I Is the collapse indispensable?

[A. Neumaier]

To get the definite outcome, one must further assume that the improper mixture is converted to a proper mixture, which is the same as assuming collapse.

Could you please (possibly in a new thread) explain these terms and how you think the conclusion follows, according to your understanding, so that your statement can be critically discussed?

 

[atyy]

Which standard texts are you referring to? Who but you decided that they are the standard?

These two books, together with the commented reprints in Wheeler and Zurek, are the modern standard! They devote considerable space to the foundations, whereas typical textbooks on quantum mechanics only have short sections where they parrot what they glean from elsewhere, often from the long past,

That is not correct. Wheeler and Zurek are research papers. Incidentally, Zurek states standard QM with collapse. His attempt at Quantum Darwinism is a research attempt to remove collapse. Removing collapse from QM is not standard, but a matter of research till this day, and is BTSM.

Standard texts include Landau & Lifshitz; Cohen-Tannoudji, Diu & Laloe; Nielsen & Chuang; Sakurai; Weinberg; Holevo.

 

[kith]

So you do accept the collapse as necessary, just not its physicality!

I wouldn't say that the use of projection operators necessarily needs a justification. If we use the quantum description for the full filtering system, the final state is a superposition which contains a term where the particle leaves the apparatus and a term where the particle is trapped inside the apparatus. The second term has zero overlap with states localized outside the apparatus. So the Born rule gives a probability of zero for all outcomes of all future measurements for the second term. So the predictions are the same whether we use both terms or only the first. In this case, the use of a projection operator to get rid of the second term is purely a matter of convenience.

I thought that you and I had already reached agreement on something similar in an older thread. The conclusion I remember is that we need more sophisticated situations like Bell tests to analyze whether collapse is necessary or not.

 

[atyy]

I wouldn't say that the use of projection operators necessarily needs a justification. If we use the quantum description for the full filtering system, the final state is a superposition which contains a term where the particle leaves the apparatus and a term where the particle is trapped inside the apparatus. The second term has zero overlap with states localized outside the apparatus. So the Born rule gives a probability of zero for all outcomes of all future measurements for the second term. So the predictions are the same whether we use both terms or only the first. In this case, the use of a projection operator to get rid of the second term is purely a matter of convenience.

I thought that you and I had already reached agreement on something similar in an older thread. The conclusion I remember is that we need more sophisticated situations like Bell tests to analyze whether collapse is necessary or not.

I thought we had agreed that collapse was not necessary provided that successive measurements were not made, eg. using something like the https://en.wikipedia.org/wiki/Deferred_Measurement_Principle.

 

[A. Neumaier]

Standard texts include Landau & Lifshitz; Cohen-Tannoudji, Diu & Laloe; Nielsen & Chuang; Sakurai; Weinberg; Holevo.

Why are they today's standard regarding the foundations, while Ballentine and Peres are not? What is the criterion that makes them standard?

None of these except perhaps Holevo (would have to check) specializes on the foundations but treats it in a very short way, that disqualifies it as a standard. For example Nielsen & Chuang devote just 16 pages (Section 2.2) to the topic, out of a total of over 600 pages. And even in these pages they cover a lot of ground, not just the postulates and their discussion.

The collapse is a frequently used textbook device simply because it is a convenient starting point, allowing one to bridge the abyss of quantum foundations in a few words, in agreement with the early history but without having to spend time on getting it fully correct.

You never find it discussed in a quantum field theory book, which is the true foundation of modern theoretical physics. Here everything is in terms of (in principle measurable) correlation functions, which is enough for all uses of quantum mechanics and quantum field theory in the applications.

 

[A. Neumaier]

Wheeler and Zurek are research papers.

Wheeler and Zurek is a commented reprint collection of research papers displaying the spectrum of serious alternatives, and their historical origin. It is unique in this respect (superseding an older treatise by Jammer).

This comprehensiveness and uniqueness makes it a standard. It displays the disagreement on basic issues, versions of which were already then treated as definite statements in many textbooks, among them some you cited - proving that the textbooks selected for convenience rather than representing an agreement (suggested by calling it a ''standard'').

 

[atyy]

Why are they today's standard regarding the foundations, while Ballentine and Peres are not? What is the criterion that makes them standard?

None of these except perhaps Holevo (would have to check) specializes on the foundations but treats it in a very short way, that disqualifies it as a standard. For example Nielsen & Chuang devote just 16 pages (Section 2.2) to the topic, out of a total of over 600 pages. And even in these pages they cover a lot of ground, not just the postulates and their discussion.

The collapse is a frequently used textbook device simply because it is a convenient starting point, allowing one to bridge the abyss of quantum foundations in a few words, without having to spend time on getting it correct.

You never find it discussed in a quantum field theory book, which is the true foundation of modern theoretical physics. Here everything is in terms of (in principle measurable) correlation functions, which is enough for all uses of quantum mechanics and quantum field theory in the applications.

Collapse is found in 2 QFT books: Weinberg and Dimock.

In addition to Holevo, Paul Busch's books on foundations contain collapse as a postulate.

If Ballentine and Peres were right, the measurement problem would be solved. You can see from the Laloe's http://arxiv.org/abs/quant-ph/0209123 and Wallace's http://arxiv.org/abs/0712.0149 that there is no consensus as to whether any interpretation can solve the measurement problem.

 

[A. Neumaier]

Collapse is found in 2 QFT books: Weinberg and Dimock.

Can you please give page numbers?

 

[A. Neumaier]

there is no consensus as to whether any interpretation can solve the measurement problem.

But no consensus also means no standard. In this case there is no obvious right or wrong, while you took sides and declared your favorite to be right (''the standard'') and the others wrong.

 

[atyy]

But no consensus also means no standard. In this case there is no obvious right or wrong, while you took sides and declared your favorite to be right (''the standard'') and the others wrong.

No, what it means is that standard quantum mechanics works. It makes successful predictions consistent with all observations to date. That is why quantum mechanics with collapse is standard - and yes it is obviously right in the sense of making successful predictions.

Now the question is whether the others can do just as well. Since there is no consensus as to whether they can, standard quantum mechanics remains the standard and consensus.

 

[atyy]

But no consensus also means no standard. In this case there is no obvious right or wrong, while you took sides and declared your favorite to be right (''the standard'') and the others wrong.

Furthermore, no consensus does mean there is a standard - the standard version with collapse and with the measurement problem.

 

[ddd123]

I went to read where atyy's speaks of in Weinberg and was a little disappointed. Weinberg recaps the standard collapse postulate in a paragraph about basic quantum theory which he takes as his own. It is then never mentioned again (it's a QFT book).

 

[stevendaryl]

Maybe he means something like this? http://arxiv.org/abs/hep-th/0205105

That paper is interesting, but it's hard for me to believe that it is correct.

 

[vanhees71]

vanhees71 is wrong for the following reasons.

1. Unitary evolution and the "filtering" that he imagines will allow the projection to be derived cannot do it, because the unitary evolution and partial trace caused by the "filtering" only produce an improper mixture. To get the definite outcome, one must further assume that the improper mixture is converted to a proper mixture, which is the same as assuming collapse. Ballentine and Peres are probably missing this assumption in their erroneous books.

The projectors are an effective description of the filtering process. As I said, it's hard to imagine to be able to describe the filtering process in all microscopic detail. I Ballentine and Peres are erroneous, then quantum theory itself is erroneous. There's no single empirical hint for that.

2. The "locality" of QFT that is enforced by the "local" interactions has the meaning of "no superluminal transmission of classical information" (and a little more). It does not mean local interactions and local causality. vanhees71 consistently confuses multiple meanings of "local".

3. Collapse is consistent with the "locality" of quantum field theory. It is not consistent with relativistic causality, but neither is quantum field theory.

Local QFT IS by construction consistent with relativistic causality, an instanteneous collapse obviously not! So there is a contradiction in the foundations, if you assume the collapse to be a physical process. I prefer to abandon the collapse and stick to minimally interpreted relativistic local QFT.

 

[vanhees71]

I'm afraid this thread is at risk of getting closed. Would be a pity if nobody actually answered or provided a source with an answer to atyy's point I quoted in post #104.

Well, I guess it's high time to close it. We are now at a point where the very foundations, common to all metaphysics ("interpretations") added on top, are called erroneous, which is simply ridiculous, because it means we don't even have a consensus on the physical part of the theory. How then can we expect to solve the more complicated metaphysical issues?

 

[kith]

I thought we had agreed that collapse was not necessary provided that successive measurements were not made, eg. using something like the https://en.wikipedia.org/wiki/Deferred_Measurement_Principle.

Yes. But a crucial point of the discussion was that using this terminology, filtering experiments like the one I described above shouldn't be considered measurements and therefore need no collapse. So I don't get the discussion between vanhees71 and you. To me, it looks like you two are talking at cross purposes.

 

[zonde]

That paper is interesting, but it's hard for me to believe that it is correct.

The paper says:
"Incidentally, since it is impossible to prepare a state with a definite number of photons, and since such an uncertainty for any given process cannot be made arbitrarily small, we may even argue that it is not possible to give a physical reality (not even locally) to any observable, except to the charges and masses (that are the invariants of the theory)."
It seems that the reasoning relies on detection loophole (photons can't be paired up to arbitrary high level).
I would say that this view is falsified by experiment. Say loophole free quantum steering experiment: http://arxiv.org/abs/1111.0760
And electron based experimental violations of Bell inequalities are outside the scope of the paper.

 

[ddd123]

Local QFT IS by construction consistent with relativistic causality, an instanteneous collapse obviously not! So there is a contradiction in the foundations, if you assume the collapse to be a physical process. I prefer to abandon the collapse and stick to minimally interpreted relativistic local QFT.

How does it explain Bell pairs phenomenology though? A paper I linked was deemed "probably wrong" by steveandaryl.

 

[atyy]

Yes. But a crucial point of the discussion was that using this terminology, filtering experiments like the one I described above shouldn't be considered measurements and therefore need no collapse. So I don't get the discussion between vanhees71 and you. To me, it looks like you two are talking at cross purposes.

That could well be. As far as I can tell, I don't have a technical disagreement with you. I still do have technical disagreements with vanhees71.

 

[zonde]

Well, I guess it's high time to close it. We are now at a point where the very foundations, common to all metaphysics ("interpretations") added on top, are called erroneous, which is simply ridiculous, because it means we don't even have a consensus on the physical part of the theory.

I don't want to take side in your discussion with atyy (I don't have clear viewpoint on collapse) but as I see atyy have valid reasons to doubt Ballentine approach.
Ballentine in his book says:
"There is no such difficulty with interpretation B [that is used by Ballentine], according to which the state vector is an abstract quantity that characterizes the probability distributions of the dynamical variables of an ensemble of similarly prepared systems."
So it seems like Ballentine is promoting sort of LHV. And we know now that it does not work.

 

[vanhees71]

No he doesn't. There are no hidden variables in this approach. He just takes Born's Rule as another postulate without attempting to derive it from some other principles, i.e., in his minimal interpretation the quantum mechanical state just describes probabilities for the outcome of measurements, i.e., they refer to ensembles of equally prepared systems. Quantum theory in this interpretation is just silent on what happens in measurement processes, which makes a lot of sense since what happens in all microscopic detail in the interaction between the system and the measurement apparatus depends on the setup of the specific apparatur used, and you cannot make general statements on what microscopically happens when you use some device to measure a quantity.

 

[atyy]

I don't want to take side in your discussion with atyy (I don't have clear viewpoint on collapse) but as I see atyy have valid reasons to doubt Ballentine approach.
Ballentine in his book says:
"There is no such difficulty with interpretation B [that is used by Ballentine], according to which the state vector is an abstract quantity that characterizes the probability distributions of the dynamical variables of an ensemble of similarly prepared systems."
So it seems like Ballentine is promoting sort of LHV. And we know now that it does not work.

Ballentine does not assume LHV.

Ballentine's error is that he claims Copenhagen is wrong. He also claims that collapse is wrong. His error is that he surreptitiously introduces collapse in the form of assuming that improper mixtures are proper mixtures, which is the same assumption as collapse.

 

[vanhees71]

Where does Ballentine do that? The minimal interpretation is a flavor of the Copenhagen interpretation. It's just silent about what happens during a measurement, i.e., it doesn't introduce a collapse, and that's a feature, not a bug as you claim. I still do not understand, what the collapse assumption should be good for at all. So why should one insist on something making only trouble in the quantum paradise?

 

[atyy]

Where does Ballentine do that? The minimal interpretation is a flavor of the Copenhagen interpretation. It's just silent about what happens during a measurement, i.e., it doesn't introduce a collapse, and that's a feature, not a bug as you claim. I still do not understand, what the collapse assumption should be good for at all. So why should one insist on something making only trouble in the quantum paradise?

Ballentine claims Copenhagen is wrong when he writes on p241, section 9.5 "Some evidence that the state vector retains its integrity, and is not subject to any “reduction” process, is provided by the spin recombination experiments that are possible with the single crystal neutron interferometer (see Sec. 5.5)."

He claims there is experimental evidence against Copenhagen.

The collapse is necessary, because Ballentine even introduces it in Eq 9.28. His error is that he claims that it can be derived from unitary evolution and a partial trace. In fact that cannot be done. The partial trace produces an improper mixture. Collapse is the conversion of the improper mixture to a proper mixture.

 

[kith]

@atyy and @vanhees71, your personal discussion about collapse (and Ballentine) pops up in so many threads and never seems to go anywhere. Couldn't you make a separate thread for it where you carefully state your arguments and objections? Then we wouldn't have so much reiteration in other threads. And when new aspects emerge somewhere, you could relate them more easily to what you have already discussed.

 

[zonde]

Ballentine does not assume LHV.

Are you sure? Have you read "9.3 The Interpretation of a State Vector"?
Here he dismisses Copenhagen and describes state vector of combined object and measuring apparatus as "probability distributions of the dynamical variables of an ensemble of similarly prepared systems".
Considering that measuring apparatus has to have macroscopically distinct states that can't be in supperposition it is a proper mixture. But that statement about interpretation of state vector is stated as general statement and not specific to that combined object and measuring apparatus state vector. So basically it seems like he says that any state is proper mixture i.e. LHV type model.

 

[A. Neumaier]

in the form of assuming that improper mixtures are proper mixtures, which is the same assumption as collapse.

I had already asked you to specify in detail your understanding of these terms and your reasoning for the conclusion, so that it can be critically discussed. Simply repeating this statement in a black box fashion as the sole justification for accusing a respectable author of making a fundamental error is not helpful at all.

 

[kith]

Considering that measuring apparatus has to have macroscopically distinct states that can't be in supperposition [...]

Ballentine's position is that there's no problem with having a superposition of macroscopically distinct states for the measurement apparatus because for him, states always refer to ensembles of objects and not to the individual object. He explicitly talks about such superpositions somewhere in his interpretation section.

 

[zonde]

Ballentine's position is that there's no problem with having a superposition of macroscopically distinct states for the measurement apparatus because for him, states always refer to ensembles of objects and not to the individual object. He explicitly talks about such superpositions somewhere in his interpretation section.

Yes, and it is explained in chapter 9.3 why he doesn't see a problem there. It's because supperposition does not apply to individual system. So ensemble of systems represents proper mixture.

 

[kith]

Yes, and it is explained in chapter 9.3 why he doesn't see a problem there. It's because supperposition does not apply to individual system. So ensemble of systems represents proper mixture.

It is evident from the mathematical definitions that superpositions and mixed states are different and Ballentine isn't claiming that they are the same. But this is off-topic here.

 

[atyy]

Are you sure? Have you read "9.3 The Interpretation of a State Vector"?
Here he dismisses Copenhagen and describes state vector of combined object and measuring apparatus as "probability distributions of the dynamical variables of an ensemble of similarly prepared systems".
Considering that measuring apparatus has to have macroscopically distinct states that can't be in supperposition it is a proper mixture. But that statement about interpretation of state vector is stated as general statement and not specific to that combined object and measuring apparatus state vector. So basically it seems like he says that any state is proper mixture i.e. LHV type model.

Ballentine may surreptitiously assume hidden variables. It is hard to say. Certainly his 1970 review makes that error. bhobba believes Ballentine corrects it in his 1998 book.

However, Ballentine explicitly rejects LHV although he is sympathetic to HV (Bohmian Mechanics), and yet believes his interpretation is agnostic to HV.

 

[zonde]

I would like to ask related question.
As I understand any treatment of measurement in QM assumes that measurement apparatus can be approximated as a pure state undergoing unitary evolution.
Is this right?

 

[A. Neumaier]

any treatment of measurement in QM assumes that measurement apparatus can be approximated as a pure state undergoing unitary evolution.
Is this right?

No. Most treatments of measurement are agnostic about the properties of the measurement device.

Then there are Wigner's friend type arguments that assume what you just wrote; they lead to an infinite regress.

Finally there are more realistic statistical mechanics treatments of the measurement problem. There the measurement device is assumed to be in a mixed state, more precisely a state close to a thermal equilbrium state.

 

[rkastner]

Well, that's the Peres and Ballentine claim. Is it correct that with only unitary evolution you can derive collapse? Till this day you have never exhibited a derivation, neither have Peres nor Ballentine. It's a pity that quantum mechanics is still not understood even by experts.

I have provided a possible physical understanding of the QM formalism. I'm not going to be dogmatic and say it's the only possible or correct interpretation, but as I've noted, TI provides a physical basis for the Born Rule--you just read it off the dynamics. The transactional process corresponds to Von Neumann's measurement transition from a pure to a mixed state (weighted set of projection operators where the weights are the Born Rule probs for each outcome described by each projection operator). The direct action theory underlying TI provides the missing link which introduces non-unitarity through absorber response.
Dismissals of TI are generally based on an aversion to considering the advanced solutions as physical, and unwillingness to consider the possibilist development of TI (PTI) are generally based on uncritically equating "physically real" with "spacetime object". How does anyone know that spacetime is the whole story for what exists? It's just a metaphysical assumption, a ground rule that physicists have worked with--but it is not mandatory. Since relativity applies to spacetime, and quantum systems display nonlocality that seems to violate relativity, a natural inference is that quantum systems do not live in spacetime, but that they still have physical reality, in that they can lead in a lawful (even if indeterministic) way to observable (spacetime) phenomena. Niels Bohr even said that 'quantum jumps transcend the frame of space and time' (I can look up the reference if someone needs it).
Also, re the idea that the quantum state is just an instrument for predicting probabilities, here's an experiment that purports to demonstrate the reality of the wavefunction and its collapse: http://scitechdaily.com/quantum-exp...-wavefunction-collapse-for-a-single-particle/

 

[bhobba]

Also, re the idea that the quantum state is just an instrument for predicting probabilities, here's an experiment that purports to demonstrate the reality of the wavefunction and its collapse: http://scitechdaily.com/quantum-exp...-wavefunction-collapse-for-a-single-particle/

Those kinds of experiments have been discussed a lot here.

Mostly they seem to be a misunderstanding of weak measurements.

If they did what was claimed it would be VERY big news.

That links claims about Einstein are also incorrect. He ascribed to the Ensemble interpretation and after his tussles with Bohr accepted QM as correct - but to his dying day incomplete. But as a populist article that sort of thing is only to be expected.

Thanks
Bill

 

[atyy]

Those kinds of experiments have been discussed a lot here.

Mostly they seem to be a misunderstanding of weak measurements.

If they did what was claimed it would be VERY big news.

That links claims about Einstein are also incorrect. He ascribed to the Ensemble interpretation and after his tussles with Bohr accepted QM as correct - but to his dying day incomplete. But as a populist article that sort of thing is only to be expected.

Thanks
Bill

It is important to stress that Einstein's "ensemble interpretation" is not the same as that in Ballentine's 1998 book. Einstein's ensemble interpretation was a hidden variable interpretation. Furthermore, Einstein hoped for a local hidden variable interpretation. We now know that local hidden variables are not consistent with all the predictions of quantum mechanics. Nonlocal hidden variables are still are possibility, and have been explicitly demonstrated for non-relativistic quantum mechanics, and some relativistic quantum theories.

 

[bhobba]

It is important to stress that Einstein's "ensemble interpretation" is not the same as that in Ballentine's 1998 book. Einstein's ensemble interpretation was a hidden variable interpretation.

Yes. Even Ballentines famous 1970 article is as well. He changed it for the books.

Thanks
Bill

 

[atyy]

Yes. Even Ballentines famous 1970 article is as well. He changed it for the books.

Thanks
Bill

And since there seems to be needless controversy in this thread, maybe I can recapitulate the bhobba's Ensemble interpretation includes an axiom that is equivalent to collapse - the improper mixture of the reduced density matrix can be taken as a proper mixture.

 

[bhobba]

And since there seems to be needless controversy in this thread, maybe I can recapitulate the bhobba's Ensemble interpretation includes an axiom that is equivalent to collapse - the improper mixture of the reduced density matrix can be taken as a proper mixture.

That's my ignorance ensemble interpretation. And I don't think it has collapse. Taking an improper mixture as a proper one is not the same as collapse. But its likely semantics and arguing about such is not something that thrills me so I will leave it at that.

Thanks
Bill

 

[atyy]

That's my ignorance ensemble interpretation. And I don't think it has collapse. Taking an improper mixture as a proper one is not the same as collapse. But its likely semantics and arguing about such is not something that thrills me so I will leave it at that.

Thanks
Bill

Regardless, one cannot do away with collapse as a postulate and replace it with nothing. For example, if collapse is taken away as a postulate, one possible replacement is the postulate that an improper mixture is proper.

 

[bhobba]

Regardless, one cannot do away with collapse as a postulate and replace it with nothing. For example, if collapse is taken away as a postulate, one possible replacement is the postulate that an improper mixture is proper.

I don't want to argue about this. My take is as I said very early on. We have interpretations where no collapse occurs eg MW, and those where it for sure happens eg GRW. Both explain how an improper mixture becomes a proper one so collapse is not a necessary part. As I said I don't want to argue the point - so that's all I will say.

Thanks
Bill

 

[rubi]

Challenge: Derive the generalized Born rule from the Born rule, but without using collapse!

Lemma (conditional probability) For commuting projections $P_A$ and $P_B$, the conditional probability $P(B|A)$ in the state $\rho$ is given by
$P(B|A) = \mathrm{Tr}\left(\frac{P_A \rho P_A}{\mathrm{Tr}(P_A \rho P_A)} P_B\right) = \frac{\mathrm{Tr}\left(P_B P_A \rho P_A P_B\right)}{\mathrm{Tr}(P_A \rho P_A)}$.
Proof. The conditional probability is defined by $P(B|A) = \frac{P(A\wedge B)}{P(A)}$. The projection operator for $A\wedge B$ is given by $P_{A\wedge B}=P_A P_B$. The Born rule tells us that $P(A)=\mathrm{Tr}(\rho P_A)$ and
$P(A\wedge B) = \mathrm{Tr}(\rho P_{A\wedge B}) = \mathrm{Tr}(\rho P_A P_B) = \mathrm{Tr}(\rho P_A^2 P_B) = \mathrm{Tr} (\rho P_A P_B P_A) = \mathrm{Tr}(P_A \rho P_A P_B) = \mathrm{Tr}(P_B P_A \rho P_A P_B)$. Now use linearity of the trace to get the formula for the conditional probability.

Now we want to apply this formula to sequential measurements. Let's say we want the probability to find observable $Y$ in the set $O_{Y}$ at time $t_2$ after having found observable $X$ in the set $O_{X}$ at time $t_1 < t_2$. This corresponds to measuring the Heisenberg observables $X(t_1)$ and $Y(t_2)$ in the same sets. Let $\pi_X$ and $\pi_Y$ be the projection valued measures of $X$ and $Y$. The corresponding projection valued measures of $X(t_1)$ and $Y(t_2)$ are given by $U(t_1)^\dagger \pi_X U(t_1)$ and $U(t_2)^\dagger \pi_Y U(t_2)$. Thus we are interested in the projections $P_A = U(t_1)^\dagger \pi_X(O_X) U(t_1)$ and $P_B = U(t_2)^\dagger \pi_Y(O_Y) U(t_2)$. We assume that these operators commute, which is true up to arbitrarily small corrections for instance for filtering type measurements or after a sufficient amount of decoherence has occured. Thus we can apply the lemma and get:
$P(Y(t_2)\in O_Y | X(t_1) \in O_X) = \frac{\mathrm{Tr}(P_B P_A \rho P_A P_B)}{\mathrm{Tr}(P_A \rho P_A)} = \frac{\mathrm{Tr}(\pi_Y(O_Y) U(t_2-t_1)^\dagger \pi_X(O_X) U(t_1)^\dagger \rho U(t_1) \pi_X(O_X) U(t_2-t_1) \pi_Y(O_Y))}{\mathrm{Tr}(\pi_X(O_X) U(t_1)^\dagger \rho U(t_1) \pi_X(O_X))}$
This is the right formula and we only assumed the Born rule and decoherence/filtering measurements.

 

[atyy]

Lemma (conditional probability) For commuting projections $P_A$ and $P_B$, the conditional probability $P(B|A)$ in the state $\rho$ is given by
$P(B|A) = \mathrm{Tr}\left(\frac{P_A \rho P_A}{\mathrm{Tr}(P_A \rho P_A)} P_B\right) = \frac{\mathrm{Tr}\left(P_B P_A \rho P_A P_B\right)}{\mathrm{Tr}(P_A \rho P_A)}$.
Proof. The conditional probability is defined by $P(B|A) = \frac{P(A\wedge B)}{P(A)}$. The projection operator for $A\wedge B$ is given by $P_{A\wedge B}=P_A P_B$. The Born rule tells us that $P(A)=\mathrm{Tr}(\rho P_A)$ and
$P(A\wedge B) = \mathrm{Tr}(\rho P_{A\wedge B}) = \mathrm{Tr}(\rho P_A P_B) = \mathrm{Tr}(\rho P_A^2 P_B) = \mathrm{Tr} (\rho P_A P_B P_A) = \mathrm{Tr}(P_A \rho P_A P_B) = \mathrm{Tr}(P_B P_A \rho P_A P_B)$. Now use linearity of the trace to get the formula for the conditional probability.

Now we want to apply this formula to sequential measurements. Let's say we want the probability to find observable $Y$ in the set $O_{Y}$ at time $t_2$ after having found observable $X$ in the set $O_{X}$ at time $t_1 < t_2$. This corresponds to measuring the Heisenberg observables $X(t_1)$ and $Y(t_2)$ in the same sets. Let $\pi_X$ and $\pi_Y$ be the projection valued measures of $X$ and $Y$. The corresponding projection valued measures of $X(t_1)$ and $Y(t_2)$ are given by $U(t_1)^\dagger \pi_X U(t_1)$ and $U(t_2)^\dagger \pi_Y U(t_2)$. Thus we are interested in the projections $P_A = U(t_1)^\dagger \pi_X(O_X) U(t_1)$ and $P_B = U(t_2)^\dagger \pi_Y(O_Y) U(t_2)$. We assume that these operators commute, which is true up to arbitrarily small corrections for instance for filtering type measurements or after a sufficient amount of decoherence has occured. Thus we can apply the lemma and get:
$P(Y(t_2)\in O_Y | X(t_1) \in O_X) = \frac{\mathrm{Tr}(P_B P_A \rho P_A P_B)}{\mathrm{Tr}(P_A \rho P_A)} = \frac{\mathrm{Tr}(\pi_Y(O_Y) U(t_2-t_1)^\dagger \pi_X(O_X) U(t_1)^\dagger \rho U(t_1) \pi_X(O_X) U(t_2-t_1) \pi_Y(O_Y))}{\mathrm{Tr}(\pi_X(O_X) U(t_1)^\dagger \rho U(t_1) \pi_X(O_X))}$
This is the right formula and we only assumed the Born rule and decoherence/filtering measurements.

How about non-commuting operators?

 

[rubi]

How about non-commuting operators?

The operators $X$ and $Y$ don't need to commute, as long as $X(t_1)$ and $Y(t_2)$ do. This is exactly what decoherence causes to high precision (environmental superselection). If it weren't the case, the event $A\wedge B$ would be ill-defined, since there is no orthogonal projection that corresponds to it.

Edit: To be more clear: $X$ and $Y$ are not directly the operators that measure the particle/..., but rather the operators that measure the position of the pointers of the measurement devices. If they didn't commute, it would mean that we couldn't read off the positions of the pointers simultaneously, i.e. the measurement devices would be in some macroscopic superposition.

Edit2: If we restrict X and Y to pointer observables, we don't even need to invoke decoherence here. A good measurement device is approximately classical, i.e. its pointer observable must commute with the pointer observables of other classical measurement devices up to at most tiny corrections. Otherwise, they would themselves be quantum objects and exhibit quantum behavior. For example, the position and momentum of a particle can never be known up to arbitrary precision, but the locations of the pointers of the position and momentum measurement devices can still be known exactly (since the measurement devices are assumed to be classical objects). Hence, they are supposed to commute. Non-commuting observables don't qualify as observables corresponding to pointers of classical measurement devices.

 

[atyy]

The operators $X$ and $Y$ don't need to commute, as long as $X(t_1)$ and $Y(t_2)$ do. This is exactly what decoherence causes to high precision (environmental superselection). If it weren't the case, the event $A\wedge B$ would be ill-defined, since there is no orthogonal projection that corresponds to it.

Edit: To be more clear: $X$ and $Y$ are not directly the operators that measure the particle/..., but rather the operators that measure the position of the pointers of the measurement devices. If they didn't commute, it would mean that we couldn't read off the positions of the pointers simultaneously, i.e. the measurement devices would be in some macroscopic superposition.

Edit2: If we restrict X and Y to pointer observables, we don't even need to invoke decoherence here. A good measurement device is approximately classical, i.e. its pointer observable must commute with the pointer observables of other classical measurement devices up to at most tiny corrections. Otherwise, they would themselves be quantum objects and exhibit quantum behavior. For example, the position and momentum of a particle can never be known up to arbitrary precision, but the locations of the pointers of the position and momentum measurement devices can still be known exactly (since the measurement devices are assumed to be classical objects). Hence, they are supposed to commute. Non-commuting observables don't qualify as observables corresponding to pointers of classical measurement devices.

That should probably work, because it is a variation of two traditional ways of avoiding collapse that do work

(1) Use the deferred measurement principle
(2) Restrict to commuting observables.

The two things are related because the deferred measurement principle changes sequential measurement of non-commuting observables to simultaneous measurement of commuting observables.

 

[stevendaryl]

Here's a sense in which collapse is unnecessary. Suppose we formalize the notion of the "macroscopic state" of the universe at a particular moment. It might, for instance, be a coarse-grained description of the mass-energy-momentum density, the spin density, the charge/current density, the values of various fields, etc. But only to a certain level of accuracy (the level of accuracy can be chosen keeping the uncertainty principle in mind). Then QM can be used to compute probabilities for macroscopic histories (or more accurately, the conditional probability that an initial history up to a specific moment will evolve into a certain more complete history). My conjecture is that in theory, it is unnecessary to invoke wave function collapse to compute these probabilities. The collapse, in this view, would come in as a short-cut, or approximation, to computing these probabilities, ignoring interference terms between macroscopically distinguishable intermediate states.

 

[rubi]

That should probably work, because it is a variation of two traditional ways of avoiding collapse that do work

(1) Use the deferred measurement principle
(2) Restrict to commuting observables.

The two things are related because the deferred measurement principle changes sequential measurement of non-commuting observables to simultaneous measurement of commuting observables.

It shows that every quantum theory that requires collapse can be converted into one that evolves purely unitarily and makes the same predictions. Here is the recipe:
We start with a Hilbert space $\mathcal H$, a unitary time evolution $U(t)$ and a set of (possibly non-commuting) observables $(X_i)_{i=1}^n$. We define the Hilbert space $\hat{\mathcal H} = \mathcal H\otimes\underbrace{\mathcal H \otimes \cdots \mathcal H}_{n \,\text{times}}$. We define the time evolution $\hat U(t) \psi\otimes\phi_1\otimes\cdots\otimes\phi_n = (U(t)\psi)\otimes\phi_1\otimes\cdots\otimes\phi_n$ and the pointer observables $\hat X_i \psi\otimes\phi_1\otimes\cdots\otimes\phi_n = \psi\otimes\phi_1\otimes\cdots\otimes (X_i\phi_i)\otimes\cdots\otimes\phi_n$. First, we note that $\left[\hat X_i,\hat X_j\right]=0$, so we can apply the previous result. Now, for every observable $X_i$ with $X_i\xi_{i k} = \lambda_{i k}\xi_{i k}$ (I assume discrete spectrum here, so I don't have to dive into direct integrals), we introduce the unitary von Neumann measurements $U_i \left(\sum_k c_k\xi_{i k}\right)\otimes\phi_1\otimes\cdots\otimes\phi_n = \sum_k c_k \xi_{i k} \otimes\phi_1\otimes\cdots\otimes \xi_{i k} \otimes\cdots\otimes\phi_n$. Whenever a measurement of an observable $X_i$ is performed, we apply the corresponding unitary operator $U_i$ to the state. Thus, all time evolutions are given by unitary operators (either $\hat U(t)$ or $U_i$) and thus the whole system evolves unitarily. Moreover, all predictions of QM with collapse, including joint and conditional probabilities, are reproduced exactly, without ever having to use the collapse postulate.

Of course, this is the least realistic model of measurement devices possible, but one can always put more effort in better models.

 

[atyy]

It shows that every quantum theory that requires collapse can be converted into one that evolves purely unitarily and makes the same predictions. Here is the recipe:
We start with a Hilbert space $\mathcal H$, a unitary time evolution $U(t)$ and a set of (possibly non-commuting) observables $(X_i)_{i=1}^n$. We define the Hilbert space $\hat{\mathcal H} = \mathcal H\otimes\underbrace{\mathcal H \otimes \cdots \mathcal H}_{n \,\text{times}}$. We define the time evolution $\hat U(t) \psi\otimes\phi_1\otimes\cdots\otimes\phi_n = (U(t)\psi)\otimes\phi_1\otimes\cdots\otimes\phi_n$ and the pointer observables $\hat X_i \psi\otimes\phi_1\otimes\cdots\otimes\phi_n = \psi\otimes\phi_1\otimes\cdots\otimes (X_i\phi_i)\otimes\cdots\otimes\phi_n$. First, we note that $\left[\hat X_i,\hat X_j\right]=0$, so we can apply the previous result. Now, for every observable $X_i$ with $X_i\xi_{i k} = \lambda_{i k}\xi_{i k}$ (I assume discrete spectrum here, so I don't have to dive into direct integrals), we introduce the unitary von Neumann measurements $U_i \left(\sum_k c_k\xi_{i k}\right)\otimes\phi_1\otimes\cdots\otimes\phi_n = \sum_k c_k \xi_{i k} \otimes\phi_1\otimes\cdots\otimes \xi_{i k} \otimes\cdots\otimes\phi_n$. Whenever a measurement of an observable $X_i$ is performed, we apply the corresponding unitary operator $U_i$ to the state. Thus, all time evolutions are given by unitary operators (either $\hat U(t)$ or $U_i$) and thus the whole system evolves unitarily. Moreover, all predictions of QM with collapse, including joint and conditional probabilities, are reproduced exactly, without ever having to use the collapse postulate.

Of course, this is the least realistic model of measurement devices possible, but one can always put more effort in better models.

Yes. The argument you have given is a generalization of using the deferred measurement principle to avoid collapse, which we have already agreed works. I have no problem with deriving the generalized Born rule without collapse for commuting observables. Ballentine does that, and I believe his argument is fine. The part of his argument I did not buy was his attempt to extend the argument to non-commuting observables. The important part of your argument is to circumvent the need for sequential measurement of non-commuting operators, which is fine, but it should be stated as a non-standard assumption.

Also, it does not support the idea that there is a wave function of the universe the evolves unitarily without collapse, because one still needs something in addition to the wave function, eg. the classical apparatus.

 

[A. Neumaier]

To get the definite outcome, one must further assume that the improper mixture is converted to a proper mixture, which is the same as assuming collapse. Ballentine and Peres are probably missing this assumption in their erroneous books.

I had already asked you to specify in detail your understanding of these terms and your reasoning for the conclusion, so that it can be critically discussed. Simply repeating this statement in a black box fashion as the sole justification for accusing a respectable author of making a fundamental error is not helpful at all.

I am not an outsider. I am stating that the standard texts are right.
Ballentine and Peres are the outsiders.

You are an outsider - where is your record of publications in the foundations of quantum mechanics? Or at least you are using a pseudonym so that you appear to be an outsider. This in itself would not be problematic. But you are making erroneous accusations based on a lack of sufficient understanding. This is very problematic.

Let's consider a system of one spin.

A pure state means that we have prepared many copies of the system of one spin, and each copy of the system is in the same pure state. For example, every copy of the single spin is pointing up.

A proper mixed state means that we have prepared many copies of the system of one spin, and each copy of the system is in a different pure state. For example, some copies of the single spin are pointing up and some are pointing down.

To illustrate an improper mixture, we have to consider a system of two spins. The system is in a pure state, which means we have many copies of the system of two spins, and each copy of the system of two spins is in the same pure state. If the pure state is one in which the two spins are entangled, then the state of a subsystem of one spin is an improper mixed state. We call the state of the subsystem a mixed state, because the subsystem in every copy behaves as if it is a proper mixed state. However, it is an improper mixed state, because if we consider the system of two spins, it is pure. So while we cannot distinguish between the proper and improper mixed state by looking at a subsystem of one spin, we can if we look at the entire system of two spins.

These explanations are valid for the Copenhagen interpretation but are meaningless in the context of the minimal (statistical) interpretation. In the minimal (statistical) interpretation discussed by Ballentine and Peres, a single system has no associated state at all. Thus your statements ''each copy of the system is in the same [resp. a different] pure state'' do not apply in their interpretation. You are seeing errors in their book only because you project your own Copenhagen-like interpretation (where a single system has a state) into a different interpretation that explicitly denies this. If a single system has no state, there is nothing that could collapse, hence there is no collapse. Upon projecting away one of the spins, an ensemble of 2-spin system in an entangled pure state automatically is an ensemble in a mixed stated of the subsystem, without anything mysterious having to be in between. Looking at conditional expectations is all that is needed to verify this. No collapse is needed.

Thus not Ballentine and Peres but your understanding of their exposition is faulty. You should apologize for having discredited highly respectable experts on the foundations of quantum mechanics on insufficient grounds.

 

[atyy]

It shows that every quantum theory that requires collapse can be converted into one that evolves purely unitarily and makes the same predictions. Here is the recipe:
We start with a Hilbert space $\mathcal H$, a unitary time evolution $U(t)$ and a set of (possibly non-commuting) observables $(X_i)_{i=1}^n$. We define the Hilbert space $\hat{\mathcal H} = \mathcal H\otimes\underbrace{\mathcal H \otimes \cdots \mathcal H}_{n \,\text{times}}$. We define the time evolution $\hat U(t) \psi\otimes\phi_1\otimes\cdots\otimes\phi_n = (U(t)\psi)\otimes\phi_1\otimes\cdots\otimes\phi_n$ and the pointer observables $\hat X_i \psi\otimes\phi_1\otimes\cdots\otimes\phi_n = \psi\otimes\phi_1\otimes\cdots\otimes (X_i\phi_i)\otimes\cdots\otimes\phi_n$. First, we note that $\left[\hat X_i,\hat X_j\right]=0$, so we can apply the previous result. Now, for every observable $X_i$ with $X_i\xi_{i k} = \lambda_{i k}\xi_{i k}$ (I assume discrete spectrum here, so I don't have to dive into direct integrals), we introduce the unitary von Neumann measurements $U_i \left(\sum_k c_k\xi_{i k}\right)\otimes\phi_1\otimes\cdots\otimes\phi_n = \sum_k c_k \xi_{i k} \otimes\phi_1\otimes\cdots\otimes \xi_{i k} \otimes\cdots\otimes\phi_n$. Whenever a measurement of an observable $X_i$ is performed, we apply the corresponding unitary operator $U_i$ to the state. Thus, all time evolutions are given by unitary operators (either $\hat U(t)$ or $U_i$) and thus the whole system evolves unitarily. Moreover, all predictions of QM with collapse, including joint and conditional probabilities, are reproduced exactly, without ever having to use the collapse postulate.

Of course, this is the least realistic model of measurement devices possible, but one can always put more effort in better models.

I replied to this two posts up (#148). But now I am unsure whether you are right about needing only unitary evolution.

First let me say what I do agree with
1) The generalized Born rule can be derived for commuting observables without assuming collapse
2) By using measuring ancilla, one can replace non-commuting observables with commuting observables, at least for the purposes of deriving their joint probability distributions.

But can one really do without collapse? The reason I am doubtful is that at each measurement in the Schroedinger picture, the rule that is used is the Born rule, not the generalized Born rule. So by using successive applications of the Born rule, one does not get the joint probability, which the generalized Born rule gives. So rather I would say that although the generalized Born rule can be derived without collapse as a postulate, the generalized Born rule implies collapse.