Where Can I Find Comprehensive Notes on Quantum Mechanics Interpretations?

[bhobba]

I got a little lost in the last few posts of this discussion. How do you respond to vanhees71's claim about the projection postulate being inequivalent to collapse (which is what he calls extraneous)?

As usual the correct answer is found in Ballentine - which is just one reason why IMHO it is THE book on QM.

The projection postulate is actually a special type of observation where the system is not destroyed by the observation, and it's probably better viewed as a pure state preparation procedure. For that it is easily seen that the Born rule and physical continuity implies the projection postulate.

Collapse is totally extraneous in the Statistical interpretation, and is arguably extraneous in most versions of Copenhagen.

Thanks
Bill

 

[atyy]

@Eloheim, there are two versions of the Born Rule.

In the first version, stated by Weinberg, if we make a measurement of O, the system that is in state ψ will collapse into an eigenstate |o_i> with eigenvalue o_i, with probability |<o_i|ψ>|².

In the second version, if we make a measurement of O, the system that is in state ψ will give the result o_i with probability |<o_i|ψ>|². So there is no collapse in the second version. If this version is used, most textbooks add collapse as a distinct postulate.

As I understand it, "collapse" and the "projection postulate" are the same, but vanhees71 is using the second version of the Born rule without collapse. Both versions of the Born rule cannot be derived from the Schroedinger equation.

I do accept that there are correct interpretations without collapse, such as Bohmian mechanics and probably also many-worlds. What I am skeptical about is whether the ensemble interpretation without collapse is correct, if one allows filtering type measurements as a means of state preparation.

 

[atyy]

As usual the correct answer is found in Ballentine - which is just one reason why IMHO it is THE book on QM.

The projection postulate is actually a special type of observation where the system is not destroyed by the observation, and it's probably better viewed as a pure state preparation procedure. For that it is easily seen that the Born rule and physical continuity implies the projection postulate.

Collapse is totally extraneous in the Statistical interpretation, and is arguably extraneous in most versions Copenhagen.

Thanks
Bill

What is physical continuity in the Statistical Interpretation? Is it something like Bohmian trajectories?

BTW, I saw your post before this one too, thanks. I'll take a look at the other way Ballentine does it.

 

[bhobba]

What is physical continuity in the Statistical Interpretation?

Its simple.

Some measurements are more like filtering in that let's say it has n outcomes, then n different quantum states are also produced associated with each outcome. Let them have numbers 1 to n as the outcomes so the observable is ∑ i |bi><bi|. The Stern-Gerlach experiment is like that - with two outcomes. Suppose k is the outcome. Now let's do the same observation immediately after. Then, from physical continuity, you reasonably expect to get the same outcome, and whatever state is associated with the outcome to have changed by a negligible amount. Again Stern-Gerlach is like that. This assumption, a little math, and the Born rule, shows the state must be the eigenvector associated with the outcome of the observations observable ie |bk><bk| - which, drum-roll, wait for it, is the projection postulate.

Really the projection postulate only applies to 'filtering' type observations like the above. Observations are much more general than that.

Thanks
Bill

 

[kith]

I don't think it makes sense to talk about "collapse", "state reduction" or the "projection postulate" in the ensemble interpretation because all of them were introduced to avoid macroscopic superpositions and these are not problematic in the ensemble interpretation.

What happens is that after the experiment, the observer redefines what he considers to be part of the system and what constitutes the environment. He doesn't have to do this, it is simply a practical matter.

But thinking further, we don't have something to talk about in the first place, if the observer isn't allowed to define what the system of interest is. You can call this an additional postulate but anytime you apply a scientific theory to a specific experimental situation you implicitly use this postulate. So it is nothing specific to the ensemble interpretation or even QM. In the MWI, you get the factorization problem if you don't use it.

 

[atyy]

Its simple.

Some measurements are more like filtering in that let's say it has n outcomes, then n different quantum states are also produced associated with each outcome. Let them have numbers 1 to n as the outcomes so the observable is ∑ i |bi><bi|. The Stern-Gerlach experiment is like that - with two outcomes. Suppose k is the outcome. Now let's do the same observation immediately after. Then, from physical continuity, you reasonably expect to get the same outcome, and whatever state is associated with the outcome to have changed by a negligible amount. Again Stern-Gerlach is like that. This assumption, a little math, and the Born rule, shows the state must be the eigenvector associated with the outcome of the observations observable ie |bk><bk| - which, drum-roll, wait for it, is the projection postulate.

Really the projection postulate only applies to 'filtering' type observations like the above. Observations are much more general than that.

Thanks
Bill

Yes, that works. But is it bhobba's Statistical Interpretation or Ballentine's? I don't think Ballentine introduced a "physical continuity" or "immediate repetition of a measurement yields the same result" assumption.

 

[audioloop]

All things are subject to interpretation whichever interpretation prevails at a given time is a function of power and not truth.
-Friedrich Nietzsche

 

[bhobba]

Yes, that works. But is it bhobba's Statistical Interpretation or Ballentine's? I don't think Ballentine introduced a "physical continuity" or "immediate repetition of a measurement yields the same result" assumption.

The continuity assumption is in Ballentine somewhere - although I can't recall exactly where.

Thanks
Bill

 

[atyy]

The continuity assumption is in Ballentine somewhere - although I can't recall exactly where.

Thanks
Bill

Well, if it is it's still hilarious, since in standard texts (like Dirac's) the projection postulate is linked to physical continuity (I believe that's Dirac's term). And if the projection postulate can be derived in Ballentine's framework because he did postulate physical continuity, then his rejection of the projection postulate is wrong, since non-unitary evolution can occur in his framework.

Yet another problem with Ballentine is his comment "The infinities of quantum field theory, of which we have here seen only the first, are somewhat of a hidden scandal. The infinities of quantum field theory, of which we have here seen only the first, are somewhat of a hidden scandal. Most physicists seem content to ignore them because there are procedures (the so-called renormalization theory) which allow us to avoid the infinities in many practical cases. One prominent physicist who was not complacent about the infinities was Dirac." For most physicists, this has been solved at the conceptual level by Wilson in the 1970s http://quantumfrontiers.com/2013/06/18/we-are-all-wilsonians-now/ .

 

[bhobba]

The infinities of quantum field theory, of which we have here seen only the first, are somewhat of a hidden scandal.

I just love his textbook - overall it's the finest QM book I have ever read and and a BIG effect on me.

But perfect he aren't - the above being a case in point.

It took me while to get it, but once you understand it renormalisation is no more of a scandal than the issues in bog standard EM with point particles which also lead to problems:
http://arxiv.org/pdf/gr-qc/9912045v1.pdf

The issue is simply that its a low energy approximation to a more complete theory so you must have a cutoff. What makes renormalizable theories so nice is you don't need to know the actual cutoff - you simply assume that some such, large cutoff, exists and you can calculate everything. Non renormalizable theories like gravity need an explicit cutoff beyond which the theory is not valid.

Renormalisation is explained VERY well here:
http://arxiv.org/pdf/hep-th/0212049.pdf

I had to go through it a few times, but once I fully understood it I realized renormalisation is not an issue at all.

Thanks
Bill

 

[atyy]

If you want something free, then:
http://lanl.arxiv.org/abs/quant-ph/0209123

Looking at this, it looks like what Laloë calls the "correlation interpretation" is similar in spirit to what Ballentine was aiming for. In particular, Laloë's Eq 37 (taking into account footnote 41) is the same as Ballentine's Eq 9.30. Laloë says "Equation (37) can be seen as a consequence of the wave packet reduction postulate of quantum mechanics, since we obtained it in this way. But it is also possible to take it as a starting point, as a postulate in itself: it then provides the probability of any sequence of measurements, in a perfectly unambiguous way, without resorting, either to the wave packet reduction, or even to the Schrödinger equation itself. The latter is actually contained in the Heisenberg evolution of projection operators, but it remains true that a direct calculation of the evolution of |ψ> is not really necessary. As for the wave packet reduction, it is also contained in a way in the trace operation of (37), but even less explicitly. If one just uses formula (37), no conflict of postulates takes place, no discontinuous jump of any mathematical quantity; why not then give up entirely the other postulates and just use this single formula for all predictions of results?"

The difference between Laloë's and Ballentine's presentation leading up to the formula is that Laloë acknowledges the projection postulate in its derivation, and that if one gets rid of the projection postulate, then Eq 37 must be postulated directly. Ballentine seems less clear about what is derived, and what is postulated in the steps leading up to his Eq 9.30. Ballentine does say that something new is being defined in this section (on p248 he says "We can still use (9.22) as a definition of the joint probability Prob(A&B|C)"), but I suspect that Ballentine has in this chapter an unacknowledged use of the projection postulate in taking the decohered improper mixture to be a proper mixture on p244, as well as in writing down the density matrix of Eq 9.28 (or he has made use of bhobba's physical continuity, from which the projection postulate can be derived, but if so his rejection of the projection postulate is not justified). ? Comments please!

 

[bhobba]

but if so his rejection of the projection postulate is not justified).

He doesn't reject the projection postulate. He says it only applies to filtering type measurements and for those it follows from continuity.

Thanks
Bill

 

[atyy]

He doesn't reject the projection postulate. He says it only applies to filtering type measurements and for those it follows from continuity.

Thanks
Bill

I think Ballentine's book would make more sense if that were the case. Ballentine does mention Dirac's use of collapse with what seems to be lack of disapproval right after Ballentine's Eq 9.28, which is where I think Ballentine has used the projection postulate. Then presumably more general measurements (of POVMs) would lead to the generalized collapse rule.

In a philosophy which takes QM to be a limited theory (no wave function of the universe, and classical systems are needed for its application), as I think the Ballentine aims for, there is nothing wrong with collapse. If one really insists on unitary dynamics for the whole universe, I think one is led almost inexorably to many-worlds, if one remains within quantum theory. Bohmian mechanics is inconsistent with quantum theory if it is applied to the whole universe because then it is more natural to have non-equilibrium dynamics, and deviations from quantum theory.

As you have often pointed out, a reason that there may be nothing wrong with collapse is that it seems analogous to statistical updating. I'm not sure how complete the analogy is, but of several different recent constructions, one which seems pretty nice is Leifer and Spekkens's http://arxiv.org/abs/1107.5849 in which they explicitly say "A positive operator valued measure (POVM) is a CPT map from a quantum input to a classical output (the measurement outcome)", and propose that wave function collapse is analogous to a form of belief propagation.

2 cheers for collapse in the ensemble interpretation

 

[kith]

He doesn't reject the projection postulate. He says it only applies to filtering type measurements and for those it follows from continuity.

He doesn't reject it as a formal tool but he rejects it as a postulate. In the paragraph directly before (9.28) he says "If we consider the result of the subsequent S measurement on only that subensemble for which R ∈ ∆_a and ignore the rest, we shall be determining the conditional probability (9.27)." (bolding mine)

So the reason why we use a projected density matrix is that we chose to ignore what happens to the subensemble which is blocked inside the measurement apparatus. We don't have to do this. We could drag it along as long as we want by applying the appropriate time evolution. It just doesn't tell us anything about the subensemble outside the apparatus which is what we use for further experiments. So why should we do this?

 

[atyy]

He doesn't reject it as a formal tool but he rejects it as a postulate. In the paragraph directly before (9.28) he says "If we consider the result of the subsequent S measurement on only that subensemble for which R ∈ ∆_a and ignore the rest, we shall be determining the conditional probability (9.27)." (bolding mine)

So the reason why we use the projection postulate is that we chose to ignore what happens to the subensemble which is blocked inside the measurement apparatus. We don't have to do this. We could drag it along as long as we want by applying the appropriate time evolution. It just doesn't tell us anything about the subensemble outside the apparatus which is what we use for further experiments. So why should we do this?

After 9.27 Ballentine says "Alternatively, we can use (9.22) to define Prob(B|A&ρ) in terms of the other two factors, both of which are known."

Looking at 9.22, it is Prob(A&B|C) = Prob(A|C) Prob(B|A&C).

However, later he says "If the operators R and S do not commute, then (9.26) does not apply. We can still use (9.22) as a definition of the joint probability Prob(A&B|C)"

So yes, in deriving the expression for Prob(B|A&C) in 9.28 he does not need the projection postulate, if he uses 9.22. However, later he uses Prob(B|A&C) to define Prob(A&B|C) in 9.22. So in fact to truly derive 9.30, he must use the projection postulate to get 9.28. Of course, he could just postulate 9.30 directly. The whole discussion is also fine if all steps leading to 9.30 are just hand-wavy, but he doesn't really present it clearly that way.

Laloë's discussion is clearer, because he derives 9.30 explicitly using the projection postulate. Then he says, if we don't want to have a projection postulate, we can treat 9.30 as a postulate instead of considering it derived. (Laloë's Eq 37 is Ballentine's 9.30 - the free paper that Demystifier linked to in post #5. One has to take into account Laloë's footnote 41 to get Laloë's Eq 37 into the same form as Ballentine's 9.30.)

 

[kith]

I don't think it makes sense to base the discussion on Laloë's paper because he (a) doesn't seem to distinguish between collapse and the Born rule by using the term "wave packet reduction" for both and (b) associates states with individual systems throughout section 6.1.

Ballentine's reasoning applies only to ensembles. Ignoring a subensemble which isn't used in future experiments is a practical matter but ignoring a part of the state of a single system is essentially a fundamental postulate. Both refer to the same formula (9.28).

 

[atyy]

I don't think it makes sense to base the discussion on Laloë's paper because he (a) doesn't seem to distinguish between collapse and the Born rule by using the term "wave packet reduction" for both and (b) associates states with individual systems throughout section 6.1.

Ballentine's reasoning applies only to ensembles. Ignoring a subensemble which isn't used in future experiments is a practical matter but ignoring a part of the state of a single system is essentially a fundamental postulate. Both refer to the same formula (9.28).

Ballentine needs a rule that specifies the state of a subensemble in 9.28. This rule must either be postulated, or derived. I believe his derivation of 9.28 fails, if 9.28 is considered as step in the derivation of 9.30. If 9.28 is not derived, then it is postulated, which is essentially a projection postulate.

Apart from nitpicking Ballentine's mathenatics, a more general argument I would give is that naive textbook QM has a projection postulate, which is used in describing filtering measurements as a means of state preparation. So if one describes such processes, then one must either use the projection postulate, or a replacement to the projection postulate. Another way to see that the projection postulate is needed is to derive the ensemble interpretation from Bohmian mechanics. Unless the ensemble interpretation is aiming for many-worlds, where there are no hidden variables and unitary evolution is everything and quantum mechanics is complete, I don't see how the Ballentine can be correct without the projection postulate or a replacement for it.

If bhobba is correct that Ballentine has has a "physical continuity" postulate, from which wave function collapse (or state reduction, or whatever one wants to call it) can be derived, then Ballentine's 9.28 can be justified from wave function collapse. Basically, I believe that bhobba's "Ballentine's Statistical Interpretation" interpretation is a correct interpretation of quantum mechanics.

 

[audioloop]

Does anyone know good notes many-world interpretation, relative facts, many histories etc. in it.

thanks

Relative-State Formulation of Quantum Mechanics.

http://plato.stanford.edu/entries/qm-everett/



.

 

[kith]

Ballentine needs a rule that specifies the state of a subensemble in 9.28. This rule must either be postulated, or derived. I believe his derivation of 9.28 fails, if 9.28 is considered as step in the derivation of 9.30. If 9.28 is not derived, then it is postulated, which is essentially a projection postulate.

The logic of the section is "if the (unitary) dynamics leads to a situation where only the subensemble ρ' of 9.28 leaves the apparatus, 9.30 is true for a susequent measurement on this subensemble". So 9.28 is derived in the sense that we can write down such unitary dynamics.

Apart from nitpicking Ballentine's mathenatics, a more general argument I would give is that naive textbook QM has a projection postulate, which is used in describing filtering measurements as a means of state preparation. So if one describes such processes, then one must either use the projection postulate, or a replacement to the projection postulate.

Textbooks usually don't treat the measurement apparatus as a QM system. If we do so, the unitary evolution of the combined state leads to a macroscopic superposition. Such a superposition doesn't correspond to the outcome in a single run of the experiment. If we want to assign it to the individual system, we are forced to introduce either the projection postulate or different simultaneous realities (many worlds). But if the superposition only corresponds to many runs of the experiments, no such problem arises.

If bhobba is correct that Ballentine has has a "physical continuity" postulate [...]

I don't think Ballentine uses something like this. I would be interested in an exact quotation from the book.

 

[atyy]

The logic of the section is "if the (unitary) dynamics leads to a situation where only the subensemble ρ' of 9.28 leaves the apparatus, 9.30 is true for a susequent measurement on this subensemble". So 9.28 is derived in the sense that we can write down such unitary dynamics.

But in that case, wouldn't ρ' be the whole ensemble, not a subensemble? Could the unitary dynamics describe the tree he draws in Fig. 9.3 where ρ' is a subensemble?

Textbooks usually don't treat the measurement apparatus as a QM system. If we do so, the unitary evolution of the combined state leads to a macroscopic superposition. Such a superposition doesn't correspond to the outcome in a single run of the experiment. If we want to assign it to the individual system, we are forced to introduce either the projection postulate or different simultaneous realities (many worlds). But if the superposition only corresponds to many runs of the experiments, no such problem arises.

I guess if one doesn't describe successive measurements, then one doesn't need the projection postulate to specify the post-measurement state. One still needs a collapse of sorts to say there are definite outcomes, but it seems satisfactory to say it's magic and the theory doesn't explain it.

I don't think Ballentine uses something like this. I would be interested in an exact quotation from the book.

Yes, bhobba couldn't find it. https://www.physicsforums.com/showpost.php?p=4614245&postcount=52

 

[kith]

But in that case, wouldn't ρ' be the whole ensemble, not a subensemble?

Let's make this explicit. Our initial state is ρ⊗ρ_A, where ρ refers to the system and ρ_A to the apparatus. We are interested in situations with final state U(ρ⊗ρ_A) = ρ'⊗ρ'_A + ρ''⊗ρ''_A where ρ' is the same as in 9.28. Furthermore, let U be such that the first term describes a subensemble of systems and apparatuses where the system leaves the apparatus. Let the second term describe a subensemble where the system is trapped inside the apparatus. Then, only ρ' is relevant for a subsequent measurement of another observable B.

Could the unitary dynamics describe the tree he draws in Fig. 9.3 where ρ' is a subensemble?

The figure displays all possible combinations of the successive application of 3 filters in the fixed directions z, u and x. It doesn't correspond to the physical situation because it doesn't specify which values get transmitted by the filters and which are blocked.

I guess if one doesn't describe successive measurements, then one doesn't need the projection postulate to specify the post-measurement state. One still needs a collapse of sorts to say there are definite outcomes, but it seems satisfactory to say it's magic and the theory doesn't explain it.

Yes because this is not specific to QM. If you have a phase space distribution and use classical Hamiltonian dynamics to describe the influence of a measurement apparatus, you will also end up with another phase space distribution and not with a definite outcome.

 

[atyy]

Let's make this explicit. Our initial state is ρ⊗ρ_A, where ρ refers to the system and ρ_A to the apparatus. We are interested in situations with final state U(ρ⊗ρ_A) = ρ'⊗ρ'_A + ρ''⊗ρ''_A where ρ' is the same as in 9.28. Furthermore, let U be such that the first term describes a subensemble of systems and apparatuses where the system leaves the apparatus. Let the second term describe a subensemble where the system is trapped inside the apparatus. Then, only ρ' is relevant for a subsequent measurement of another observable B.

Is ρ'⊗ρ'_A + ρ''⊗ρ''_A the same as ρ^r in Eq 4 of http://www.physics.arizona.edu/~cronin/Research/Lab/some%20decoherence%20refs/zurek%20phys%20today.pdf ? If so, isn't it a mixed state?

 

[kith]

Is ρ'⊗ρ'_A + ρ''⊗ρ''_A the same as ρ^r in Eq 4 of http://www.physics.arizona.edu/~cronin/Research/Lab/some%20decoherence%20refs/zurek%20phys%20today.pdf?

No. As you say, Zurek's equation (4) describes a mixed state. Applying a unitary operator U doesn't change the purity, so the final state in my last post is still pure if the initial state was pure (which we can assume for simplicity).

Above equation (5) Zurek writes "When the off-diagonal terms are absent, one can safely maintain that the apparatus and the system are each separately in a definite but unknown state [...]". The ensemble interpretation doesn't want to maintain this.

 

[atyy]

No. As you say, Zurek's equation (4) describes a mixed state. Applying a unitary operator U doesn't change the purity, so the final state in my last post is still pure if the initial state was pure (which we can assume for simplicity).

Above equation (5) Zurek writes "When the off-diagonal terms are absent, one can safely maintain that the apparatus and the system are each separately in a definite but unknown state [...]". The ensemble interpretation doesn't want to maintain this.

So ρ'⊗ρ'_A + ρ''⊗ρ''_A is ρ^c in Eq 3 of http://www.physics.arizona.edu/~cronin/Research/Lab/some%20decoherence%20refs/zurek%20phys%20today.pdf ?

 

[bhobba]

The ensemble interpretation doesn't want to maintain this.

Yes - but the Ignorance Ensemble Interpretation I hold to does (I agree that Ballentine's version doesn't require decoherence, and indeed probably has issues if you try to use it as is):
http://philsci-archive.pitt.edu/5439/1/Decoherence_Essay_arXiv_version.pdf

'Ignorance interpretation: The mixed states by taking the partial trace over the environment can be interpreted as a proper mixture. Note that this is essentially a collapse postulate.'

'Postulating that although the system-apparatus is in an improper mixed state, we can interpret it as a proper mixed state superficially solves the problem of outcomes, but does not explain why this happens, how or when. This kind of interpretation is sometimes called the ensemble, or ignorance interpretation. Although the state is supposed to describe an individual quantum system, one claims that since we can only infer probabilities from multiple measurements, the reduced density operator is supposed to describe an ensemble of quantum systems, of which each member is in a definite state.'

Thanks
Bill

 

[vanhees71]

@Eloheim, there are two versions of the Born Rule.

In the first version, stated by Weinberg, if we make a measurement of O, the system that is in state ψ will collapse into an eigenstate |o_i> with eigenvalue o_i, with probability |<o_i|ψ>|².

In the second version, if we make a measurement of O, the system that is in state ψ will give the result o_i with probability |<o_i|ψ>|². So there is no collapse in the second version. If this version is used, most textbooks add collapse as a distinct postulate.

As I understand it, "collapse" and the "projection postulate" are the same, but vanhees71 is using the second version of the Born rule without collapse. Both versions of the Born rule cannot be derived from the Schroedinger equation.

I do accept that there are correct interpretations without collapse, such as Bohmian mechanics and probably also many-worlds. What I am skeptical about is whether the ensemble interpretation without collapse is correct, if one allows filtering type measurements as a means of state preparation.

I never understood, why the state must collapse to the eigenstate, while when we apply QT to real-lab experiments we use the second version only. QT doesn't say, what happens to the measured system but that's given by the measurement apparatus we use to measure the observable. The Born postulate just states that the probability to find a value as you said above (tacitly assuming that the eigenvalue is non-degenerate, i.e., the eigenspace of this eigenvalue is one-dimensional).

There are, of course, special cases, where you (can) do an ideal von Neumann filter measurement. The most famous example is the Stern-Gerlach experiment, which nowadays can be done with practically arbitrary precision using neutrons.

The good old original setup by Stern and Gerlach is, however, better to discuss this in principle: You use an oven with a little opening to get a particle beam that can be described by a mixed state (thermal in the restframe of the gas, making up the particle beam). The particles then go through an inhomogeneous magnetic field. One solve the corresponding dynamical problem in very good approximation analytically and even exactly to arbitrary precision numerically, see e.g.,

G. Potel et al, PRA 71, 052106 (2005).
http://arxiv.org/abs/quant-ph/0409206

You end up with well-separated partial beams that are "sorted" (with high precision) after their spin components given by the direction of the magnetic field, usually chosen as the $z$ direction. In other words, after running trough the magnet you have a quantum state, where the position and spin-z component are entangled, and you can get a particle beam in a (nearly) pure spin state by just forgetting all unwanted partial beams, blocking them by some absorber material. There is no collapse necessary but just to put some absorber material in the way of the "unwanted" partial beams. Of course, you may now ask, how the absorption process happens microscopically in terms of quantum theory, and this might not be a simple issue, but experiment clearly shows that you can block particles, and nothing hints at some "collapse mechanism" that may ly outside of the quantum dynamics of a particle interacting with the particles in the absorber.

 

[atyy]

I never understood, why the state must collapse to the eigenstate, while when we apply QT to real-lab experiments we use the second version only. QT doesn't say, what happens to the measured system but that's given by the measurement apparatus we use to measure the observable. The Born postulate just states that the probability to find a value as you said above (tacitly assuming that the eigenvalue is non-degenerate, i.e., the eigenspace of this eigenvalue is one-dimensional).

There are, of course, special cases, where you (can) do an ideal von Neumann filter measurement. The most famous example is the Stern-Gerlach experiment, which nowadays can be done with practically arbitrary precision using neutrons.

The good old original setup by Stern and Gerlach is, however, better to discuss this in principle: You use an oven with a little opening to get a particle beam that can be described by a mixed state (thermal in the restframe of the gas, making up the particle beam). The particles then go through an inhomogeneous magnetic field. One solve the corresponding dynamical problem in very good approximation analytically and even exactly to arbitrary precision numerically, see e.g.,

G. Potel et al, PRA 71, 052106 (2005).
http://arxiv.org/abs/quant-ph/0409206

You end up with well-separated partial beams that are "sorted" (with high precision) after their spin components given by the direction of the magnetic field, usually chosen as the $z$ direction. In other words, after running trough the magnet you have a quantum state, where the position and spin-z component are entangled, and you can get a particle beam in a (nearly) pure spin state by just forgetting all unwanted partial beams, blocking them by some absorber material. There is no collapse necessary but just to put some absorber material in the way of the "unwanted" partial beams. Of course, you may now ask, how the absorption process happens microscopically in terms of quantum theory, and this might not be a simple issue, but experiment clearly shows that you can block particles, and nothing hints at some "collapse mechanism" that may ly outside of the quantum dynamics of a particle interacting with the particles in the absorber.

It would be nice to have an example of how this is done for successive measurements. The reason I am skeptical is that by including the apparatus and environment in the unitary evolution, to get the state for a subsystem, one has to trace out degrees of freedom, which in general will result in an improper mixed state, not a pure state. If we interpret the improper mixed state as a proper mixed state, we can get a pure state for a sub-ensemble. So I think the assumption still has to be made that an improper mixed state can be treated as a proper mixed state.