Ballentine on the quantum Zeno paradox

[Demystifier]

His book has far less errors than most textbooks describing quantum mechanics.

I very much agree with that statement, but that's exactly why I find important to discuss the rare errors in it.

 

[atyy]

So basically, Ballentine does not believe in the quantum Zeno paradox because he does not believe in collapse.
But it seems that he does not understand that effective collapse can almost be "explained" by modern understanding of decoherence, and it seems to be because he is not aware of the importance of decoherence.

The reason for such a suspicion comes from another part of his (otherwise great) book:
Sec. 9.3 - The Interpretation of a state vector
Subsection - The measurement theorem for general states
After Eq. (9.13) he writes:
"The terms with alpha_r1 notequal alpha _r2 indicate a coherent superposition of macroscopically distinct indicator vectors ... It is clear that the nondiagonal terms in (9.13) cannot vanish ..."
But it seems to me that someone who were familiar with decoherence would immediately recognize that they CAN vanish, due to decoherence. Nevertheless, he does not even mention decoherence - at this place at which a "Modern Introduction" to QM should.

Any comments?

Ballentine does mention decoherence in chapter 9, page 244 "This ρ is obtained form the total state operator |Ψ><Ψ| by taking the partial trace over the degrees of freedom of the environment. If the difference between the effects of taking paths ABD and ACD on the environment is so great that |e₁> and |e₂> are orthogonal, then the state reduces to the incoherent mixture ρ_inc (9.18)."

I suspect his error is that he rejects the projection or collapse postulate, but does not replace it with another postulate, ie, he is not entitled to treat the improper mixture reduced density matrix as a proper mixture without the projection postulate, or by postulating explicitly that the two are the same. Similar thoughts that there is a hidden use of the projection postulate are found in

http://arxiv.org/abs/quant-ph/0312059 (p9) "The reduced density matrix looks like a mixed state density matrix because, if one actually measured an observable of the system, one would expect to get a definite outcome with a certain probability; in terms of measurement statistics, this is equivalent to the situation in which the system is in one of the states from the set of possible outcomes from the beginning, that is, before the measurement. As Pessoa (1998, p. 432) puts it, “taking a partial trace amounts to the statistical version of the projection postulate.”"

http://philsci-archive.pitt.edu/5439/1/Decoherence_Essay_arXiv_version.pdf (p37) "Ignorance interpretation: The mixed states we find by taking the partial trace over the environment can be interpreted as a proper mixture. Note that this is essentially a collapse postulate."

 

[Demystifier]

I suspect his error is that he rejects the projection or collapse postulate, but does not replace it with another postulate,

Good point!

 

[audioloop]

In that paper they are replying invalid statements done by Ballentine about their work.

The collapse is needed for any consistent formulation of QM. That is why standard textbooks use it.

until it is not ruled out experimentally. stand.

Are collapse models testable with quantum oscillating systems? The case of neutrinos, kaons, chiral molecules
http://www.nature.com/srep/2013/130606/srep01952/full/srep01952.html?WT.ec_id=SREP-639-20130701

"The effect is stronger for neutral mesons, but still beyond experimental reach. Instead, chiral molecules can offer promising candidates for testing collapse models"


------
Observation of a kilogram-scale oscillator near its quantum ground state
http://eprints.gla.ac.uk/32707/1/ID32707.pdf

"allow us to prepare kilogram test masses in different quantum states, and to study their quantum dynamics"


-----
Effective Field Theory Approach to Gravitationally Induced Decoherence
http://prl.aps.org/abstract/PRL/v111/i2/e021302
http://arxiv.org/pdf/1211.4751v1.pdf


.

 

[bhobba]

The collapse is needed for any consistent formulation of QM. That is why standard textbooks use it.

Collapse is a purely interpretive assumption (in the sense of what it means). Of course it exists in the QM formalism (in the sense of the system being in an eigenstate of the observable after) for filtering type observations (in many cases the original system is destroyed by the observation), where, as a result of an observation, the state instantaneously changes. But its meaning is open ie in some interpretations like Ensemble or Copenhagen - its simply a theoretical concept like probability, useful in helping to predict certain results, but actually existing out there - nope. And like probability the fact it changes instantaneously is of zero concern, any more than when you throw a dice the probability changes from 1/6th on each face to a dead cert for one face and zero for the rest.

And Ballentine indeed has a few rare errors, such as implicitly assuming interpretations where the state is considered a compete description of a Quantum System, it must be real (in fact most versions of Copenhagen explicitly assume its a subjective state of knowledge, and it matters not one whit that such instantaneously changes), but his view that collapse only really applies to filtering type observations isn't one of them.

Thanks
Bill

 

[bhobba]

I suspect his error is that he rejects the projection or collapse postulate, but does not replace it with another postulate

I don't believe he rejects it, but rather formally shows it only applies to filtering type observations.

That aside, and IMHO it's not that big a deal, it indeed is a big issue he rejects dechoherence as an explanation for APPARENT collapse and only alludes to it it in a round about way in his textbook, because his interpretation cries out for it. Indeed where he does mention it, its more or less forced on him - you can't really escape it - but he tries to. Rather strange really.

Maybe it skates a bit close to the Achilles Heel of his interpretation - namely exactly how is an actual outcome selected, and even more basic - why do we get any outcome at all.

Thanks
Bill..

 

[strangerep]

I suspect his error is that he rejects the projection or collapse postulate, but does not replace it with another postulate,

I see no "error" there. As I read it, he does replace the CP, not by another postulate, but by an analytical treatment of measurement by an apparatus -- in his sections 9.2 et seq.

Similar thoughts that there is a hidden use of the projection postulate
[...]
"As Pessoa (1998, p. 432) puts it, `taking a partial trace amounts to the statistical version of the projection postulate.'"

I don't think this is a "use" of the PP. The fact that taking a partial trace is legitimate when dealing with an observable that is trivial on that component of a composite system comes from the basic QM maths (i.e., the use of tensor product Hilbert spaces), hence is not itself a postulate.

 

[strangerep]

[...] it indeed is a big issue [Ballentine] rejects dechoherence as an explanation for APPARENT collapse and only alludes to it it in a round about way in his textbook,

I don't see how this gives an apparent collapse. IIUC, decoherence (interaction with a thermal environment) esssentially just cause the off-diagonal terms in the state operator to decay very fast. I.e., it doesn't determine a final specific outcome, but rather reduces a quantum-probabilistic situation to one of classical probability.

Or am I missing something?

[...] Indeed where he does mention it, its more or less forced on him - you can't really escape it - but he tries to.

Where precisely are you referring to in Ballentine? (I didn't p244 that way, but maybe you had somewhere else in mind?)

Maybe it skates a bit close to the Achilles Heel of his interpretation - namely exactly how is an actual outcome selected, and even more basic - why do we get any outcome at all.

As to why we get any outcome at all, I don't see that any interpretation explains that properly -- it always seems to be some variation on "it's magic!".

 

[atyy]

I see no "error" there. As I read it, he does replace the CP, not by another postulate, but by an analytical treatment of measurement by an apparatus -- in his sections 9.2 et seq.

Yes, that's fine.

I don't think this is a "use" of the PP. The fact that taking a partial trace is legitimate when dealing with an observable that is trivial on that component of a composite system comes from the basic QM maths (i.e., the use of tensor product Hilbert spaces), hence is not itself a postulate.

Yes, but then if it is used to specify the state of a sub-ensemble (when using a filtering measurement as state preparation), the density matrix must represent a proper mixture, whereas a reduced density matrix is an improper mixture. The assumption that an improper mixture can be treated as a proper mixture is an additional assumption (equivalent to collapse).

 

[bhobba]

I don't see how this gives an apparent collapse.

It has been discussed innumerable times - no need to go through it again.

The basic idea is mathematically and observationally an improper mixed state is indistinguishable from a proper one. That's what is meant - and I think apparent is a very apt description of it, and since I have seen others such as dymystifyer use it, I am not the only one. If you don't think its apt - arguing about it won't change anything - semantics is a rather silly thing to argue about - simply understand that's what decoherence proponents mean.

Thanks
Bil

 

[bhobba]

Where precisely are you referring to in Ballentine? (I didn't p244 that way, but maybe you had somewhere else in mind?)

Page 241 on the spin recombination experiment. He discusses it using the decoherence paradigm and an interesting different paradigm that gives the same answer. He doesn't name it though. In fact its well known Ballentine doesn't believe decoherence is of any value interpretation wise - and in fact - within the paradigm of his interpretation its pretty useless - unless you are worried about some issues he doesn't worry about. And no I don't want to discuss what that is - if you or anyone is interested it really requires another thread.

As to why we get any outcome at all, I don't see that any interpretation explains that properly -- it always seems to be some variation on "it's magic!".

Well its a bit more reasonable in BM where particles have a well defined position and momentum while in MWI you must experience some world.

Thanks
Bill

 

[bhobba]

I see no "error" there. As I read it, he does replace the CP, not by another postulate, but by an analytical treatment of measurement by an apparatus -- in his sections 9.2 et seq.

There is no error - and indeed its replaced by other reasonable assumptions and/or analysis.

I use physical continuity, and I thought Ballentine did as well, but for the life of me can't find it in his book - may have picked it up elsewhere. Either way Ballentine reduces it to other considerations.

Thanks
Bill

 

[strangerep]

if it is used to specify the state of a sub-ensemble (when using a filtering measurement as state preparation), the density matrix must represent a proper mixture, whereas a reduced density matrix is an improper mixture. [...].

I'm missing something here, possibly because I'm too indoctrinated with Ballentine's terminology: he avoids the term "mixture" because of its ambiguity, and uses instead the terms "pure state" and "nonpure state".

You're using the term "proper mixture" to mean "nonpure state", right?
(I'd better not attempt any further response until we clear this up.)

 

[strangerep]

It has been discussed innumerable times - no need to go through it again.

It would be more helpful to give me a link to a specific thread.

[...]- simply understand that's what decoherence proponents mean.

Well, I'm not a mind-reader. That's why I raised my query -- to try and clarify "what decoherence proponents mean". When this thread was originally started, I didn't have time to participate properly -- my background on these specific points was a bit narrow. Similarly, I don't normally have time to follow every thread in the QM forum closely. But now it's the Xmas-NY break, and since this thread was re-activated, I figured I'd try to catch up on a few things.

TBH, I'm a bit disappointed that you're being so short with me. I've tried to help plenty of other people on PF in the past but I hardly ever request assistance for myself.

 

[strangerep]

I use physical continuity [...]

What do you mean by "physical continuitity". (A link to a paper or previous thread is fine if you can't be bothered explaining.)

 

[bhobba]

You're using the term "proper mixture" to mean "nonpure state", right? (I'd better not attempt any further response until we clear this up.)

Not quite.

An improper mixture is his non pure state. A proper mixture is a state where states are randomly presented for observation. If the states of a proper mixture are eigenstates of the observable then measurement problem solved - the state is there prior to observation - no collapse, nothing changes, everything sweet. If it's an improper mixture you can't tell the difference, but since it wasn't prepared the same way, but rather by decoherence (via tracing over the environment) it can't be said to be the same - there is no way to tell the difference - but you can't say for sure the state was there prior to observation. You can interpret it that way - but you can't say it is the same. That is what's meant by apparent.

There is also the issue of the pointer basis it singles out (ie the 'components' of the mixed state, which as Ballentine shows is not unique) - but that is a slightly different issue that again requires its own thread.

BTW it's all treated in the paper I constantly link to regarding this stuff:
http://philsci-archive.pitt.edu/5439/1/Decoherence_Essay_arXiv_version.pdf

It's basically a cut down version of my go-to book on this by Schlosshauer:
https://www.amazon.com/dp/3540357734/?tag=pfamazon01-20

I personally believe, like Ballentine, it should be in the library of anyone interested in QM, but then again this sort of stuff interests me a lot and may not interest those into say QFT or solid state physics.

Thanks
Bill

 

[atyy]

I'm missing something here, possibly because I'm too indoctrinated with Ballentine's terminology: he avoids the term "mixture" because of its ambiguity, and uses instead the terms "pure state" and "nonpure state".

You're using the term "proper mixture" to mean "nonpure state", right?
(I'd better not attempt any further response until we clear this up.)

In my understanding, there are two sorts of mixed states (which hopefully are the same as "nonpure states").

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. This is the sort of mixture we use in quantum statistical mechanics.

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.

 

[atyy]

TBH, I'm a bit disappointed that you're being so short with me. I've tried to help plenty of other people on PF in the past but I hardly ever request assistance for myself.

Sorry that's my fault I just had a long discussion in some thread with bhobba about this, and it turned out that basically we agreed on all technical details, but I didn't like the terminology of "apparent collapse".

As I understand, the terminology "apparent collapse" as used by some decoherence folks does not imply that when decoherence is used in any interpretation with collapse, that the need for collapse is removed. The terminology seems most appropriate to me in the many-worlds interpretation.

 

[atyy]

@strangerep, the paper bhobba linked to http://philsci-archive.pitt.edu/5439/1/Decoherence_Essay_arXiv_version.pdf has explicit examples of proper and improper mixtures and their density matrices in section 1.2.3.

 

[strangerep]

@strangerep, the paper bhobba linked to http://philsci-archive.pitt.edu/5439/1/Decoherence_Essay_arXiv_version.pdf [Bas Hensen] has explicit examples of proper and improper mixtures and their density matrices in section 1.2.3.

Yes, I was reviewing it while you were composing your previous messages...

Previously, I found it difficult to get into Hensen because he uses a version of QM with collapse-to-eigenstate as a fundamental postulate. But his description of the distinction between "proper" and "improper" mixed states is clear. Observationally, they're the same: what Ballentine calls a "nonpure state". The difference is in how they were prepared. I guess that's the whole point: using a thermal coupling to the environment, and the forgetting about the environment by tracing it out, produces a state which is observationally indistinguishable from a "proper" mixed state.

I.e., "reduction" vs "apparent reduction", though not "apparent collapse".

Thus far, I have no problem with it, and... I continue to be puzzled why there's often such fuss and long-winded discussion about environmental decoherence. For me that mechanism became compelling about 10 yrs ago when I read the paper of Ford & O'Connell in which they show analytically how coupling to a thermal field causes the off-diagonal terms in the density matrix to decay extremely fast.

ISTM, Ballentine uses a summarized form of a similar idea, but in his case the fluctuations (i.e., interaction with the environment) result in incoherent phase changes in the beams on opposite sides of the apparatus in his fig 9.2.

So... I see no conflict between Ballentine's treatment in his section 9.5, and the now-well-known mechanism called "decoherence".

But such reduction from a pure density matrix by interaction-with-environment, yielding a nonpure density matrix, is not what I understand by "collapse-to-eigenstate post-measurement". So I guess my problem was with the way I'd seen the words "collapse" and "reduction" used elsewhere (seemingly interchangeably). Good to get that sorted out.

Cheers.

Edit:

[...] but I didn't like the terminology of "apparent collapse".

I think I don't like it either.

 

[bhobba]

TBH, I'm a bit disappointed that you're being so short with me. I've tried to help plenty of other people on PF in the past but I hardly ever request assistance for myself.

Sorry mate - accept my apologies.

Its just I seem to go through the same stuff over and over - and you do indeed help me and others quite a bit.

So fire away and I will do my best.

Thanks
Bill

 

[bhobba]

What do you mean by "physical continuitity". (A link to a paper or previous thread is fine if you can't be bothered explaining.)

Its associated with the idea of 'filtering' measurements from Ballentine - page 246. These are measurements that don't destroy the state but rather change it as a result of the measurement.

I thought he used my argument but can't find it there so it may be my imagination.

Here it is. From physical continuity we expect the same measurement just after such a measurement will give the same result. Also we expect the state to change insignificantly. Let the observable of the observation be ∑ yi bi><bi|. Suppose yi is the outcome. If P is the state after the observation then Trace (P |bi><bi|) = 1. A little math shows P must be|bi><bi| (I will post the detail of you like). This is the projection postulate ie the state after the observation is the corresponding eigenvector of the outcome.

Thanks
Bill

 

[strangerep]

Its associated with the idea of 'filtering' measurements from Ballentine - page 246. These are measurements that don't destroy the state but rather change it as a result of the measurement.

[...] From physical continuity we expect the same measurement just after such a measurement will give the same result. Also we expect the state to change insignificantly. Let the observable of the observation be ∑ yi bi><bi|. Suppose yi is the outcome. If P is the state after the observation then Trace (P |bi><bi|) = 1. A little math shows P must be|bi><bi| (I will post the detail of you like). This is the projection postulate ie the state after the observation is the corresponding eigenvector of the outcome.

OK, I think I understand what you're talking about, though perhaps "idempotency of filtering operators" may be a more specific phrase than "physical continuity" in this context.

It's curious that Ballentine [p246, bottom] cites the Stern-Gerlach setup as example of "measurement of the filtering type, in which the ensemble of systems generated by the ρ-state preparation is separated into subensembles according to the value of the dynamical variable R". He apparently considers that "measurement" has occurred immediately after the beam has passed the non-uniform magnetic field and divided into 2 beams. However, I maintain that "measurement" has not occurred until a silver atom (or whatever) has impacted a final detector and caused a count to increment. Before that, one has merely applied an operator (implemented by interaction with the magnetic field), obtaining a new state but not yet a number.

 

[atyy]

It's curious that Ballentine [p246, bottom] cites the Stern-Gerlach setup as example of "measurement of the filtering type, in which the ensemble of systems generated by the ρ-state preparation is separated into subensembles according to the value of the dynamical variable R". He apparently considers that "measurement" has occurred immediately after the beam has passed the non-uniform magnetic field and divided into 2 beams. However, I maintain that "measurement" has not occurred until a silver atom (or whatever) has impacted a final detector and caused a count to increment. Before that, one has merely applied an operator (implemented by interaction with the magnetic field), obtaining a new state but not yet a number.

I'm not sure if this is what Ballentine had in mind, but I pictured here something like Fig 2 of http://www.physics.arizona.edu/~cronin/Research/Lab/some%20decoherence%20refs/zurek%20phys%20today.pdf , where he puts a detector in the path, and then has decoherence and collapse so that the detector reads a definite outcome. Ballentine mentions decoherence on p245, so perhaps he's thinking of something similar on p246.