Ballentine's Ensemble Interpretation Of QM

[strangerep]

My question is whether the filtering method of state preparation follows from the axioms in Chapters 2 (and 3), or whether it is an additional axiom.

I wouldn't call it an axiom, but rather an application of projection operators, hence a special case.

But for non-commuting R,S, are 9.28 and 9.29 then new axioms?

By my reading, no. Instead, he has reinterpreted the 4th Cox axiom for probability (which is the same equation as (9.22)), i.e.,
$$
\def\Pr{\text{Prob}}
\Pr(A\&B|C) ~=~ \Pr(A|C) \; \Pr(B|A\&C)
$$ into
$$\Pr(A\&B|C) ~:=~ \Pr(A|C) \; \Pr(B|A\&C) ~.$$
The only difference here is that I've changed "$=$" into "$:=$". In classical probability, both sides of the original equation make sense, but not so in QM for noncommuting quantities. Here, Ballentine generalizes the concept, though still restricted (iiuc) to the case of filtering-type operations.

 

[kith]

My question is whether the filtering method of state preparation follows from the axioms in Chapters 2 (and 3), or whether it is an additional axiom.

I would say that it's neither of these possibilities but merely a choice of the observer which allows him to redefine the system.

In a Stern-Gerlach experiment, the composite system of the particle and the Stern-Gerlach apparatus evolves unitarily from an initital product state ρ to an entangled macroscopic superposition state ρ' = p_upρ_up + p_downρ_down. If we block particles with spin down, only ρ_up is a state were particles leave the apparatus. This means only ρ_up is relevant for future experiments.

Nothing in the formalism picks out ρ_up. The full unitary time evolution gives us the state of the composite system at all times. But ρ_down is superflous for future measurements on particles which leave the apparatus. Considering only ρ_up leads to the same predictions but the calculations are a lot easier because it is a product state, so we can separate the particle from the apparatus again. In the other case we have to drag all the previous history of the particle with us.

So I would say the picking of a subensemble in filtering is not a fundamental process but a convenient choice to simplify calculations.

 

[kith]

I'm just saying that the interpretations seem to fall into three general classes based on their top priority. For CI and ensemble, the top priority is to hold firm to the idea that physics is a tool for predicting outcomes that we the physicists can perceive as outcomes, and theories organize the information we use to do that, but theories can mislead us if we take them too seriously.

Yes, I agree with your post almost completely. Still I want to highlight the conceptual differences between the CI and the ensemble interpretation because this whole talk about classical apparatuses leads people to think that classical mechanics is necessary for QM. What is necessary to talk about experiments in QM is an observer, not some special device which cannot be described by QM.

 

[atyy]

I wouldn't call it an axiom, but rather an application of projection operators, hence a special case.

There doesn't seem to be an axiom which says that one is allowed to act on the state with a projection operator.

The axioms in chapters 2 & 3 seem to be
(1) observables correspond to operators
(2) given a state, we can calculate the expectation value of an observable
(3) the time evolution of the state is governed by the Hamiltonian

Am I missing some axioms, or can the action of a projection operator be derived from these?

By my reading, no. Instead, he has reinterpreted the 4th Cox axiom for probability (which is the same equation as (9.22)), i.e.,
$$
\def\Pr{\text{Prob}}
\Pr(A\&B|C) ~=~ \Pr(A|C) \; \Pr(B|A\&C)
$$ into
$$\Pr(A\&B|C) ~:=~ \Pr(A|C) \; \Pr(B|A\&C) ~.$$
The only difference here is that I've changed "$=$" into "$:=$". In classical probability, both sides of the original equation make sense, but not so in QM for noncommuting quantities. Here, Ballentine generalizes the concept, though still restricted (iiuc) to the case of filtering-type operations.

OK, that's my understanding too.

I would say that it's neither of these possibilities but merely a choice of the observer which allows him to redefine the system.

In a Stern-Gerlach experiment, the composite system of the particle and the Stern-Gerlach apparatus evolves unitarily from an initital product state ρ to an entangled macroscopic superposition state ρ' = p_upρ_up + p_downρ_down. If we block particles with spin down, only ρ_up is a state were particles leave the apparatus. This means only ρ_up is relevant for future experiments.

Nothing in the formalism picks out ρ_up. The full unitary time evolution gives us the state of the composite system at all times. But ρ_down is superflous for future measurements on particles which leave the apparatus. Considering only ρ_up leads to the same predictions but the calculations are a lot easier because it is a product state, so we can separate the particle from the apparatus again. In the other case we have to drag all the previous history of the particle with us.

So I would say the picking of a subensemble in filtering is not a fundamental process but a convenient choice to simplify calculations.

Actually, back to the successive measurements we discussed a few posts back. If we start in a pure state, after a measurement doesn't the state generally become a mixed state? This would be non-unitary evolution, so isn't this state reduction?

Yes, I agree with your post almost completely. Still I want to highlight the conceptual differences between the CI and the ensemble interpretation because this whole talk about classical apparatuses leads people to think that classical mechanics is necessary for QM. What is necessary to talk about experiments in QM is an observer, not some special device which cannot be described by QM.

If the observer is not an ensemble, and QM only applies to ensembles, then the observer is not governed by QM.

 

[strangerep]

There doesn't seem to be an axiom which says that one is allowed to act on the state with a projection operator.
[...]

A pure state operator is a projection operator. More generally, a projection operator may correspond to a dynamical variable that takes on only the values 0 and 1. See section 2.2 (just after eq(2.2)), and the discussion in section 2.3 about general states and pure states.

If we start in a pure state, after a measurement doesn't the state generally become a mixed state? This would be non-unitary evolution, so isn't this state reduction?

It's unitary if one analyzes the experiment properly by modelling the apparatus as well. See Ballentine section 9.2, where he explains how and why one does not need the usual principle of "collapse to an eigenvector". (That part of the discussion begins around halfway down p233.)

 

[Ken G]

Yes, I agree with your post almost completely. Still I want to highlight the conceptual differences between the CI and the ensemble interpretation because this whole talk about classical apparatuses leads people to think that classical mechanics is necessary for QM. What is necessary to talk about experiments in QM is an observer, not some special device which cannot be described by QM.

I agree, it's not so much classical mechanics one needs, it is decoherence. We manipulate and test mixed states,not superposition states. Physics must simplify to be effective, and our best simplification tool is reductionism, so we must separate the quantum system from the apparatus and the apparatus from the observer. Reality doesn't do that, but quantum mechanics does (the way we do it in practice, and the way we verify that it works), and decoherence is the reason we don't notice the disconnect, a disconnect that is there even in the ensemble interpretation. So it's not quantum vs. classical that is the "Heisenberg divide", it is the full system vs. the projected subsystem, but it's more or less the same in the end (so CI could easily be translated to fit these alternative words). In any of the interpretations, we simply don't test the formal theory we write down, we test something else, and that is the source of the "measurement problem." That's also why none of the interpretations do away with that problem, but the ensemble approach pushes it the farthest out of sight, while other approaches (like BM and MWI) make it an organic structural element. CI, on the other hand, pays homage to it, but doesn't really know what to do with it, and that's actually why I like CI-- I think we should not know what to do with the measurement problem as long as we insist on reductionism.

 

[kith]

Actually, back to the successive measurements we discussed a few posts back. If we start in a pure state, after a measurement doesn't the state generally become a mixed state?

My post is about successive measurements and it answers your question.

If the observer is not an ensemble, and QM only applies to ensembles, then the observer is not governed by QM.

The observer can be considered to be part of an ensemble and you can include him in the unitary time evolution. You simply can't test hypotheses about such a system without an additional observer who is again excluded from the description.

 

[atyy]

A pure state operator is a projection operator. More generally, a projection operator may correspond to a dynamical variable that takes on only the values 0 and 1. See section 2.2 (just after eq(2.2)), and the discussion in section 2.3 about general states and pure states.

But that just says that pure states exist. It doesn't tell you how they are prepared. I can understand there is no collapse, if there is an axiom that says that one can prepare pure states by filtering, ie. one moves collapse from measurement to preparation. Is that what is being done?

It's unitary if one analyzes the experiment properly by modelling the apparatus as well. See Ballentine section 9.2, where he explains how and why one does not need the usual principle of "collapse to an eigenvector". (That part of the discussion begins around halfway down p233.)

It's the same in interpretations with collapse. But then there cannot be successive measurements, ie. a measurement is by definition the final step in the analysis.

 

[bohm2]

Perhaps, by "operationalism", bohm2 simply meant the attitude that "if the formula works, I don't need to use it to inspire any naive pictures."... So perhaps we can identify three levels of realism, not two: naive realism, where we take our pictures literally, inspirational realism, where we look for organizational pictures that may or may not be literally real (think, virtual particles as an example), and operationalism, where we just solve equations and "go through the motions." That doesn't seem to be the way D'Espagnat uses the terms, but it does seem like what bohm2 means, unless I'm mistaken.

You are correct. I would not classify guidance for progress via symmetry, beauty, elegance and the Pythagorean standpoint (e.g. Tegmark) as operationalism. I think Tegmark goes too far to argue that there is nothing but math as I think it's clear that this is false. For how do mathematical entities lead to phenomenology? It's the latter "shut up and calculate" approach that I find too restrictive and dead end. With respect to the ensemble and CI interpretation, there are some versions that can be considered to take this approach. As Lucien Hardy and Robert Spekkens write:

But operationalism is not enough. Explanations do not bottom out with detectors going ‘click’. Rather, the existence of detectors that click is the sort of thing that we can and should look to science to explain.

 

[Len M]

You are correct. I would not classify guidance for progress via symmetry, beauty, elegance and the Pythagorean standpoint (e.g. Tegmark) as operationalism.

But the formal definition of scientific realism (you referred to the desirability of adopting a form of scientific realism in your post #293) states that there is a reality totally independent of phenomena along with the hypothesis that we do have access to the said reality in that we can say something "true" concerning it. From this standpoint we then proceed to build up a representation of that independent reality, the nature of that representation being dependent on the particular flavour of realism chosen. You seem (I think) to put forward the view that such a scientific realism is needed in order to prevent physics from becoming a dead end - that state of affairs being encompassed within the stance of operationalism. D’Espagnat is saying that no such realism is required to take physics forward.

D’Espagnat’s remarks concerning the mathematics does not refer to “Tegmark like” approaches to physics, rather he is making clear that the representative element associated with mathematical realism need not be invoked at all by operationalists – the representative element that Tegmark invokes at a later stage serves no purpose for them, rather it is the manipulation of the mathematics itself that leads to new and imaginative models. Operationalists wouldn't claim that the model represents independent reality as per Tegmark, they would simply want to show that it works as a predictive mathematical model of phenomena.

Perhaps I’m unclear over what you mean by scientific realism. If by this term you just mean pictures having no pretensions of being representational elements of independent reality then I think we are just describing this scientific process (involving these pictures) by different names, my description being operationalism. If however you are referring to scientific realism in the formal manner as being indicative of what may exist within independent reality then it would seem to suggest that you are saying that physics needs this potential “reality” in which to grow. I don’t think that physics needs such a potential “reality” in which to proceed because for me that potential is as far from what may actually exist within independent reality as anyone can imagine, simply because a miss is as good as a mile. How close or how far a model may be from independent reality is a meaningless question to me because we are never going to know, but as d’Espagnat has illustrated, it is possible for models to be created on the basis that the pictures and mathematics used are not taken to represent independent reality at all, as in the case of his example of Dirac and his sea of holes.

For me and I would say d’Espagnat, operationalism is not simply shut up and calculate, that surely is a just the application of the model to technology. Rather operationalism is physics with no regard for what may be a true picture (or not) outside of phenomena. Instead, the picture is a means to an end in creating the verifiable mathematical predictive model. That of course doesn't exclude meaning being given to pictures of phenomena acting upon phenomena, it's simply that operationalism is entirely unconcerned with what the phenomena may or may not be within independent reality (or, as per radical idealism, may not even be concerned over there needing to be an independent reality)

 

[kye]

I agree, it's not so much classical mechanics one needs, it is decoherence. We manipulate and test mixed states,not superposition states. Physics must

Ken, isn't mixed states also in superposition? you just treat certain portion of the pure states which makes it mixed states, but it should still be in superposition but there is no longer interfering terms in the density matrix. Do you equate mixed states with born rule applied and the state taking on classical eigenvalues?

simplify to be effective, and our best simplification tool is reductionism, so we must separate the quantum system from the apparatus and the apparatus from the observer. Reality doesn't do that, but quantum mechanics does (the way we do it in practice, and the way we verify that it works), and decoherence is the reason we don't notice the disconnect, a disconnect that is there even in the ensemble interpretation. So it's not quantum vs. classical that is the "Heisenberg divide", it is the full system vs. the projected subsystem, but it's more or less the same in the end (so CI could easily be translated to fit these alternative words). In any of the interpretations, we simply don't test the formal theory we write down, we test something else, and that is the source of the "measurement problem." That's also why none of the interpretations do away with that problem, but the ensemble approach pushes it the farthest out of sight, while other approaches (like BM and MWI) make it an organic structural element. CI, on the other hand, pays homage to it, but doesn't really know what to do with it, and that's actually why I like CI-- I think we should not know what to do with the measurement problem as long as we insist on reductionism.

 

[Ken G]

Ken, isn't mixed states also in superposition?

Only if you embed them in a larger system than the physicist actually deals with when recording outcomes. The mixed state is the projection of the superposition, but that's all we deal with in physics experiments. The superposition is purely conceptual, all we test is the projection onto our experience, which is a mixed state before we look, and a "collapsed" state after we look. Whether we look or not is not so important, because we know all about the difference between mixed states and collapsed states from classical physics (and from playing poker), it is that tricky superposition that is in our mathematics, and we need it to explain what we get, but the superposition is not the outcome of the experiment.

Do you equate mixed states with born rule applied and the state taking on classical eigenvalues?

Yes, it is the decohered projection onto what we experience when we do measurements. It can be explained using the formal theory, but most of that formal theory is never actually tested, all we test is that is makes sense of what we see.

 

[Dale]

This "interpretations" thread has run its course.