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

[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:)

 

[atyy]

@kith and @Fredrik, a couple more thoughts on collapse in the Ensemble Interpretation. Above I outlined two ways that collapse may be avoided.

1) Couple to ancillae, and push all measurements to the end of time. However, this means that successive measurements are not allowed, which is fine. However, Ballentine does describe successive measurements. He even says that a filtering measurement is a method of state preparation, which implies successive measurements since one would presumably make a measurement on the state prepared by the previous measurement.

2) Use the Stinespring theorem, which shows that every completely positive map, including collapse, can be described as unitary evolution on a larger Hilbert space. I don't think this eliminates the need for a postulate equivalent to collapse, because we know that this map occurs only with some probability, given by the Born rule. Although the map can be described as unitary evolution, I don't see how its probability is obtained without the form of the Born rule that includes collapse (unless Many-Worlds works, but that's not Ballentine's Ensemble Interpretation).

 

[gill1109]

@kith and @Fredrik, a couple more thoughts on collapse in the Ensemble interpretation. Above I outlined two ways that collapse may be avoided.

1) Couple to ancillae, and push all measurements to the end of time. However, this means that successive measurements are not allowed, which is fine. However, Ballentine does describe successive measurements. He even says that a filtering measurement is a method of state preparation, which implies successive measurements since one would presumably make a measurement on the state prepared by the previous measurement.

2) Use the Stinespring theorem, which shows that every completely positive map, including collapse, can be described as unitary evolution on a larger Hilbert space. I don't think this eliminates the need for collapse, because we know that this map occurs only with some probability, given by the Born rule. Although the map can be described as unitary evolution, I don't see how its probability is obtained without the form of the Born rule that includes collapse (unless many-worlds works, but that's not Ballentine's ensemble interpretation).

Exactly. This is exactly Belavkin's point: these mathematical tricks don't solve the Schrödinger cat logical conundrum. Taking Born's law and the Heisenberg cut seriously, does solve it. But so far, for most practical minded physicists, there was not an issue, since they know how to get the right answer and they don't mind if there are some logical issues with the framework they are using. "it's not their problem". This is called by Bell the FAPP trap (FAPP = for all practical purposes). Many other practical minded physicists thinks that everything is solved by the MWI (many worlds interpretation). Personally I see that as more of a smokescreen than a solution, but that is just my hang-up, being a mathematician and not a practical minded physicist.

 

[vanhees71]

Again, I'm to naive to understand the necessity of a collapse at all! Take a "classical" situation of througing a die. Without further knowledge about its properties, I use the maximum-entropy method to associate a probability distribution. Of course, the least-prejudice distribution in the Shannon-Jaynes information-theoretical sense is that the occurrence of any certain value is p_k=1/6, k \in \{1,2,3,4,5,6 \}.

Now I through the die once and get "3". Is now my probability distribution collapsed somehow to p_3=1 and all other p_k=0? I don't think that anybody would argue in that way.

Now, of course my probability distribution is assumed on the uncomplete knowledge about the die. In this case I assumed, I don't know anything about it and used the maximum-entropy principle to make a prediction of the probability distribution based on the objective principle of "least prejudice" (which of course has a certain well-defined meaning). Now, how to check this prediction? Of course, I've to through the die many times indepenently and always under the same conditions and make the statistics, how often the outcomes occur. Then I may come to the conclusion that the die is somehow biased towards a different distribution, and henceforth I use a new probability distribution based on the gained (statistical) knowledge about the die. Would you now say, somehow the probability distribution is collapsed to the new one, and that this is a physical process involving the die? I'd not say that anybody would argue in that way.

That's done only in quantum theory. There you also have a very precise definition of how to associate probability distributions to the outcome of measurents, given a certain (equivalence class of) preparation procedures of the (quantum) system under investigation. E.g., if you prepare a complete set of observables to have determined values, you have prepared the system in a unique pure state (represented by a ray in Hilbert space). Now I perform a measurement of any other observable. In the following let's assume that we perform an ideal von Neumann filter measurement.

Is now some collapse occurring in the sense of some flavors of the Copenhagen interpretation? If so, when does it occur?

Note that there are two possible scenarios: (a) I measure the quantity and filter out all systems having a certain given value of this observable. According to standard quantum theory, such a system, I have to describe by a pure state, corresponding to the very eigenvector which is given by the projection of the original state onto the eigenspace of the measured eigenvalue. Has now the state collapsed to the new pure state and, if yes, when and how does this collapse occur?

(b) I measure the quantity in the sense of an ideal filter measurement but do not take notice of the measured outcome. Then, again according to standard quantum theory in the minimal interpretation, I associate a mixed state with the then newly prepared system. Has now the state collapsed to the new mixed state?

I'd answer "no" to all these collapse questions. For me the association of states is an objective association of mathematical objects like state kets or statstistical operators based on a known preparation procedure of the quantum system. This gives me certain probabilities to find a certain value of any observable of the system, implying that the value of this observable is indetermined if the system is not in an eigenstate of the self-adjoint operator assiciated with this observable. The measurement is simply an interaction with a measurement apparatus which somehow stores the value of the measured observable to read off this value. Nothing else happens than this physical interactions, and nothing collapses in the Copenhagen sense as a physical process in nature. It may be that the system can be followed further after the measurement (e.g., when an ideal filter measurement has been performed), and the interaction with the measurement apparatus (and probably an associated filter proceess has been done) can be seen as a new preparation of the system, which I can describe further quantum mechanically in terms (of a pure or mixed) state. In this latter case, there is no collapse either, but just the application of well-defined rules of how to describe our knowledge about the system, which can be checked experimentally. That's all what physics is about: A description of our possible objective knowledge about physical systems. No more and no less. Any further questions some philosophers are after like an "ontology of quantum systems" is metaphysics and not my business as a physicist. This I leave to the philosophers!

More often you even destroy the system you measure, e.g., using photons as quantum systems usually they get absorbed finally by the detector. Are you the saying there must occur a collapse, although there's not even the photon left to be described in any way? I don't think so. I also leave it to philosophers to determine, what is the "ontology of photons" beyond the description we use as physicists in terms of QED.

 

[bhobba]

Again, I'm to naive to understand the necessity of a collapse at all!

Same here.

There is no collapse axiom in QM.

The 2 axioms in Ballentine say nothing about it at all.

Its an invention of some interpretations.

In the ignorance ensemble interpretation one simply takes an improper mixture to be a proper one. No collapse involved. It has problems, as all interpretations do, but collapse aren't one of them.

I think using Gleason's Theorem clarifies this quite a bit:
https://www.physicsforums.com/showthread.php?t=758125

A state is simply a logical consequence of mapping outcomes to POVM's - its simply an aid in calculating an outcomes probabilities.

Thanks
Bill

 

[gill1109]

When you didn't destroy the system you were measuring, then if you measure it the von Neumann way, then people usually understand that the particles which are associated with measurement outcome "a" are now in the pure state with state vector = corresponding eigenvector. That is what people call "collapse". What's in a name? The Born law tells us which proportion of particles correspond to which particular eigenvalue. The collapse postulate says that this is in effect a way to "prepare" particles in particular pure states.

Sure, you can leave to the philosophers to puzzle about what might actually be going on. The working physicist just needs a way to translate back and forth between things he or she can do in the lab and pieces of mathematical machinery. One of those pieces has nowadays got the name "collapse". "A rose by any other name would smell as sweet". You can call it "preparation by measurement" if you prefer.

Does Ballentine tell you how to prepare a quantum system in the pure state corresponding to the eigenstate of some observable? If so, he has a collapse postulate but he gives it another name, so as not to upset people's feelings.

The modern student of quantum information theory certainly does need to know this notion. Read Nielsen and Chuang. And most everything in there has been implemented in the lab, too. It's not just theory.

Send polarized photons through a piece of polarized glass so that all the ones which get through are polarized in the same direction. What do you call that? You can say "the polarization of each photon was measured in a particular direction. Those for which the outcome was "I'm polarized that way" are allowed through. Those for which the outcome was "I have the perpendicular direction of polarization" get absorbed.

 

[atyy]

In the ignorance ensemble interpretation one simply takes an improper mixture to be a proper one. No collapse involved. It has problems, as all interpretations do, but collapse aren't one of them.

That's fine. What I am saying is that if collapse is removed as a postulate, then the presentation of quantum mechanics is incomplete unless it is replaced by another postulate.

Ballentine fails to state that an improper mixture can be treated as a proper one, so he is missing a postulate.

So here you are agreeing with me, not vanhees71.

 

[bhobba]

The collapse postulate says that this is in effect a way to "prepare" particles in particular pure states.

The issue is its not a postulate.

The state is simply an aid in calculating the probabilities of an outcome of an observation and these days is associated with a preparation procedure. Different state - different preparations. After an observation that didn't destroy the object its now prepared differently - obviously it will have a different state.

Thanks
Bill

 

[atyy]

The issue is its not a postulate.

The state is simply an aid in calculating the probabilities of an outcome of an observation and these days is associated with a preparation procedure. Different state - different preparations. After an observation that didn't destroy the object its now prepared differently - obviously it will have a different state.

Thanks
Bill

The question is: can you derive that an improper mixture is equivalent to a proper mixture?

In a selective measurement, the state |ψ> collapses to some state |j>. This is a completely-positive map, so by Stinespring's theorem you can think of it as unitary evolution on a larger Hilbert space. However, Stinespring's theorem does not tell you that this collapse or unitary evolution is probabilistic, but in fact it is. And it occurs with a probability given by the Born rule with collapse.

 

[bhobba]

Does Ballentine tell you how to prepare a quantum system in the pure state corresponding to the eigenstate of some observable? If so, he has a collapse postulate but he gives it another name, so as not to upset people's feelings..

He doesn't nor does he have to.

He simply associates a state with a preparation procedure.

If you have a filtering type observation all you have done is prepared the system differently so of course it is in a different state.

However that's Ballentine.

I go further and base QM on one axiom:
An observation/measurement with possible outcomes i = 1, 2, 3 ... is described by a POVM Ei such that the probability of outcome i is determined by Ei, and only by Ei, in particular it does not depend on what POVM it is part of.

A state is not even mentioned. Then Gleason is applied to show a positive operator of unit trace P exists such that the probability is Trace (P Ei). Its simply an aid to calculation in predicting the probability of outcomes. What possible difference could such changing when a system is prepared differently make?

Thanks
Bill

 

[Fredrik]

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.

Without a way to map preparation procedures to the mathematical things that are supposed to represent them in the theory, we don't have a theory. So yes, we need something like the rule that says that after a non-destructive measurement of an observable $A$ with non-degenerate result $a$, the state is $|a\rangle$.

I just don't approve of the term "collapse". That's a term invented by people who were thinking about pure states as representing what you called the system's "real state". It strongly suggests that we're talking about some kind of physical process that isn't part of standard QM, and that changes the "real state" from an arbitrary state vector to an eigenvector.

People who don't think about pure states this way find this pretty absurd. But I think they too sometimes use terminology that's too suggestive. For example, someone said that collapse is just a selection of a subensemble on which to perform measurements. (For example, split a beam in two with a Stern-Gerlach magnet, and use only one beam in the experiment). This makes even less sense to me, unless we assume that every particle has a well-defined position at all times, regardless of its wavefunction.

I prefer to be more neutral about what's actually happening, and just point out that we're only talking about a mapping between preparation procedures and states. If we view the rule as only approximate, then it's perfectly consistent with the rest of QM (because of decoherence).

 

[bhobba]

Without a way to map preparation procedures to the mathematical things that are supposed to represent them in the theory, we don't have a theory. So yes, we need something like the rule that says that after a non-destructive measurement of an observable $A$ with non-degenerate result $a$, the state is $|a\rangle$.

It follows from continuity and the Born rule - it's not a separate axiom.

All a change in it means is the system was prepared differently.

Thanks
Bill

 

[gill1109]

It follows from continuity and the Born rule - it's not a separate axiom.

All a change in it means is the system was prepared differently.

Thanks
Bill

Please tell me your two rules, Bill. I am not familiar with Ballentine. Obviously I should be ... but sorry. I learned some kind of QM from many sources including Nielsen and Chuang. If your two Ballentine axioms allow me to build a quantum computer, then they are enough, and they include a "hidden" version of "collapse". It's just not named as such. As a mathematician, I couldn't care less what you name and what you don't name and what names you use. Presumably the theory is the same and the mapping between theory and the lab is the same ... unless Ballentine is actually incomplete and actually needs another axiom, but didn't realize it.

You can obviously make "collapse" redundant as long as you have "Born law for measurement" and you allow "preparation of a particle in an eigenstate". "Collapse" is just those two operations strung together, and not taking any notice of the measurement outcome. If we don't care about what is actually going on (that's for philosophers or even according to some philosophies - eg Bohr's - a total waste of time) then it doesn't matter what names we give to things and what basic operations we put into the mathematical formalism as long as we can generate all the things which we actually meet in the lab by putting those operations together in any way we like.

 

[atyy]

It follows from continuity and the Born rule - it's not a separate axiom.

All a change in it means is the system was prepared differently.

Thanks
Bill

The point is you need an additional assumption. Here by continuity, you mean "for projective measurements, immediate repetition of the measurement yields the same result". So the basic point is that Ballentine is missing a postulate whether it is collapse, or equivalence of proper and improper mixtures, or immediate repetition of a projective measurement gives the same result. It's not that I want to nitpick Ballentine, except that he clearly ridicules collapse in his book, and makes it seem like he can do away with it in his Ensemble Interpretation without adding an equivalent postulate.

 

[atyy]

Without a way to map preparation procedures to the mathematical things that are supposed to represent them in the theory, we don't have a theory. So yes, we need something like the rule that says that after a non-destructive measurement of an observable $A$ with non-degenerate result $a$, the state is $|a\rangle$.

I just don't approve of the term "collapse". That's a term invented by people who were thinking about pure states as representing what you called the system's "real state". It strongly suggests that we're talking about some kind of physical process that isn't part of standard QM, and that changes the "real state" from an arbitrary state vector to an eigenvector.

People who don't think about pure states this way find this pretty absurd. But I think they too sometimes use terminology that's too suggestive. For example, someone said that collapse is just a selection of a subensemble on which to perform measurements. (For example, split a beam in two with a Stern-Gerlach magnet, and use only one beam in the experiment). This makes even less sense to me, unless we assume that every particle has a well-defined position at all times, regardless of its wavefunction.

I prefer to be more neutral about what's actually happening, and just point out that we're only talking about a mapping between preparation procedures and states. If we view the rule as only approximate, then it's perfectly consistent with the rest of QM (because of decoherence).

OK, I think we basically agree. I don't really care what it's named, some people prefer "state reduction". And yes, once we take this Heisenberg cut in the Copenhagen interpretation, we are agnostic as to whether the quantum state is a "real state".

The only quibble is that I think the collapse postulated is as exact as the Born rule itself. But this is quibbling, since the ingredients that go into the Born rule like the Heisenberg cut, the notion of measuring device, and the notion of a definite outcome are fuzzy.

 

[kith]

Collapse is only needed for successive measurements (successive irreversible marks on the detector on the macroscopic side of the cut).

If we can put the cut wherever we want to, why not include the detectors on the microscopic side? This would mean that the need for collapse depends on the description, so collapse would not be a fundamental notion but a tool we use in the calculations.

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.

Now this sounds more like the crucial difference is the observer. If the observer sets up two spin measurement apparatuses and just looks at the final result, no collapse is needed. If he checks the intermediate result, collapse is needed. Do I get you right here?