Errors in Ballentine (QM Textbook)?

[Demystifier]

A system is described by an observable algebra (realized usually by a set of self-adjoint operators) on an appropriate Hilbert space. The properties of the system is described by the statistical operator, representing its state. What else should there be?

So there are two things, the state (which is deterministic) and the observables (which are random). I have two questions.

1) Do observables have random values when they are not measured?

2) If observables are random, how is it compatible with the Heisenberg picture where the observable operator has a deterministic evolution with time?

 

[vanhees71]

1) doesn't make sense, and it's not what quantum theory says. Quantum theory tells you the probability to find a certain value when measuring an observable, given the state the measured system is prepared in.

2) Quantum theory doesn't depend on the picture of time evolution used. The observable (probabilistic) predictions of quantum theory are always in the picture-independent matrix elements of the statistical operator,
$$\rho(t,o,o') \langle o,t|\hat{\rho}(t)|o,t' \rangle.$$
where $|o,t \rangle$ is a common eigenvector of a complete set of compatible observables $O$.

Both the equations of motion for the states (statistical operators) and the self-adjoint operators representing observables in an arbitrary picture of time evolution are of course deterministic.

 

[Demystifier]

1) doesn't make sense, and it's not what quantum theory says. Quantum theory tells you the probability to find a certain value when measuring an observable, given the state the measured system is prepared in.

Does quantum theory say anything about those values when they are not measured?

2) Quantum theory doesn't depend on the picture of time evolution used. The observable (probabilistic) predictions of quantum theory are always in the picture-independent matrix elements of the statistical operator,
$$\rho(t,o,o') \langle o,t|\hat{\rho}(t)|o,t' \rangle.$$
where $|o,t \rangle$ is a common eigenvector of a complete set of compatible observables $O$.

Both the equations of motion for the states (statistical operators) and the self-adjoint operators representing observables in an arbitrary picture of time evolution are of course deterministic.

So there are 3 things, not 2. The state (which is deterministic), the observable operator (which is also deterministic), and the value of the observable operator (which is random). Is that right?

 

[vanhees71]

Quantum theory predicts the values of observables when measured. Physics doesn't care about unobserved things.

An observable operator doesn't take values. It's a linear mapping $\mathcal{H} \rightarrow \mathcal{H}$.

I'm a bit puzzled why we are discussing these completely basic undisputed facts about QT all of a sudden.

 

[Demystifier]

Physics doesn't care about unobserved things.

So why do you care that conserved charge exists even when it is not measured?

 

[Demystifier]

I'm a bit puzzled why we are discussing these completely basic undisputed facts about QT all of a sudden.

That's my strategy of phishing, to catch you in an inconsistency.
Now I think I know what exactly is inconsistent in your interpretation, it's inconsistent double standards of relevancy.

 

[MathematicalPhysicist]

So why do you care that conserved charge exists even when it is not measured?

Some might say that an unobserved universe doesn't exist; for an observer to exists in the first place it or he needs a universe to exists in and for such a universe to exists it needs an observer in it that will observe/notice its existence.

But this is just philosophy...
Life goes in circles anyways.

 

[kith]

I'm not happy with calling it a postulate for the said reason. It's the definition of a special (usually idealized) kind of experiments. It's rather a question of how to apply the theory to a specific kind of preparation-observation procedures in each individual case of such a kind of experiment.

I have quoted the general version of the postulate in the other thread. Do you think the state-after-measurement rule (2.93) there shouldn't be included as part of a postulate because it can be derived from the other postulates? (I'm not sure if it makes sense to keep these two threads separate)

 

[PeterDonis]

There is only unitary evolution of the quantum state when considering a closed system.

The term "closed system", at least as it is used in the 7 Basic Rules Insights article, does not include any system on which a measurement is being made. So your statement here, while true, is irrelevant to what happens when a measurement is made, which is the case under discussion.

There is no randomness in the evolution of the quantum state of a closed system at all.

See my bolded addition above. With that addition, you are simply stating a property of unitary evolution of a closed system, as that term is defined above. But again, that property is not relevant to what happens when a measurement is made.

 

[PeterDonis]

Everyone, please bear in mind that this thread is about the specific discussion of errors in Ballentine, and that we are in the regular QM forum, where the accepted statement of the postulates of QM is that given in the 7 Basic Rules Insights article. We are investigating the possibility that the Ballentine reference in that article might need to be corrected, but that is not intended to open the doors to a general discussion of everyone's views of QM. Also please bear in mind that interpretation discussions belong in the interpretations subforum, not this one.

 

[PeterDonis]

I'm not sure if it makes sense to keep these two threads separate

This thread is specifically about Ballentine, as I noted in my previous post just now. And as I also noted in that post, the relevant version of any postulate for purposes of this thread is what is in the 7 Basic Rules Insights article, not any other source.

 

[PeterDonis]

Since there seems to be enough information indicating that we will need to make some corrections to the 7 Basic Rules Insights article, I have moved posts specifically on that topic to the discussion thread for the article, here:

https://www.physicsforums.com/threads/the-7-basic-rules-of-quantum-mechanics.971724/

Please move further discussion of what corrections need to be made to the Insights article to that thread.

 

[kith]

I think this is a separate question from the one I described above.

In principle, yes. In practice, part of the critique of the projection postulate is that it isn't general. This discussion has been going on at PF without resolution for a long time and currently, I think the best road to identify the core of the issue is the general case. I think this thread and it's spin-off have contributed quite a bit here, so in any case thanks for starting them.

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).

I would appreciate this but I can only contribute limited time and limited expertise.

 

[atyy]

The term "closed system", at least as it is used in the 7 Basic Rules Insights article, does not include any system on which a measurement is being made. So your statement here, while true, is irrelevant to what happens when a measurement is made, which is the case under discussion.

I suspect that @vanhees71 refers to a closed system, because he believes that we can in principle include the observer and measurement apparatus in the quantum state, so that there is only unitary evolution. This is also my reading of what Ballentine means in his textbook, given his criticism of standard QM. I believe that postulating unitary evolution without state reduction is not correct unless one introduces additional postulates (eg. as attempted by many worlds, hidden variables, which also remain non-standard).

 

[PeterDonis]

In practice, part of the critique of the projection postulate is that it isn't general. This discussion has been going on at PF without resolution for a long time and currently, I think the best road to identify the core of the issue is the general case.

As I noted in post #82 a little bit ago, I have moved that discussion to the comment thread on the Insights article.

 

[PeterDonis]

I suspect that @vanhees71 refers to a closed system, because he believes that we can in principle include the observer and measurement apparatus in the quantum state, so that there is only unitary evolution.

But just including the observer and measurement apparatus is not enough. You also have to include the environment, which potentially can include the entire rest of the universe.

I think at this point things become highly interpretation-dependent.

This is also my reading of what Ballentine means in his textbook, given his criticism of standard QM.

I'm not sure Ballentine's viewpoint is that of "only unitary evolution", because he believes quantum measurements have single outcomes. You can't get single outcomes out of only unitary evolution. I'm not sure Ballentine is taking any of the alternative viewpoints you mention (many worlds, hidden variables, etc.), but it doesn't seem to me like he is taking an "only unitary evolution" viewpoint either.

 

[strangerep]

I'm not sure Ballentine's viewpoint is that of "only unitary evolution", because he believes quantum measurements have single outcomes.

Please give a specific reference that supports your account of what Ballentine supposedly believes.

[Sorry, but I don't like unsupported verbaling. In the spirit of the PF rules, such claims need to be supported by appropriate references.]

 

[PeterDonis]

Please give a specific reference that supports your account of what Ballentine supposedly believes.

Um, his entire textbook?

Seriously, I'm not sure where to start, since the assumption that individual measurements have single outcomes seems to me to be there in pretty much everything he says. Certainly it seems to be a necessary assumption of the ensemble interpretation that he explicitly adopts. If you really aren't seeing that when you read his textbook, then I can try to pick out particular passages that give me that impression.

 

[atyy]

I'm not sure Ballentine's viewpoint is that of "only unitary evolution", because he believes quantum measurements have single outcomes. You can't get single outcomes out of only unitary evolution. I'm not sure Ballentine is taking any of the alternative viewpoints you mention (many worlds, hidden variables, etc.), but it doesn't seem to me like he is taking an "only unitary evolution" viewpoint either.

When I mean his point of view is only unitary evolution, I mean only unitary evolution as part of the first 6 postulates in the 7 Basic Rules, ie. including the Born Rule but excluding state reduction, and probably without hidden variables. I think this is the most plausible reading of his text, because of what he says in the section "The measurement theorem for general states" which contains Eq 9.10 to 9.13 (interpretation apart, the mathematics is essentially the same as Zurek's Eq 1-5 and Eq 6, without Eq 7 in https://arxiv.org/abs/quant-ph/0306072). In this section Ballentine uses only unitary evolution, shows that state reduction is not the outcome of unitary evolution, and uses that an argument against accepting state reduction.

The Born rule gives single measurement outcomes, but it does not give quantum states corresponding to the single measurement outcomes, which is why I think Ballentine wrongly rejects the state reduction postulate without replacing it with anything else.

The other plausible reading, but I think less likely, is that Ballentine assumes hidden variables, since he refers to Einstein's Ensemble interpretation. But like you, I think this is not likely what he means (otherwise vanhees71 would not read Ballentine as a minimal statistical interpretation).

 

[PeterDonis]

That's precisely the question. Which additional postulates do you mean?

Since your post took this question far beyond just a question about Ballentine specifically, and well over the line into interpretation, I have moved it to the other thread in the interpretations forum where collapse is being discussed:

https://www.physicsforums.com/threads/difference-between-collapse-and-projection.998545/post-6445336

 

[vanhees71]

Since your post took this question far beyond just a question about Ballentine specifically, and well over the line into interpretation, I have moved it to the other thread in the interpretations forum where collapse is being discussed:

https://www.physicsforums.com/threads/difference-between-collapse-and-projection.998545/post-6445336

Fine with me, but if this splits in zillions of subthreads it's hard to follow. I think the claim that Ballentine's book is "wrong" is just the claim that the ensemble interpretation is "wrong". So why not keeping the postings in one thread such that the context of the arguments is clear.

But I think, I made my argument now several times, and don't need to repeat it further anyway.

 

[PeterDonis]

I think the claim that Ballentine's book is "wrong" is just the claim that the ensemble interpretation is "wrong".

And that discussion belongs in the thread in the interpretations subforum, which is where I have moved all posts along those lines.

why not keeping the postings in one thread such that the context of the arguments is clear.

Because interpretation discussions are always matters of opinion. If Ballentine prefers the ensemble interpretation, that's his business. No one can say he's "wrong" for doing that; doing it is not an "error" in his textbook. But if there are errors in Ballentine about minimal QM itself, apart from any interpretation, that is what this thread, which is in the regular QM forum, is about.

 

[atyy]

There are ways to derive the state reduction rule without hidden variables. They involve using a simultaneous measurement (for which the Born rule applies) to define a sequential measurement (which requires state reduction), and requiring consistency between the simultaneous and sequential calculations.

Heuristically, one can see this in Bell tests, where if the measurement is simultaneous in one frame, it is sequential in another, then it can be seen that state reduction is required for consistency.

Another example is found in using an instrument to define state reduction, where a simultaneous measurement on the apparatus and the system is considered to be equivalent to sequential measurements on the system, eg. section 6.2.3 on Conditional output states in
https://arxiv.org/abs/0810.3536
Guide to Mathematical Concepts of Quantum Theory
Teiko Heinosaari, Mario Ziman

However, these are not compatible with Ballentine's assertion that unitary evolution alone should disallow state reduction and they are not compatible with Ballentine's criticism of Messiah's statement that the measurement unpredictably disturbs the system. In the above notes by Heinosaari and Ziman, section 6.3.1 says No information without disturbance. If one is not continually enlarging the Hilbert space with each measurement, then the state reduction postulate does give correct quantum mechanics.

 

[A. Neumaier]

@A. Neumaier will have to clarify that part, as I think it wasn't in the drafts I read, or I missed it. However, the possibility is the edition and page numbers are correct, and that A. Neumaier read that as a derivation of effective state reduction, because that is what Ballentine intends 9.21 to be. At this point, Ballentine believes that Interpretation A has a state reduction, and he is trying to explain why Interpretation A seems to work most of the time.

See my comments here.

 

[PeterDonis]

Moderator's note: A discussion about the ensemble interpretation and the PBR theorem has been moved to a new thread in the interpretations subforum:

https://www.physicsforums.com/threa...ion-inconsistent-with-the-pbr-theorem.998624/

 

[atyy]

See my comments here.

Thanks. I think the revised comments have essentially the same meaning as the original comments.

With respect to the subject of the OP of this thread, I think Ballentine's derivation in that section is problematic, as it
(i) is in the context of wrongly assuming that the standard interpretation has a state reduction, where the standard interpretation has none.
(ii) on p244, Ballentine says about his derivation that "This “reduction” of the state is not a new fundamental process, and, contrary to the impression given in some of the older literature, it has nothing specifically to do with measurement."

Well, perhaps it is not a new fundamental process (we don't care about that in the orthodox interpretation, as the state is just a way of calculating probabilities of measurement outcomes), but given that Nielsen and Chuang still state reduction as a postulate, explicitly acknowledging that its derivation is controversial, it still remains correct to state it as a postulate. And even if one derives state reduction in the orthodox interpretation by defining it via consistency of simultaneous and sequential measurements (reference in post #93), there the state reduction is specifically related to measurement, and specifically with the measurement outcome.

 

[strangerep]

Um, his entire textbook? [...]

Oh, I see now what you meant. I misunderstood you before.

(It's probably time for me to exit this thread, take a Bex, and have a good lie down.)

 

[A. Neumaier]

Thanks. I think the revised comments have essentially the same meaning as the original comments.

With respect to the subject of the OP of this thread, I think Ballentine's derivation in that section is problematic, as it
(i) is in the context of wrongly assuming that the standard interpretation has a state reduction, where the standard interpretation has none.
(ii) on p244, Ballentine says about his derivation that "This “reduction” of the state is not a new fundamental process, and, contrary to the impression given in some of the older literature, it has nothing specifically to do with measurement."

Well, perhaps it is not a new fundamental process (we don't care about that in the orthodox interpretation, as the state is just a way of calculating probabilities of measurement outcomes), but given that Nielsen and Chuang still state reduction as a postulate, explicitly acknowledging that its derivation is controversial, it still remains correct to state it as a postulate. And even if one derives state reduction in the orthodox interpretation by defining it via consistency of simultaneous and sequential measurements (reference in post #93), there the state reduction is specifically related to measurement, and specifically with the measurement outcome.

I agree.

 

[vanhees71]

Heuristically, one can see this in Bell tests, where if the measurement is simultaneous in one frame, it is sequential in another, then it can be seen that state reduction is required for consistency.

This is self-contradictory: According to local relativistic QFT (in this case particularly QED) describes all findings correctly. This implies that there can be no causal effect between measurement events that are spacelike separated (that's a mathematical statement!). So there can be no state reduction through the measurement at one place affecting causally the outcome of the (in some frame) later measurement at the other place.

Since nature is frame-independent if there's no state reduction in one frame, there cannot be one in any other.

 

[atyy]

This is self-contradictory: According to local relativistic QFT (in this case particularly QED) describes all findings correctly. This implies that there can be no causal effect between measurement events that are spacelike separated (that's a mathematical statement!). So there can be no state reduction through the measurement at one place affecting causally the outcome of the (in some frame) later measurement at the other place.

Since nature is frame-independent if there's no state reduction in one frame, there cannot be one in any other.

Quantum mechanics is not about cause and effect. It is only about predicting the probabilities of measurement outcomes.

 

[Demystifier]

Quantum mechanics is not about cause and effect. It is only about predicting the probabilities of measurement outcomes.

No, standard (orthodox) quantum mechanics is about proving locality, whatever it takes. Sometimes it takes cause and effect, sometimes it takes denying cause and effect, sometimes it takes objective reality, sometimes it takes denying objective reality.

 

[martinbn]

No, standard (orthodox) quantum mechanics is about proving locality, whatever it takes. Sometimes it takes cause and effect, sometimes it takes denying cause and effect, sometimes it takes objective reality, sometimes it takes denying objective reality.

Ha! The thief is screaming "thief"!

 

[Demystifier]

Ha! The thief is screaming "thief"!

Please elaborate!

 

[martinbn]

Please elaborate!

Well, it seems that all fans of BohminanMech are only interested in proving non-locality.

 

[Demystifier]

Well, it seems that all fans of BohminanMech are only interested in proving non-locality.

Maybe, but to do it we don't need to contradict ourselves, which orthodox guys do. Sure, we must assume the existence of something not directly seen in experiments, and we are not very happy with that, but we believe it's a much smaller sin than self-contradiction.