Notes on qm interpretation

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.

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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/

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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⊗ρA, where ρ refers to the system and ρA to the apparatus. We are interested in situations with final state U(ρ⊗ρA) = 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⊗ρA, where ρ refers to the system and ρA to the apparatus. We are interested in situations with final state U(ρ⊗ρA) = 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 ρr in Eq 4 of http://www.physics.arizona.edu/~cronin/Research/Lab/some decoherence refs/zurek phys today.pdf ? If so, isn't it a mixed state?

kith
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 ρc in Eq 3 of http://www.physics.arizona.edu/~cronin/Research/Lab/some decoherence refs/zurek phys today.pdf ?

bhobba
Mentor
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

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vanhees71
Gold Member
@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 |oi> with eigenvalue oi, with probability |<oi|ψ>|2.

In the second version, if we make a measurement of O, the system that is in state ψ will give the result oi with probability |<oi|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.