Insights The 7 Basic Rules of Quantum Mechanics

[atyy]

The claimed error in Ballentine is that he omits the projection postulate. The 7 Basic Rules aren't clear on whether this is possible. They state "The most general kind of quantum measurement and the resulting prepared state is described by so-called positive operator valued measures (POVMs)." What does "described" mean here? Do we need a postulate which generalizes the projection postulate or can this description be derived from the other postulates?

I think this is a separate question from the one I described above. The 7 Basic Rules, as stated, don't use the more general POVM formalism and so they are limited in application.

An early draft of the statement "Basdevant 2016; ... and for measurements not defined by self-adjoint operators but by POVMs." in the 7 Basic Rules was suggested by me. In my original suggestion (which included many of the textbooks referenced in A. Neumaier's final version, but did not refer to Ballentine) , I intended it to mean that the projection postulate is not the most general state reduction postulate, and was thinking that rule 7 can be replaced by something like the state reduction postulate in Nielsen and Chuang. I did not intend to suggest that state reduction can be derived from the other 6 postulates alone. Nielsen and Chuang also have the interesting statement that derivations of the Born rule and state reduction postulate remain controversial, and they have therefore included both in their postulates.

 

[vanhees71]

The claimed error in Ballentine is that he omits the projection postulate. The 7 Basic Rules aren't clear on whether this is possible. They state "The most general kind of quantum measurement and the resulting prepared state is described by so-called positive operator valued measures (POVMs)." What does "described" mean here? Do we need a postulate which generalizes the projection postulate or can this description be derived from the other postulates?

So I don't think that the question of whether Ballentine contains this error can be resolved by the 7 Basic Rules.

We neither need the projection postulate nor a generalization, because what's happening to the system and its description when interacting with a measurement or filter device depends on the specific experimental setup. It's only an opinion that Ballentine's ensemble interpretation without the projection postulate of some generalization of it were incomplete. I you consider real-world experiments, you have a preparation procedure which you have to describe well enough as the initial state of the system in the quantum formalism. Then you have some Hamiltonian describing the system's dynamics and then measure it. What's predicted by QT are the probabilities for the outcome of these measurements.

If you want to know the state of the system after these measurements you must consider this again as a preparation procedure (if you cannot include the interaction with the measurement devices with sufficient accuracy in the Hamiltonian describing the time evolution of the system). Whether or not you perform a more or less well realized projection measurement (corresponding to the collapse postulate) or not depends on the setup and cannot be generally postulated.

 

[PeterDonis]

We neither need the projection postulate nor a generalization

If you want to know the state of the system after these measurements you must consider this again as a preparation procedure

Don't these two statements contradict each other? Rule 7 in the Insights article points out, correctly, that the projection in the projection postulate is a preparation procedure.

 

[vanhees71]

Yes, and in this formulation it's ok. It's not a general postulate but describes a preparation procedure, i.e., it's referring to specific experimental setups and not to a general description of the behavior of a quantum system as the "dynamical postulates" (unitary time evolution) do.

In other words, the projection postulate is the description of a specific kind of preparation procedure and not a fundamental postulate of the quantum formalism. So there's no contradiction in my statement but it's the statement!

 

[PeterDonis]

it's referring to specific experimental setups and not to a general description of the behavior of a quantum system as the "dynamical postulates" (unitary time evolution) do

According to the 7 Basic Rules as given in the Insights article, unitary time evolution only applies to an isolated quantum system. So it is also only referring to a specific experimental setup. Quantum systems in general are not isolated; you have to make a special effort to set up an isolated quantum system in the lab.

 

[atyy]

Yes, and in this formulation it's ok. It's not a general postulate but describes a preparation procedure, i.e., it's referring to specific experimental setups and not to a general description of the behavior of a quantum system as the "dynamical postulates" (unitary time evolution) do.

In other words, the projection postulate is the description of a specific kind of preparation procedure and not a fundamental postulate of the quantum formalism. So there's no contradiction in my statement but it's the statement!

Yes, it's a preparation procedure. However, it is a preparation procedure that also uses the measurement outcome to label the state prepared.

 

[vanhees71]

That's of course also right, and you also have to make a special effort to realize projection postulates.

The point is that the foundation of quantum mechanics (as the foundation of classical mechanics) refers to closed systems. The behavior of open systems then is derived with many different methods (coarse-graining a la Kadanoff, Baym et al, projection formalism a la Zwanzig et al, influence functional formalism a la Feynman, Vernon, Caldeira, Leggett et al,...).

 

[A. Neumaier]

(Sigh.) I see now that I should have made time to proof-read that Insights article instead of ignoring it. @A. Neumaier mentions eq(9.21) and p243f, which seem to me to be incorrect references. Although Arnold says (in the comments) that what he wrote about Ballentine was designed to be compatible with what he says in his book, I think it's a bit misleading and open to misinterpretation. But before discussing that, we need Arnold to check those references. (?)

You can proofread it now and post your comments here.

Agreed. I had raised the possibility earlier that something might have changed between editions of Ballentine, but that turned out not to be the case. So it looks like we'll need to make some corrections to the article.

In the Insight article, I had originally stated in the second paragraph:

[Even Ballentine 1998, who rejects rule (7) = his process (9.9) as fundamental, derives it in the form (9.21) as an effective rule.]

I now replaced it by the more accurate

[Even Ballentine 1998, who rejects rule (7) = his process (9.9) as fundamental, derives it at the bottom of p.243 as an effective rule.]

On p.241, Ballentine writes: ''Some evidence that the state vector retains its integrity, and is not subject
to any “reduction” process, is provided by [...]''. No state reduction is his basic credo that he wants to support here. He says on the next page that state reduction should produce a mixed state, (9.18), and on p.243 that in a spin recombination experiment, only the pure state (9.21) is compatible with the experimental results. This is his ''evidence''. Since there was no measurement at the point B/C of investigation - only unitary 2-state dynamics happens -, this is no surprise, anyone would agree. It is not a situation where state reduction should be invoked. Thus his ''evidence'' is bogus.

On the other hand, at the end of page 243 he says

Thus we see that the so-called “reduced” state is physically significant in certain circumstances. But it is only a phenomenological description of an effect on the system (the neutron and spectrometer) due to its environment (the cause of the noise fluctuations)''.

This is the effective rule referred to in the Insight article.

 

[A. Neumaier]

The 7 Basic Rules, as stated, don't use the more general POVM formalism and so they are limited in application. Perhaps we need to either augment the article or do a follow-up article to cover how the rules need to be generalized to the POVM formalism. If there is interest in doing that, I'll start a separate thread on that topic (and post a link to it here).

Postulate 7 in the Insight article was explicitly restricted to the special case of projective von Neumann experiments. In the formal comments to the rule, the more general case of POVM measurements is mentioned but not detailed.

Indeed, POVMs also feature state reduction under measurement, though not projective ones. Instead, the posterior state after a measurement is obtained from the prior state by the application of the POVM operator corresponding to the measurement result obtained. For a discussion of POVMs in terms of a single basic postulate see my paper Born's rule and measurement.

 

[PeterDonis]

@A. Neumaier, thanks for the clarifications!

Since there was no measurement at the point B/C of investigation - only unitary 2-state dynamics happens -, this is no surprise, anyone would agree. It is not a situation where state reduction should be invoked. Thus his ''evidence'' is bogus.

Yes, I agree with this. I think the experiment he describes is interesting because of the fact that coherence is maintained during the passage of the neutron through a solid object, but I agree it doesn't involve any measurement at B/C so it doesn't tell us anything about state reduction as a result of measurement.

 

[strangerep]

You can proofread it now and post your comments here.

I'm still seeing the old version, so I'll wait for the new version to appear and then proofread it.

 

[A. Neumaier]

I'm still seeing the old version, so I'll wait for the new version to appear and then proofread it.

Strange. The new version is online for 18 hours. Maybe you got a cached version. Note that I only edited a few words in that sentence.

 

[PeterDonis]

The new version is online for 18 hours.

I'm seeing the new version.

 

[strangerep]

You can proofread it now and post your comments here.

It's still dated May 11, 2019, but I now see your modified sentence.
[ @Greg Bernhardt: is there a way for a "last-modified" date to be automatically included in these Insights, as well as the original date?]

I now replaced it by the more accurate

Even Ballentine 1998, who rejects rule (7) = his process (9.9) as fundamental, derives it at the bottom of p.243 as an effective rule.

I see no such derivation at the bottom of p243. Rather, the last paragraph on that page talks about how an imperfect apparatus could give rise to the "reduced" state eq(9.18) by environmental decoherence mechanisms. This is not a "non-destructive projective measurement" of the type addressed by Rule 7. Hence it is incorrect to link the two, as you currently do.

On p.241, Ballentine writes: ''Some evidence that the state vector retains its integrity, and is not subject
to any “reduction” process, is provided by [...]''. No state reduction is his basic credo that he wants to support here. He says on the next page that state reduction should produce a mixed state, (9.18), and on p.243 that in a spin recombination experiment, only the pure state (9.21) is compatible with the experimental results. This is his ''evidence''. Since there was no measurement at the point B/C of investigation - only unitary 2-state dynamics happens -, this is no surprise, anyone would agree. It is not a situation where state reduction should be invoked. Thus his ''evidence'' is bogus.

I think you misread Ballentine's sect 9.5. As I read it, Ballentine's point (starting at the 2nd paragraph on p242) is this: IF one supposed that all coherence were lost between the wavefunctions at points B and C, then the spin state should be (9.18), i,e., $$\rho^{inc} ~=~ \frac12 \; \Big( |+\rangle \langle +| ~+~ |-\rangle \langle -|\Big).$$ But then, the spin-recombination experiment (with sufficiently good apparatus) described on the rest of p242 and over onto the top of p243, would reveal one's error.

That's what he means by "evidence" (in my humble opinion, of course, since I'm not a mind reader, though neither is anyone else around here, afaik). In other words, IF one (mistakenly) assumed reduction at points B and C, the actual experiment furnishes evidence of one's mistake.

 

[A. Neumaier]

[ @Greg Bernhardt: is there a way for a "last-modified" date to be automatically included in these Insights, as well as the original date?]

Such an addition would be nice indeed.

 

[gill1109]

Continue reading...

I miss qBism as one of the interpretations. It is certainly is presently quite popular. It is a bit more than "shut up and calculate" because it is the claim that this is all that physics ought to do, namely tell an agent what they ought to believe given what they presently know.

I also read in the description of the Quantum Mechanics forum that there was a separate physics forum for Interpretations of Quantum Mechanics, but it seems that this never materialised.

So questions can be put about foundational issues, after all?

 

[bhobba]

So questions can be put about foundational issues, after all?

Of course, Gill. There is a very active subforum on the QM forum about foundational and interpretation issues. The only rule is our general rule against purely philosophical posts. It is recognised that it will occasionally be tough to avoid such problems, so mentors will keep an eye on it to ensure it doesn't get out of hand. I want to emphasise we have the philosophy rule, not because we are anti-philosophy on this forum. We had a sub-forum on it for many years. It just became low quality, and we do not have the mentors expert to ensure it is of the appropriate standard.

Arnold has recently posted an interesting paper on his interpretation:
https://www.physicsforums.com/threads/quantum-mechanics-via-quantum-tomography.1007993/

Thanks
Bill

 

[Demystifier]

a separate physics forum for Interpretations of Quantum Mechanics, but it seems that this never materialised.

 

[vanhees71]

WHAT? We've more postings about interpretation than about the "real" QT (pun intended)!

 

[PeterDonis]

I also read in the description of the Quantum Mechanics forum that there was a separate physics forum for Interpretations of Quantum Mechanics, but it seems that this never materialised.

Yes, it did:

https://www.physicsforums.com/forums/quantum-interpretations-and-foundations.292/

 

[Pari338]

The article is based on a first draft by @atyy and several improved versions by @tom.stoer. Other significant contributors to the discussions included @fresh_42, @kith, @stevendaryl, and @vanhees71.
I slightly expanded the final version and added headings and links to make it suitable as an insight article. Maybe the participants of the discussion 20 months ago can confirm their continued support or voice disagreements with this public version.

Nice i am new in physics its very helpful for me