The Born Rule in Many-Worlds

[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.

 

[atyy]

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?

In an improper mixture, there is no notion of a sub-ensemble. Since the experiment clearly shows that sub-ensembles exist, you have to add the notion of a sub-ensemble to the improper mixture. Adding this notion is the statement that an improper mixture can be treated like a proper mixture, or collapse.

Edit: One can have sub-ensembles for the pure state ensemble if one assumes hidden variables. But if one does this then the interpretation is not minimal. At any rate, at this point one must add something: equivalence of proper and improper mixtures, collapse, or hidden variables in order to define the notion of a subensemble.

Edit: Once a Heisenberg cut has been made, and if one is agnostic about the reality of the wave function, then there is no problem with collapse, since you are just collapsing an unreal thing.

 

[Fredrik]

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.

I've been wondering what you guys meant by proper and improper mixed states. The distinction and terminology seem odd to me. The distinction only makes sense to someone who believes that a pure state represents the system's "real state". (Such of person is an MWI-advocate, whether they understand it or not). Such a person would say that a mixed state is only used when we don't know what the correct pure state is.

Regarding the terminology, it seems to make at least as much sense to call the first kind "improper" and the other "proper", because the first kind involves a degree of ignorance that's been introduced artificially, and the second kind involves something fundamentally unknowable. But I guess the terms "proper" and "improper" are only supposed to be labels anyway. The terms might as well be "blue" and "green".

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.

It doesn't allow you to say that they're not the same. So the "postulate" would have to be implicit in the complete lack of additional postulates on top of QM.

You appear to be saying that even the minimal ensemble/statistical/Copenhagen interpretation automatically (implicitly) includes collapse. Since you're saying that without explaining what you mean by "collapse", it looks like you're referring to an exact collapse, which requires modifications of the theory. In that case, I strongly disagree. "Collapse" in a minimal ensemble/statistical interpretation is just decoherence, and that's of course included, since the "minimal interpretation" isn't really an interpretation. It's just QM without unnecessary assumptions.

 

[Jilang]

I am sure that I must be missing something obvious here. My understanding is the wave function evolution is a reversible process. When there is some sort of event that increases entropy would that not make the evolution non-reversible? Isn't that the cut?

 

[stevendaryl]

I am sure that I must be missing something obvious here. My understanding is the wave function evolution is a reversible process. When there is some sort of event that increases entropy would that not make the evolution non-reversible? Isn't that the cut?

There is never a precise, objective moment where anything irreversible happens. It's just that as time goes on, and a particle interacts with more and more particles, the practical possibility of reversing the interaction drops exponentially fast.

 

[atyy]

I've been wondering what you guys meant by proper and improper mixed states. The distinction and terminology seem odd to me. The distinction only makes sense to someone who believes that a pure state represents the system's "real state". (Such of person is an MWI-advocate, whether they understand it or not). Such a person would say that a mixed state is only used when we don't know what the correct pure state is.

Regarding the terminology, it seems to make at least as much sense to call the first kind "improper" and the other "proper", because the first kind involves a degree of ignorance that's been introduced artificially, and the second kind involves something fundamentally unknowable. But I guess the terms "proper" and "improper" are only supposed to be labels anyway. The terms might as well be "blue" and "green".

Yes, "proper" and "improper" are just labels. The definition does assume that a pure state represents the maximum information one can have about a quantum system, without recourse to hidden variables. The pure state is privileged because it obeys unitary Schroedinger evolution. Yes, this is a bit like a secret Many-Worlds. But it isn't technically, because there is a classical/quantum cut, and collapse. The proper mixed state, being constructed from pure states, is also privileged because each component in the proper mix obeys Schroedinger evolution. The improper mixed state is not, because its evolution is governed by Schroedinger evolution of the pure state of the total system, and the subsystem does not usually evolve by Schroedinger evolution.

It doesn't allow you to say that they're not the same. So the "postulate" would have to be implicit in the complete lack of additional postulates on top of QM.

You appear to be saying that even the minimal ensemble/statistical/Copenhagen interpretation automatically (implicitly) includes collapse. Since you're saying that without explaining what you mean by "collapse", it looks like you're referring to an exact collapse, which requires modifications of the theory. In that case, I strongly disagree. "Collapse" in a minimal ensemble/statistical interpretation is just decoherence, and that's of course included, since the "minimal interpretation" isn't really an interpretation. It's just QM without unnecessary assumptions.

Yes, I am saying that the minimal ensemble/statistical/Copenhagen interpretation must explicitly or implicitly include collapse or an equivalent axiom in order be considered correct quantum mechanics. Actually, Copenhagen explicitly includes collapse, which is the Born rule in the form that the probability to observe a state |a> given that the system is in state |ψ> is |<a|ψ>|².

The Ensemble interpretation without collapse usually says that the probability to observe the eigenvalue corresponding to state |a> given that the system is in state |ψ> is |<a|ψ>|². Thus this form of the Born rule without collapse doesn't give you the probability of the sub-ensembles that are formed. Yet we know that the probability of obtaining sub-ensemble |a> after a measurement is |<a|ψ>|². The Born rule without collapse is unable to make this prediction.

Incidentally, I should say that one who thinks that collapse or an equivalent axiom is not needed, and that only decoherence is needed is also secretly a Many-Worlds advocate, because it is trying to do everything with unitary evolution of a pure state.

 

[kith]

In an improper mixture, there is no notion of a sub-ensemble. Since the experiment clearly shows that sub-ensembles exist [...]

I don't think that these statements are obviously true. I don't see a quick way to resolve this, so... what is the definition of a sub-ensemble?

/edit: maybe this belongs in an own thread

 

[atyy]

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.

I don't think that these statements are obviously true. I don't see a quick way to resolve this, so... what is the definition of a sub-ensemble?

/edit: maybe this belongs in an own thread

Yes, what is a sub-ensemble? Maybe there are several ways to do this.

1. The way I did it above, I only defined ensemble. Then I defined a super-ensemble for a proper mixed state. So a sub-ensemble is an ensemble that is part of a super-ensemble, which leaves the notion of sub-ensemble for a pure state undefined, and the corresponding problem for an improper mixed state.

2. The other way of doing it is to say that for a pure state ensemble |a> each individual of the ensemble has a hidden variable x, so that the state is really (|a>, x), with a conventional probability distribution over x. Since there is a conventional probability distribution here, the ensemble |a> can be divided into physical sub-ensembles. We know that x must be a "Bohmian" hidden variable, and cannot be a quantum variable, because if the quantum state |a>=|b1>+b|2>, and we measure in the B basis, the sub-ensembles will be in definite states of b, even though |a> could not have been drawn from a distribution over b1 and b2. Another way of saying that the hidden variables cannot be quantum variables is that for a generic pure state, there is no classical probability distribution over [x,p], since the Wigner function has negative bits.

In option 1, since the sub-ensemble for a pure state is undefined, to derive a sub-ensemble we have to add the postulate by hand. In option 2, the sub-ensembles are defined by "Bohmian" hidden variables and even a pure state corresponds to a conventional probability distribution, so we can form the sub-ensembles. But option 2 is not minimal.

Is there another way?

 

[kith]

Yes, what is a sub-ensemble? Maybe there are several ways to do this.

1. The way I did it above, I only defined ensemble. Then I defined a super-ensemble for a proper mixed state. So a sub-ensemble is an ensemble that is part of a super-ensemble, which leaves the notion of sub-ensemble for a pure state undefined, and the corresponding problem for an improper mixed state.

How does the experiment show the existence of such sub-ensembles - or proper mixed states to begin with? Wouldn't this falsify the MWI?

I think the introduction of proper mixed states corresponds to defining the location of the Heisenberg cut. In principle, we could shift the boundary and track the correlations with the state of the instruments which produced the mixture.

 

[atyy]

How does the experiment show the existence of such sub-ensembles - or proper mixed states to begin with? Wouldn't this falsify the MWI?

Experiment shows the existence of sub-ensembles because if we take all the individual systems whose measurement produced the eigenvalue k corresponding to the vector |k>, and form an ensemble from those systems, that ensemble has state |k>. For example, if after a measurement you get 2 beams, where one beam is measured to be up and the other down, and you block the beam of particles that were measured to be down. The ensemble formed from the beam measured to be up will have state |up>. This is why in Copenhagen, the Born rule is stated that the probability to find a particle in state |k> given that it is in state |ψ> is |<k|ψ>|².

I don't know whether it would falsify Many-Worlds, but Many-Worlds is the programme of trying to derive (among other things) the collapse as only an apparent collapse due to unitary evolution of the wave function.

I think the introduction of proper mixed states corresponds to defining the location of the Heisenberg cut. In principle, we could shift the boundary and track the correlations with the state of the instruments which produced the mixture.

Yes, you can do that. But unless Many-Worlds works, we ultimately must place a cut somewhere in order to use quantum mechanics, and have measurements in which there are definite outcomes.

 

[kith]

I think the introduction of proper mixed states corresponds to defining the location of the Heisenberg cut. In principle, we could shift the boundary and track the correlations with the state of the instruments which produced the mixture.

Yes, you can do that.

Then I don't see how experiments can say anything definite about sub-ensembles.

Your definition of sub-ensembles relies on proper mixed states. The question whether a mixed state is proper or not depends on the location of the Heisenberg cut. If the experiment shows that some states are proper mixtures, it also shows where the cut is. So we would not be able to do what I have written above.

 

[atyy]

Then I don't see how experiments can say anything definite about sub-ensembles.

Your definition of sub-ensembles relies on proper mixed states. The question whether a mixed state is proper or not depends on the location of the Heisenberg cut. If the experiment shows that some states are proper mixtures, it also shows where the cut is. So we would not be able to do what I have written above.

Ok, maybe there is a problem. But my initial thought is this.

In the first case, I consider two (or more) successive measurements, so I use the Born rule with collapse.

In the second case, instead of measuring, I couple the system to an ancilla. Instead of multiple measurements, I have multiple ancillae (is that the correct plural?). Then at the end, I do a measurement of of the correlation between the ancillae. So now I have only one measurement, and I can use the Born rule without collapse.

Collapse is only needed for successive measurements (successive irreversible marks on the detector on the macroscopic side of the cut). So if we can push the time that we see the irreversible mark back so that there is only one measurement, there is no need to have collapse.

Edit: Maybe the "Principle of Delayed Measurement"?
(44:30)
http://arxiv.org/abs/0803.3237v3 footnote 15: "This is the delayed measurement principle, stating that for any quantum setup involving measurements, there is an equivalent one in which all measurements are postponed at the final stage"

 

[atyy]

@kith, continuing post #101 here.

http://arxiv.org/abs/quant-ph/0512125
" ... So a quantum operation represents, quite generally, the unitary evolution of a closed quantum system, the nonunitary evolution of an open quantum system in interaction with its environment, and evolutions that result from a combination of unitary interactions and selective or nonselective measurements.

As we have seen, the creed of the Church of the Larger Hilbert Space is that every state can be made pure, every measurement can be made ideal, and every evolution can be made unitary – on a larger Hilbert space."

It does seem that if we restrict ourselves to "quantum operations", including filtering measurements, then non-unitary operations can always be made unitary on a larger Hilbert space.

I guess your question then is: doesn't this show that collapse is not needed? My thought is that in an interpretation with a cut, for a given choice of cut, collapse is needed. If one could start off axiomatically with no cut, ie. there is a wave function of the universe, then collapse as a postulate is not needed, and might be derived.

http://mattleifer.wordpress.com/2006/04/13/the-church-of-the-smaller-hilbert-space/
Ten Commandments of the Church of the Smaller Hilbert Space by Matt Leifer :)

 

[gill1109]

As we have seen, the creed of the Church of the Larger Hilbert Space is that every state can be made pure, every measurement can be made ideal, and every evolution can be made unitary – on a larger Hilbert space."

This is not just a Creed (Dogma), it also corresponds to a Theorem. If we take states to be represented by density matrices and accept the probabilistic interpretation of mixing, and if we also accept the idea of forming composite systems by taking tensor products, then we can write down a few sensible axioms which must be satisfied by any physically possible mapping from states to states (evolution) or any physically possible mapping from states to probability distributions of measurement outcomes (measurement). And from these axioms we can *prove* that every state can be made pure, every measurement can be made ideal, and every evolution can be made unitary. See Nielsen and Chuang.

 

[gill1109]

I guess your question then is: doesn't this show that collapse is not needed? My thought is that in an interpretation with a cut, for a given choice of cut, collapse is needed. If one could start off axiomatically with no cut, ie. there is a wave function of the universe, then collapse as a postulate is not needed, and might be derived.

I am rather fond of Slava Belavkin's "eventum mechanics" which adds to the basic framework of QM also the idea that there truly is a cut, and it is something physical. So quantum collapse is something real, which actually happens; and it is truly probabilistic. ie the theory is non-deterministic.

The cut has to be situated in the Hilbert space in a way which is compatible with causality (with time). I tried to explain this theory in simple terms in
http://arxiv.org/abs/0905.2723

Many collapse theories have something else "arbitrary" such as a preferred basis. I think that the Belavkin approach is a good way of turning a bug into a feature. We enrich the usual QM framework so that there is no longer any Schrödinger cat problem. Collapse is for real. QM is only a framework. Physicists have to figure out how the framework fits to the real world. So where precisely the collapse actually happens can be theorized about, can be experimented on, ... it is a topic to be further investigated. The enriched framework (Belavkin calls it "event enhanced QM - it is QM enhanced so as to give a place to a macroscopic real reality, embedded in a larger quantum universe) is just a framework. Without any interpretational problems, any more, ie it *solves* the Schrödinger cat problem by seeing it as a metaphysical problem which has a mathematical solution.

 

[atyy]

@gill1109, thanks for the pointer to Belavkin's work! I hope to understand it some day:)