.

A fact about classical joint probabilities, if there is no causal influence between the two measurements, is that even when the probabilities don't factor, they can be understood in terms of lack of information about factorable distributions. That is, there is some more detailed description of the probabilities as follows:

P(R_A, R_B | \alpha, \beta) = \sum_\lambda P_C(\lambda) P_A(R_A | \alpha, \lambda) P_B(R_B | \beta, \lambda)

In other words, classically, we can always find some fact, represented in the formula by the value of the parameter \lambda such that if we knew that fact, we could then factor the joint probability distributions for distant, causally disconnected measurements.

I believe that the use of the word "entangled" in QM is such that it always means a composite state that cannot be factored into a product of component states.

 

[stevendaryl]

I believe that the use of the word "entangled" in QM is such that it always means a composite state that cannot be factored into a product of component states.

Actually, immediately after writing that, I realized that that's slightly wrong. For a pair of particles, the state is ALWAYS entangled in this sense, because of Bose or Fermi statistics. That is, if I have a two-electron state |\Psi \rangle, I can never write it in a product form |\Psi\rangle = |\phi\rangle |\psi\rangle, because Fermi statistics require that the state be anti-symmetric under swapping the two electrons. So the closest I can get to a product state is one of the form:

|\Psi\rangle = \sqrt{\dfrac{1}{2}}(|\phi\rangle|\psi\rangle - |\psi\rangle|\phi\rangle

So it looks like I have to amend my definition of an "entangled" to mean a state that cannot be expressed as the symmetrization of a product of component states. That's kind of a messy definition, I know.

 

[stevendaryl]

Standard proability theory does not exhibit entanglement - it fact QM's ability to do that is what distinguishes it from probability theory:
http://arxiv.org/pdf/0911.0695v1.pdf

Thanks
Bill

Actually, they are using almost the same definition of "entangled" that I am using:

We call the pure state entangled if it is not a product state.

I was using an analogous definition to distinguish between entangled and un-entangled classical probability distributions: A probability distribution is entangled if it is not a product of component probability distributions.

The difference in terminology is that the authors only apply the word "entangled" to pure states, not mixtures. But that makes the notion of "entangled" not just false for classical probability, but meaningless (or maybe trivial). The only pure states in classical probability theory are states where all probabilities are 0 and 1. It's pretty obvious that you can't have entangled pure states using only probabilities 0 and 1.

 

[Fredrik]

You are making the distinction between science as a means of making predictions, and science as something more than that. You're calling the second "wishful thinking". But both are wishful thinking, in the sense that there is no necessary reason for nature to be humanly comprehensible, or predictable, at all.

It would be wishful thinking to believe that science can find all the answers, but I haven't advocated that view. There's no wishful thinking involved in thinking that QM makes very accurate predictions about results of experiments, or in explaining what a theory must do in order to be falsifiable. But there's wishful thinking involved in thinking that QM must be "more than that" (in the sense discussed above). It doesn't bother me that people are interested in exploring that option too, but it should be viewed as a long shot.

 

[stevendaryl]

It would be wishful thinking to believe that science can find all the answers, but I haven't advocated that view. There's no wishful thinking involved in thinking that QM makes very accurate predictions about results of experiments, or in explaining what a theory must do in order to be falsifiable. But there's wishful thinking involved in thinking that QM must be "more than that" (in the sense discussed above). It doesn't bother me that people are interested in exploring that option too, but it should be viewed as a long shot.

But using the word "wishful thinking" is just not helpful. Given any unsolved problem, it's wishful thinking to a certain extent to believe that we will ever solve it. So whether something is wishful thinking is not much of a guide to what we should be working on in science.

 

[stevendaryl]

But using the word "wishful thinking" is just not helpful. Given any unsolved problem, it's wishful thinking to a certain extent to believe that we will ever solve it. So whether something is wishful thinking is not much of a guide to what we should be working on in science.

What's different about the mysteries of quantum mechanics is not whether it's wishful thinking. It's really the (almost) complete lack of progress, and (almost) complete lack of any hints as to where a solution might be found. People give up because they are tired of beating their heads against a wall. I think it's something akin to "sour grapes" to retroactively adjust your view of what science is all about so that the problems that you have no idea how to solve are excluded as not really science, in the first place. I guess there is a practical reason for drawing such a boundary, and to consider something a scientific question if there is some hope of answering it. But in lots of cases, the only way we know whether there is any hope of answering something is by trying, and either succeeding or failing.

 

[Fredrik]

If everything is quantum and there is only unitary evolution, then there would be unitary evolution of the wave function of the universe.

I assume that what you mean by "everything is quantum" is that every physical system is such that a pure state (a mathematical thing) can represent what you previously called the system's "real state" (a real-world thing). Since the universe is a physical system, it follows that we can assign a state to the universe. But to me, "everything is quantum" just means that there's no experiment in which QM will not work, and that doesn't imply that we can assign a state to the universe.

The "minimal interpretation" without collapse and without a classical quantum cut is not minimal - it is making a huge claim - that the unitary evolution of the wave function of the universe makes sense. If this were true, the minimal interpretation would have solved the problem that the many-worlds approach investigates.

If someone who advocates a minimal interpretation disagrees with this, it's not because they're making some huge assumption. It's because they disagree with you about the meaning of concepts like "collapse" or "classical/quantum cut", as I did above.

 

[atyy]

I assume that what you mean by "everything is quantum" is that every physical system is such that a pure state (a mathematical thing) can represent what you previously called the system's "real state" (a real-world thing). Since the universe is a physical system, it follows that we can assign a state to the universe. But to me, "everything is quantum" just means that there's no experiment in which QM will not work, and that doesn't imply that we can assign a state to the universe.

In your minimal interpretation, does the universe have a "real state"?

If someone who advocates a minimal interpretation disagrees with this, it's not because they're making some huge assumption. It's because they disagree with you about the meaning of concepts like "collapse" or "classical/quantum cut", as I did above.

I was replying to vanhees71 there, not to you, because I am not sure that your and vanhee71's idea of a "minimal interpretation" are the same. For example, I am pretty sure that bhobba's ensemble interpretation is not the same as Ballentine's, and there is no substantial disagreement between his Ensemble interpretation and Copenhagen. So far, I am not sure whether you and I disagree about the meaning of a "classical/quantum cut" and "collapse", maybe just the naming of the concept.

Edit: bhobba's Ensemble interpretation differs from Ballentine's because bhobba explicitly acknowledges as axioms a classical/quantum cut, and the equivalence of proper and improper density matrices. That's why I believe bhobba's interpretation makes sense, while Ballentine's is misleading or wrong.

 

[stevendaryl]

In your minimal interpretation, does the universe have a "real state"?

I can let Fredrick answer for himself, but I certainly wouldn't call any assumption about a "real state" part of a minimal interpretation. What I think of the minimal interpretation is purely an input/output relation: Set up the initial conditions, let things evolve, make a measurement. Quantum mechanics gives you the probability for each possible output (measurement results) as a function of the input (initial setup). That's minimal in that you don't need to assume anything else in order to apply QM.

I guess what's not minimal about this minimal interpretation is that it assumes that the input and output can be understood in pre-quantum terms.

 

[stevendaryl]

Edit: bhobba's Ensemble interpretation differs from Ballentine's because bhobba explicitly acknowledges as axioms a classical/quantum cut, and the equivalence of proper and improper density matrices. That's why I believe bhobba's interpretation makes sense, while Ballentine's is misleading or wrong.

I'm a little uncomfortable with the ensemble interpretation, in that it seems to me that there is an element of pretense involved. After decoherence, you perform a trace over unobservable environmental degrees of freedom, and then what's left is a density matrix that looks like a mixed state. Then you can go on to pretend that this mixed state represents an ensemble. But I call it a pretense, because you know that really, pure states never evolve into mixed states. You're pretending it's a mixed state so that you can give an ensemble interpretation.

 

[atyy]

I'm a little uncomfortable with the ensemble interpretation, in that it seems to me that there is an element of pretense involved. After decoherence, you perform a trace over unobservable environmental degrees of freedom, and then what's left is a density matrix that looks like a mixed state. Then you can go on to pretend that this mixed state represents an ensemble. But I call it a pretense, because you know that really, pure states never evolve into mixed states. You're pretending it's a mixed state so that you can give an ensemble interpretation.

It's fine, because once you make the classical/quantum cut, you acknowledge that it is all pretense - or in more conventional language - quantum mechanics is an instrumental theory and only tells us how to predict the outcomes of measurements, where a measuring device is a fundamental concept.

ie: it's fine, because we acknowledge the problems (or limitations) upfront.

 

[vanhees71]

Well, I think this threads, proves my hypothesis that there are as many interpretations of quantum theory as physicists using it ;-)).

The question, if there exists a (pure or mixed) state of the whole universe, of course, is a challenge to the ensemble representation, because you cannot prepare an ensemble of universes, because there is only one (except you adhere to some "parallel universes" picture, which in my opinion is unscientific, because by definition, you cannot observe these parallel universes at all).

I don't think that the notion of a quantum state of the entire universe makes sense, because a probabilistic description can only be checked by doing measurements of an ensemble of independently and equally prepared setups of a system, and that cannot be done.

 

[stevendaryl]

Well, I think this threads, proves my hypothesis that there are as many interpretations of quantum theory as physicists using it ;-)).

The question, if there exists a (pure or mixed) state of the whole universe, of course, is a challenge to the ensemble representation, because you cannot prepare an ensemble of universes, because there is only one (except you adhere to some "parallel universes" picture, which in my opinion is unscientific, because by definition, you cannot observe these parallel universes at all).

I don't think that the notion of a quantum state of the entire universe makes sense, because a probabilistic description can only be checked by doing measurements of an ensemble of independently and equally prepared setups of a system, and that cannot be done.

As I said in another post, people can certainly apply physics to the early universe where there were no observers or measurement devices. Of course, measurements and observations are critical in testing theories of science, but the theories themselves have a usefulness beyond testability. We can use physics for reasoning about "what-if" scenarios: What if the matter in the universe were arranged in perfect spherical symmetric? What would the gravity be like? What if the universe were filled with noninteracting dust? What if all the mass in a star were concentrated into a volume of say 60 cubic kilometers?

Saying that it's not science if there are no observers or measurement devices makes for an overly constrained notion of what counts as science.

 

[atyy]

Well, I think this threads, proves my hypothesis that there are as many interpretations of quantum theory as physicists using it ;-)).

The question, if there exists a (pure or mixed) state of the whole universe, of course, is a challenge to the ensemble representation, because you cannot prepare an ensemble of universes, because there is only one (except you adhere to some "parallel universes" picture, which in my opinion is unscientific, because by definition, you cannot observe these parallel universes at all).

I don't think that the notion of a quantum state of the entire universe makes sense, because a probabilistic description can only be checked by doing measurements of an ensemble of independently and equally prepared setups of a system, and that cannot be done.

OK, that makes sense - but in which case why do you object to a classical quantum cut? If there is no wave function of the universe, and quantum mechanics only applies to subsystems of the universe, then the cut between the measuring device and the quantum system is a classical/quantum cut.

The coarse graining doesn't eliminate the cut, because if we take the measuring device and the quantum system as a quantum system, there is nothing to coarse grain it, yet the measuring device is classical. We could coarse grain the measuring device and measured system by extending the quantum boundary once more - but there is a limit to this, since the wave function doesn't apply to the universe. So at some point, in the Ensemble interpretation, you have to make a cut - one can debate where - but there is a cut.

Strictly speaking, I don't think a successful ensemble interpretation can require a real ensemble, because otherwise the calculation of Mukhanov and Chibisov and the test by Planck (or BICEP2?) will not make sense in the ensemble interpretation.

 

[Fredrik]

In your minimal interpretation, does the universe have a "real state"?

It's impossible to assign a state (in the sense of QM) to it.

A more interesting question is if any system has a "real state". This question can be split in two: 1. Does a (pure) state in QM represent a "real state"? 2. If no, does a system have a "real state" at all?

If we are to remain truly minimal, we should leave question 1 unanswered. But since "yes" is the starting point of a many-worlds interpretation, most "minimalists" are probably thinking that the answer is probably "no".

Question 2 is of course impossible to answer without a better theory to replace QM, but it's interesting to think about it. I'm thinking that systems probably do have "real states", and that a "theory" that can describe them may have some very undesirable features. It might describe what's going on in terms of things that are unobservable in principle, and be falsifiable only in the sense that under certain conditions, it makes essentially the same predictions as QM.

I was replying to vanhees71 there, not to you, because I am not sure that your and vanhee71's idea of a "minimal interpretation" are the same. For example, I am pretty sure that bhobba's ensemble interpretation is not the same as Ballentine's, and there is no substantial disagreement between his Ensemble interpretation and Copenhagen.

Yes, there are varieties, and no standardized terminology. Ballentine doesn't even agree with Ballentine. His 1970 article is assuming that regardless of what the wavefunction is, every particle has a well-defined position at all times. I haven't seen anything like that in his book. My only issue with that part of the book is that his wording makes it sound like he's proving the ensemble interpretation.

Copenhagen is typically defined only to ridicule it, by people who have misunderstood it, so I prefer not to use that term when I can avoid it. I think that a sensible definition of Copenhagen would be identical to a sensible definition of a minimal statistical interpretation.

So far, I am not sure whether you and I disagree about the meaning of a "classical/quantum cut" and "collapse", maybe just the naming of the concept.

Probably just terminology.

 

[Fredrik]

I can let Fredrick answer for himself, but I certainly wouldn't call any assumption about a "real state" part of a minimal interpretation. What I think of the minimal interpretation is purely an input/output relation: Set up the initial conditions, let things evolve, make a measurement. Quantum mechanics gives you the probability for each possible output (measurement results) as a function of the input (initial setup). That's minimal in that you don't need to assume anything else in order to apply QM.

Agreed.

I'm a little uncomfortable with the ensemble interpretation, in that it seems to me that there is an element of pretense involved. After decoherence, you perform a trace over unobservable environmental degrees of freedom, and then what's left is a density matrix that looks like a mixed state. Then you can go on to pretend that this mixed state represents an ensemble. But I call it a pretense, because you know that really, pure states never evolve into mixed states. You're pretending it's a mixed state so that you can give an ensemble interpretation.

Pure states never evolve into mixed states under unitary time evolution (i.e. the Schrödinger equation), but only isolated systems evolve that way. If such a system has two interacting subsystems, the only way to assign states to them is through the partial trace operation, and when you do, you will find that a pure state (of a subsystem that isn't isolated) does evolve into a mixed state. Further, this evolution is irreversible in the sense that it can't be reversed by unitary evolution of that subsystem alone.

At least that's my naive understanding of the methods of positive operator valued measures and similar techniques that I only recently began to look at. So far I have only skimmed Flory's article "POVMs and superoperators" (I can't find it online...weird), and read a few pages in the book by Busch, Grabowski & Lachti.

 

[stevendaryl]

Pure states never evolve into mixed states under unitary time evolution (i.e. the Schrödinger equation), but only isolated systems evolve that way. If such a system has two interacting subsystems, the only way to assign states to them is through the partial trace operation, and when you do, you will find that a pure state (of a subsystem that isn't isolated) does evolve into a mixed state. Further, this evolution is irreversible in the sense that it can't be reversed by unitary evolution of that subsystem alone.

At least that's my naive understanding of the methods of positive operator valued measures and similar techniques that I only recently began to look at. So far I have only skimmed Flory's article "POVMs and superoperators" (I can't find it online...weird), and read a few pages in the book by Busch, Grabowski & Lachti.

Yes, you're right, tracing produces a mixed state, but the origin of the mixed state is from the fact that you're doing a trace. You might start with a description of a single electron (say), then at some later time, it interacts with the environment, and you do a trace to get a mixed state representation. But the state came from a single electron. It doesn't really represent an ensemble.

 

[kith]

I'm a little uncomfortable with the ensemble interpretation, in that it seems to me that there is an element of pretense involved. After decoherence, you perform a trace over unobservable environmental degrees of freedom, and then what's left is a density matrix that looks like a mixed state. Then you can go on to pretend that this mixed state represents an ensemble. But I call it a pretense, because you know that really, pure states never evolve into mixed states. You're pretending it's a mixed state so that you can give an ensemble interpretation.

At least in Ballentine's book, already pure states are interpreted as referring to ensembles. So in a measurement, the reduced mixed state refers to an ensemble of systems because the pure entangled state refers to an ensemble of apparatuses+systems.

 

[kith]

The coarse graining doesn't eliminate the cut, because if we take the measuring device and the quantum system as a quantum system, there is nothing to coarse grain it, yet the measuring device is classical. We could coarse grain the measuring device and measured system by extending the quantum boundary once more - but there is a limit to this, since the wave function doesn't apply to the universe. So at some point, in the Ensemble interpretation, you have to make a cut - one can debate where - but there is a cut.

But why should this cut be called quantum / classical cut? If you acknowledge that you can move the boundary, how do you verify that the far side of the cut behaves according to classical mechanics? For every possible experiment which investigates something at the far side, you could simply shift the boundary by using the quantum description of this something and you would be in the quantum domain again.

 

[atyy]

But why should this cut be called quantum / classical cut? If you acknowledge that you can move the boundary, how do you verify that the far side of the cut behaves according to classical mechanics? For every possible experiment which investigates something at the far side, you could simply shift the boundary by using the quantum description of this something and you would be in the quantum domain again.

We can call it the Heisenberg cut if you prefer, or the quantum/common-sense reality cut or the quantum/macroscopic cut (or whatever, if it is just a matter of naming).

 

[atyy]

At least in Ballentine's book, already pure states are interpreted as referring to ensembles. So in a measurement, the reduced mixed state refers to an ensemble of systems because the pure entangled state refers to an ensemble of apparatuses+systems.

The ensemble interpretation doesn't solve allow one to derive that proper and improper mixed states are the same. It must be postulated, which is equivalent to postulating collapse.

A proper mixed state is when Alice makes Ensemble A in pure state |A> and Ensemble B in pure state |B>, then she makes a Super-Ensemble C consisting of equal numbers of members of Ensemble A and Ensemble B. If she hands me C without labels A and B, I can use a mixed density matrix to describe the statistics of my measurements on C. But if in addition I receive the labels A and B, then I can divide C into two sub-ensembles, each with its own density matrix, since C was just a mixture of A and B. Here C is a "proper" mixture, which can be naturally divided into sub-ensembles.

An improper mixed state is when I have an ensemble C in a pure state, each member of which consists of a subsystem A entangled with subsystem B. If I do a partial trace over B, I get a density matrix (the reduced density matrix) which describes the statistics of all measurements that are "local" to A. This reduced density matrix for A is not a pure state, and is an "improper" mixed state. There is no natural way to partition this into sub-ensembles, since there is only one ensemble C.

 

[Jilang]

We can call it the Heisenberg cut if you prefer, or the quantum/common-sense reality cut or the quantum/macroscopic cut (or whatever, if it is just a matter of naming).

Why is is not referred to as the "no going back due to irreversible increase in entropy" cut?

 

[atyy]

Why is is not referred to as the "no going back due to irreversible increase in entropy" cut?

I think Weinberg's term is the best "common sense reality", which I notice bhobba has also adopted.

Technically, entropy can be defined on a quantum system. There is the entropy of a mixed state. There is even the entanglement entropy of a subsystem of a pure state. There are attempts to show that the second law of thermodynamics can be derived from an increase in entanglement emtropy. So I think within a Copenhagen/Ensemble interpretation where there is a cut, I would reserve the word "entropy" for something else.

In a MWI approach there is no cut, so the apparent cut would be derived from decoherence which I think leads to increased entropy in the subsystem via entanglement.

 

[atyy]

Question 2 is of course impossible to answer without a better theory to replace QM, but it's interesting to think about it. I'm thinking that systems probably do have "real states", and that a "theory" that can describe them may have some very undesirable features. It might describe what's going on in terms of things that are unobservable in principle, and be falsifiable only in the sense that under certain conditions, it makes essentially the same predictions as QM.

Ok, I think I agreed with everything you said in that post, so let me just try to use this bit to say what the measurement problem is in these terms. If we believe a theory beyond QM is possible in principle, and that such a theory has "real states" in principle, can we show that this possibility exists in principle? Historically, the problem arose because of von Neumann's erroneous proof that such a theory cannot exist even in principle. The achievement of Bohm was to providing a concrete example that the proof was wrong, ie. that there is no way to correct von Neumann's proof to make it right.

A Bohmian-type view even supports your intuition that such a theory might have very undesirable features, explaining why we prefer to use QM in practice as long as experiments allow. For example, Montina showed that "any ontological Markovian theory of quantum mechanics requires a number of variables which grows exponentially with the physical size." http://arxiv.org/abs/0711.4770

However, although a Bohmian-type theory is unwieldy and under-constrained without experimental input, it can be falsified in a sense that goes beyond reproducing the predictions of QM. This is because BM says that QM is a "quantum equilibrium" situation, and to fully solve the measurement problem BM has to postulate that at some point there was "quantum nonequlibrium", and that in principle there are experiments that will show QM to be an incorrect description of the universe.

So in a sense, the measurement problem is to show that "real states" can exist, and to construct some possibilities. BM constructs some possibilities by adding things, MWI tries to construct it by removing things.

 

[kith]

We can call it the Heisenberg cut if you prefer, or the quantum/common-sense reality cut or the quantum/macroscopic cut (or whatever, if it is just a matter of naming).

Most of these expressions suggest that there's a domain where QM is valid and a domain where QM is wrong and classical mechanics is valid instead. My point is that this statement can't be justified if there is no definitive limit to shifting the boundary.

 

[kith]

There is no natural way to partition this into sub-ensembles, since there is only one ensemble C.

If we want to perform an experiment on a sub-ensemble, we select the sub-ensemble in a physical way (by blocking one beam in a SG apparatus for example). This way obviously depends on the experimental setting. Why do we need an additional, "natural" way to partition the ensemble?

 

[atyy]

Most of these expressions suggest that there's a domain where QM is valid and a domain where QM is wrong and classical mechanics is valid instead. My point is that this statement can't be justified if there is no definitive limit to shifting the boundary.

Yes, it doesn't mean that. It means that every user of quantum mechanics known so far must make this cut, and have as a fundamental notion a measuring device that registers a macroscopic mark. MWI tries to make it such that that statement might be false for future users of QM.