Nonlocality - fact or fiction?

[akhmeteli]

Take the Schroedinger's cat thought-experiment--do you agree that if we only allow the wavefunction to evolve via unitary evolution, we'll end up with a state that's a superposition which assigns nonzero amplitude to different position eigenstates corresponding to both "dead cat" and "live cat"?

I agree. Strictly speaking, this is correct.

Without anything but unitary evolution, do you agree the wavefunction is not going to settle on one macroscopic possibility or the other?

This is correct in the same sense as the Poincare recurrence theorem is correct. On the other hand, we'll have something very similar to a macroscopic outcome in the same sense in which we have irreversibility in thermodynamics. So I don't think this is a great problem. I think we should learn how to live with the idea of Schroedinger's cat, dead or alive. I think we should accept unitary evolution in all cases, not just when we like the results. Otherwise all kinds of problems arise.

Apply the same reasoning to the wavefunction of the universe and you have the MWI.

As you can see, I don't need MWI at all, because reversibility does not scare me (even if exemplified by a cat with a totally uncertain health status :-) ).

It seems to me we should distinguish between two different aspects of the "projection postulate":

1. At the experimental level, if we want to connect the theoretical evolving wavefunction with actual experimental results, we must use the Born rule where the probability of a given outcome depends on the amplitude that the wavefunction assigns to the eigenstate associated with that outcome (the probability being the complex conjugate of the amplitude)

I guess this is a typo, as probability should be real (should be "amplitude times its conjugate")

2. At a theoretical level, the projection postulate says that each measurement "collapses" the wavefunction, converting it at the moment of measurement into the eigenstate associated with whatever outcome was seen.

Obviously you reject #2, but your earlier comments in post #78 seemed to indicate that you'd accept #1:

As an operational principle, yes, I accept #1, although I suspect this is an approximation as well. I should emphasize though that there are different definitions of the projection postulate (#2).

The MWI, too, is generally understood to accept that the Born rule must work out as an operational rule for the probabilities seen by any individual observer in the giant superposition that is the universal wavefunction, while rejecting the idea of #2 that anything special happens to the wavefunction during measurement. And as far as I can see, Bell's theorem depends only on accepting #1, not on #2...if you accept that probability can always be determined from the amplitudes using the Born rule, then if you calculate the relevant probabilities for an entangled state, you can find probabilities which violate Bell inequalities. Do you disagree?

This is certainly a good question. I guess a lot depends on the exact definitions of the Born rule and the projection postulate. In your definition it is not clear what "outcome" means, whether it is an outcome of measurement of one observable or two. Other definitions mention just one observable. Actually, in the Bell theorem, correlations are calculated, so you should average a product of, say, spin projections for two particles. One may regard the relevant procedure as two measurements: one for the first particle, and the other for the second particle (the exact order is not important). Anyway, we must appreciate that any calculation procedure should correspond to the actual experimental situation, where indeed two measurements take place, or so I guess. So you have to apply the projection postulate to know how to describe the system after the first measurement. Maybe it is possible to generalize the definition of the Born's rule to include the projection postulate, but this would be a different story. For example, for what it's worth, the Wikipedia article (http://en.wikipedia.org/wiki/Bell's_Theorem ) uses the projection postulate. Thus, I believe the projection postulate (in some form) is used as an assumption of the Bell's theorem.

 

[Fra]

At least relative to my own thinking, the standard QM formalism as it stands, is not consistent with the lack of solid objectivity. It rather contains implicitly a kind of objectivity in the selection of a hilbert space, which also results in the determinism in the probability space.

IMH, in the general case, to truly implement the ideals of truly relational models, then even the relations are relative and dynamically so, and standard QM needs fundamental revision. But most of those who think along this lines, still pictures standard QM as beeing effectively emergent so as to be consistent with current experimental evidence.

In classical relational models like, GR. The observations the observers make differ, but their relations are somehow objective. In that sense there is a classical notion of objective relations between relative observations.

But I ask, who is establishing these objective relations? Here the notion of information and information capacity constraints really does hit me in the face. I don't even find rovelli's relational QM to solve this as far as I see. He takes an IMO excellent initiative, but I think "reinterpretations only" dosen't solve the main issue.

I agree that to a certain extent, it sure is possible in some sense, that there exists a universal objectivity at some level of abstraction. But the mere fact that it's "only" possible, and not a certainty, does influence at least my actions. This is why I do not think it's correct to say objectivity doesn't exists. Because I am uncertain about statement as well. It's possible that it exists, but to me the operational question is how to make progress. What strategy do I choose to make progress on this matter?

I don't think we all need to agree.

/Fredrik

 

[JesseM]

This is correct in the same sense as the Poincare recurrence theorem is correct. On the other hand, we'll have something very similar to a macroscopic outcome in the same sense in which we have irreversibility in thermodynamics.

What do you mean? In thermodynamics, it is overwhelmingly probable that entropy will increase, and you would have to wait a vast amount of time for entropy to decrease and the original state to recur. I don't see anything analogous to this with Schroedinger's cat if you believe only in unitary evolution. Unitary evolution will continually give you large amplitudes for both the live and dead state, it's not as if the amplitude will be concentrated almost entirely on one and the amplitude of the other will be overwhelmingly small.

As you can see, I don't need MWI at all, because reversibility does not scare me (even if exemplified by a cat with a totally uncertain health status :-) ).

You give no explanation of what reversibility has to do with the problem of significant amplitudes for completely different macroscopic states!

I guess this is a typo, as probability should be real (should be "amplitude times its conjugate")

Not exactly a typo, but writing too quickly...I was indeed thinking of multiplying the amplitude by its complex conjugate, but there was a malfunction somewhere between brain and keyboard.

This is certainly a good question. I guess a lot depends on the exact definitions of the Born rule and the projection postulate. In your definition it is not clear what "outcome" means, whether it is an outcome of measurement of one observable or two. Other definitions mention just one observable. Actually, in the Bell theorem, correlations are calculated, so you should average a product of, say, spin projections for two particles. One may regard the relevant procedure as two measurements: one for the first particle, and the other for the second particle (the exact order is not important). Anyway, we must appreciate that any calculation procedure should correspond to the actual experimental situation, where indeed two measurements take place, or so I guess. So you have to apply the projection postulate to know how to describe the system after the first measurement.

My memory of exactly how multiparticle systems are dealt with mathematically in QM is fuzzy, but I had the idea that one could assign a single amplitude to different possible combinations of measurement outcomes for two particles, in which case I'd think you'd be able to use the Born rule to get the probability of that combination in a pair of simultaneous measurements, without the need to have the first measurement "collapse" the system's wavefunction via the projection postulate in order to calculate probabilities for the second measurement. This page, for example, seems to confirm my memory, assigning single amplitudes to joint events like one photon being deflected by a half-silvered mirror while the other passes through. If you think I'm completely off here, though, maybe I should go back and reread my old college textbook...

 

[akhmeteli]

What do you mean? In thermodynamics, it is overwhelmingly probable that entropy will increase, and you would have to wait a vast amount of time for entropy to decrease and the original state to recur. I don't see anything analogous to this with Schroedinger's cat if you believe only in unitary evolution. Unitary evolution will continually give you large amplitudes for both the live and dead state, it's not as if the amplitude will be concentrated almost entirely on one and the amplitude of the other will be overwhelmingly small.

I am not sure what version of the Shroedinger's cat paradox you have in mind. The original version has a time limitation. For this discussion, let us imagine though that we consider the setup over an unlimited or extremely long time period. Eventually the atom will decay, triggering the killing mechanism, and the cat will die, either peacefully, or after prolonged suffering :-) If the box has finite dimensions, the system will apparently have discrete energy eigenvalues and thus satisfy the conditions of the quantum recurrence theorem (Phys. Rev. V.107 #2, pp.337-338, 1957), so after a hopelessly long period, the cat will rise from the dead, or, to be precise, will be as close to its initial state as you wish:-). On the other hand, this scary picture can perhaps coexist with practical irreversibility.

If this does not answer your question, please advise.

You give no explanation of what reversibility has to do with the problem of significant amplitudes for completely different macroscopic states!

Again, if the above explanation in this post does not seem satisfactory, please advise.

My memory of exactly how multiparticle systems are dealt with mathematically in QM is fuzzy, but I had the idea that one could assign a single amplitude to different possible combinations of measurement outcomes for two particles, in which case I'd think you'd be able to use the Born rule to get the probability of that combination in a pair of simultaneous measurements, without the need to have the first measurement "collapse" the system's wavefunction via the projection postulate in order to calculate probabilities for the second measurement. This page, for example, seems to confirm my memory, assigning single amplitudes to joint events like one photon being deflected by a half-silvered mirror while the other passes through. If you think I'm completely off here, though, maybe I should go back and reread my old college textbook...

First of all, after a cursory look at the link that you gave, I did not understand how it is relevant, as it seems to me that they describe some procedure (it does not matter whether the procedure is correct or not), but do not say whether they use the Born's rule or the projection postulate.

Technically, perhaps it might be possible from the point of quantum theory to measure the product of, say, spin projections of two different particles (I guess we can construct the relevant hermitian operator). However, in the tests of the Bell inequalities, measurements on the two particles are done pretty much independently, as far as I understand, and I am not sure it matters if these measurements are simultaneous or not quite, as the measurements are spatially separated anyway.

 

[JesseM]

I am not sure what version of the Shroedinger's cat paradox you have in mind. The original version has a time limitation. For this discussion, let us imagine though that we consider the setup over an unlimited or extremely long time period. Eventually the atom will decay, triggering the killing mechanism, and the cat will die, either peacefully, or after prolonged suffering :-) If the box has finite dimensions, the system will apparently have discrete energy eigenvalues and thus satisfy the conditions of the quantum recurrence theorem (Phys. Rev. V.107 #2, pp.337-338, 1957), so after a hopelessly long period, the cat will rise from the dead, or, to be precise, will be as close to its initial state as you wish:-). On the other hand, this scary picture can perhaps coexist with practical irreversibility.

If this does not answer your question, please advise.

No, it doesn't answer the question at all. My question has nothing to do with what happens a zillion years after the atom decays or doesn't decay (enough time for the system to have a significant likelihood of returning to its initial state), I don't understand why you think that would be relevant--I'm just asking what happens shortly afterwards, say after an hour. At this point, if you take unitary evolution seriously without the projection postulate, the cat should be in a superposition which assigns significant amplitudes to both the "atom decayed and cat is dead" states and the "atom didn't decay and cat is alive" states. So what do you think is the actual physical truth of the matter at this point in time? Do you think the cat is really one or the other, implying that unitary evolution can't be the whole story? Or do you think that since both outcomes are assigned a significant amplitude by the wavefunction, both must be on equal footing as far as 'physical truth' is concerned--and if so, how is this different from the many-worlds interpretation? Or do you suggest some third alternative, and if so what is it?

First of all, after a cursory look at the link that you gave, I did not understand how it is relevant, as it seems to me that they describe some procedure (it does not matter whether the procedure is correct or not), but do not say whether they use the Born's rule or the projection postulate.

I only brought up that link because it seemed to support my memory that in QM when you construct the wavefunction for a multiparticle system, you can use it to assign amplitudes to combinations of measurable outcomes. This is just a question about the mathematics of the theory of QM, you said you were a physicist yourself so I figured you'd know whether my memory on this is right or wrong; if not, then I can go dig up my old textbooks to see if I can find an example of such a multiparticle amplitude. But obviously if it's true that a single amplitude can be assigned to combinations of outcomes, then we can use the Born rule to calculate the probability of measuring such combinations, without the need to worry about the projection postulate discontinuously shifting the wavefunction between measurements.

Technically, perhaps it might be possible from the point of quantum theory to measure the product of, say, spin projections of two different particles (I guess we can construct the relevant hermitian operator). However, in the tests of the Bell inequalities, measurements on the two particles are done pretty much independently, as far as I understand, and I am not sure it matters if these measurements are simultaneous or not quite, as the measurements are spatially separated anyway.

For combinations of spin measurements it might be that there is no time-dependence in the amplitudes, in which case it wouldn't really matter whether the measurements were simultaneous or not.

 

[ThomasT]

Because these interpretations claim that objective (i.e., existing even without observations) reality does not exist. Of course, these interpretations admit that there are nonlocal correlations (which is only what is truly experimentally proved), but they claim that reality itself is not nonlocal, simply because reality does not exist.

I am not saying that it makes sense to me, I am just saying what they say.

I agree that what you say they're saying doesn't make sense. However, I don't think that what you say they're saying is what they're saying.

I don't think that Bohr and Heisenberg would characterize the Copenhagen interpretation as saying that reality doesn't exist. They would say, I think, that if you want the word "reality" to have some physical meaning (rather than using it merely as an honorific and absurdly ambiguous term), then there's no reality (that is, we can have nothing, save speculatory metaphysics, to say about what exists) beyond the level of instrumental behavior.

If one says that there is a hidden reality that exists beyond our sensory apprehensions, then what is this reality? How can we know that it exists. What can we say about it unambiguously?

The essence of the Copenhagen interpretation is that the modern science of physics shouldn't proceed via the path of metaphysical speculation (even though it seems that it sometimes does). The speculatory path is the one taken by, eg., Bohmian mechanics and MWI, so the developers of the Copenhagen interpretation would say that these interpretations are unacceptable adjuncts, unnecessary baggage, with respect to the development of the quantum theory.

As for nature being nonlocal. That's an open question. And, we return to the Copenhagen interpretation with its emphasis on semantics to ask two questions when any claim about nature (such as its nonlocality) is made.
(1) What do you mean?
(2) How do you know?

 

[akhmeteli]

No, it doesn't answer the question at all. My question has nothing to do with what happens a zillion years after the atom decays or doesn't decay (enough time for the system to have a significant likelihood of returning to its initial state), I don't understand why you think that would be relevant--I'm just asking what happens shortly afterwards, say after an hour. At this point, if you take unitary evolution seriously without the projection postulate, the cat should be in a superposition which assigns significant amplitudes to both the "atom decayed and cat is dead" states and the "atom didn't decay and cat is alive" states. So what do you think is the actual physical truth of the matter at this point in time? Do you think the cat is really one or the other, implying that unitary evolution can't be the whole story? Or do you think that since both outcomes are assigned a significant amplitude by the wavefunction, both must be on equal footing as far as 'physical truth' is concerned--and if so, how is this different from the many-worlds interpretation? Or do you suggest some third alternative, and if so what is it?

Sorry, could not reply earlier - was a bit busy.

Ok, so you suggest that we consider the original version of the Schroedinger cat paradox. No objections. I am not sure though that this is a major change from what I wrote. Yes, now we start the measurement in an hour after the start of the experiment. However, the question is how long it takes to perform the measurement. If the measurement takes a very long time, again, the system can switch from one state to the other. The problem is the dichotomy "dead cat/alive cat" may be false: an alive cat can die (if, for example, the measurement provokes a radioactive decay), and a dead cat can become alive, say, as a result of a thermodynamically absurd, but mechanically inevitable process. So I do suspect that both outcomes might be "on equal footing as far as 'physical truth' is concerned", as telling one outcome from the other is not so straightforward. I guess this is quite different from MWI, as there is just one world, but no measurement is ever final.

I am not trying to offer some clearer picture of what happens in the Shroedinger's gedanken experiment, I am just trying to say that the picture that unitary evolution presents is not as absurd as it may seem (although it may seem thermodynamically absurd). Perhaps a clearer picture can be found in the article by Allahverdyan et al. quoted in my post #20 in this thread. They consider an exactly solvable model of spin measurement and show when and how the "Shroedinger's cat's" terms disappear and how this is related to irreversibility. They conclude: "The solution of our model shows that the so-called “measurement problem”, to wit, the fact that the final state... does not seem to be related unitarily to the initial state, has the same nature as the celebrated “paradox of irreversibility” in statistical mechanics,... with additional quantum features." Actually, they derive the Born rule from the unitary evolution, but as an approximation, not as a rigorous result.

I only brought up that link because it seemed to support my memory that in QM when you construct the wavefunction for a multiparticle system, you can use it to assign amplitudes to combinations of measurable outcomes. This is just a question about the mathematics of the theory of QM, you said you were a physicist yourself so I figured you'd know whether my memory on this is right or wrong; if not, then I can go dig up my old textbooks to see if I can find an example of such a multiparticle amplitude. But obviously if it's true that a single amplitude can be assigned to combinations of outcomes, then we can use the Born rule to calculate the probability of measuring such combinations, without the need to worry about the projection postulate discontinuously shifting the wavefunction between measurements.

And I tried to explain that, while it may certainly be important what is written in the link you gave or in your textbooks, another question may be more important for this discussion: why it is written, as it may well be that something is written just on the basis of the projection postulate, and it would not mean that it is downright wrong, but it would mean it is not quite relevant. For example, I would not call a textbook downright wrong just because it states that all processes are irreversible. Irreversibility may be a good approximation, but, strictly speaking, it contradicts mechanics.

Anyway, let us assume for this discussion that "you can use [the wavefunction] to assign amplitudes to combinations of measurable outcomes." But then, to conduct an experiment and test the Bell inequalities, you have to measure correlations, and one typically conducts two separate measurements - on the first and on the second particle, and consequently the projection postulate is used to calculate the probabilities that quantum mechanics is supposed to predict. Again, as I said, while maybe "we can use the Born rule to calculate the probability of measuring such combinations", as the product of spin projections is an Hermitian operator, but in practice the actual measurement consists of two separate measurements. Furthermore, maybe I should admit that I cannot be sure that it is the projection postulate and not the Born rule that generates nonlocality, which does not seem to exist in the rigorous unitary evolution picture, as both the Born rule and the projection postulate are just approximations. However, my bet is the projection postulate is the main culprit.

For combinations of spin measurements it might be that there is no time-dependence in the amplitudes, in which case it wouldn't really matter whether the measurements were simultaneous or not.

And if it does not matter, we may assume that we need two measurements, one after another (anyway, two absolutely simultaneous measurements are an abstraction), and the projection postulate is needed to calculate the prediction of quantum mechanics.

 

[akhmeteli]

You accepted "fair sampling" in neutrino detection but reject it in Bell-type experiments? What gives? How do you think they detect neutrinos in the first place?

Zz.

I admit that I have no clear idea on what form of the fair sampling assumption is used in neutrino detection and what it has to do with the fair sampling assumption in the experiments on Bell inequalities. Furthermore, if it is found tomorrow that neutrinos have zero mass after all, I won't loose any sleep over such development. I'd like also to emphasize that there is not necessarily something wrong with accepting something in one situation and rejecting the same thing in another situation. I both accept irreversibility in thermodynamics and refuse to accept it as a rigorous concept. Another example: the Boltzmann's hypothesis of molecular chaos made possible his outstanding achievements - the kinetic equation named after him and the famous H-theorem. However, strictly speaking, the hypothesis is wrong, as it introduces irreversibility where mechanics is reversible. The analogy with the fair sampling assumption is striking: in both cases some assumptions are introduced which are external with respect to the underlying mechanics, and inevitably such assumptions are, strictly speaking, wrong in both cases. At least that's my understanding.