Undergrad Murray Gell-Mann on Entanglement

[atyy]

If collapse is actually in the theory, its existence shouldn't depend on what picture we use. So if collapse is there in the Schrodinger picture, it should have a counterpart in the Heisenberg picture, some kind of an evolution for operators that doesn't satisfy the Heisenberg's equation of motion. Otherwise we can just stop using Schrodinger picture and then there is no collapse in the theory!

But otherwise, what you say makes sense to me!

Yes, you can hide the collapse by going in a sophisticated way to the Heisenberg picture - this requires a generalization of the Born rule. I have no problem with that.

There are other ways to avoid collapse, like insisting on never making sequential measurements (in principle it is possible, but almost impossible in practice).

Similarly, Bob can avoid nonlocality by insisting that Alice is not real at spacelike separation.

Many choices are possible, including accepting that the locality can be derived from nonlocality - concretely, the reduced density matrix of B (showing locality) is derived by tracing over the collapsed wave function of both A and B (showing nonlocality).

 

[ShayanJ]

this requires a generalization of the Born rule

Can you provide a reference?

 

[atyy]

Can you provide a reference?

This is not the most general form, but it will give you the right idea: Eq 37 of http://arxiv.org/abs/quant-ph/0209123.

 

[Ken G]

How do you prove that there is causation?

The scientist never proves anything, they use models successfully. Causation is something that we say is present when we find that by using that concept, it gives us power over the situation. We control one thing, in order to control something else-- that's the value of the causation concept. But like all our concepts, we tend to take them too literally, and try to apply them in situations where we gain nothing by doing so. Like entanglement.

 

[vanhees71]

The collapse is nonlocal in the sense that the wave function is assigned to a spacelike surface of simultaneity, and the wavefunction on that hypersurface collapses instantaneously.

From the nonlocal collapse, the reduced density matrix of B can be derived, from which it can be seen that the collapse does not allow superluminal signalling.

So locality can be derived from nonlocality, and nonlocality does not contradict locality.

I disagree, and that's also not in accordance with what Peres writes in the here discussed article. Again, you have to distinguish between longranged-correlations ("nonlocality" realized by entanglement also in relativistic QFT) and local interactions (realized by microcausality of the local observables and locality of the interaction Hamiltonian, as also clearly specified by Peres; he gives even a stronger argument, why relativistic QT should be realized as local relativistic QFT, then Weinberg in QT of Fields vol. I!).

 

[vanhees71]

That makes sense for QFT. But that means NRQM doesn't need collapse because Galilean transformations preserve simultaneity of the experiments.

In other words: Nothing is wrong with even a naive collapse assumption for non-relativistic QT. There you can use it without contradicting the theory itself. That's not possible for relativistic local QFT (as applied in the formulation of the Standard Model). There a naive collapse assumption contradicts the very foundations of the theory. Since NRQT is just an approximation of relativistic QFT, one shouldn't use the collapse assumption there either, but at least it's not self-contradictory as if applied for relativistic QFT.

 

[vanhees71]

This is not the most general form, but it will give you the right idea: Eq 37 of http://arxiv.org/abs/quant-ph/0209123.

Why do you say there is an extension of the Born rule? Also note that the outcome of anything physical like the said probabilities are independent of the picture of time evolution since any two pictures of time evolution are connected by unitary transformations. It may be more or less convenient to use a specific picture, but there cannot be any difference concerning the physical outcomes of the formalism due to the change of the picture.

 

[Weddgyr]

"People say loosely ,crudely,wrongly that when you measure one of the photons it does something to the other one. It doesn't."
Do most physicists working in this field agree with the above statement ?

Most physicists are trained to avoid asking the question. It's been a hugely successful program.

There aren't any answers as of today. You can assume a non-local influence if you like, but that will be philosophically in conflict with relativity, etc. You can assume a conspiracy/super-determinism of detector settings, etc, which will be another philosophical muddle. You can just accept the quantum correlations as is, without need for a causal mechanism, but you will then be in conflict with the general philosophy of the scientific method. Above all you are a physicist so whichever option you pick you will of course NOT be doing any of that philosophical crap.

 

[Simon Phoenix]

That's not possible for relativistic local QFT (as applied in the formulation of the Standard Model). There a naive collapse assumption contradicts the very foundations of the theory

OK - that's fair enough - but does assuming a naïve collapse model actually lead one to derive any inconsistent experimental results?

 

[ShayanJ]

In other words: Nothing is wrong with even a naive collapse assumption for non-relativistic QT. There you can use it without contradicting the theory itself. That's not possible for relativistic local QFT (as applied in the formulation of the Standard Model). There a naive collapse assumption contradicts the very foundations of the theory. Since NRQT is just an approximation of relativistic QFT, one shouldn't use the collapse assumption there either, but at least it's not self-contradictory as if applied for relativistic QFT.

I'm trying to make sense of the way you think about this. But I seem to lack some essential knowledge about how people like you actually use QM.
The part I know is that you prepare a large number of identical systems in identical quantum states. Of course the preparation device is not perfect and there may be some deviations from the desired state, but I don't know whether you take that into account and use a mixed state for the ensemble or just make the approximation of a perfect device and use the desired pure state.
Anyway, the next step is you measure the probability distribution of a desired observable on this ensemble. What I don't understand is, what do you do if you want to measure the probability distribution of another observable on the same ensemble. Would you assume its in the same state before the first measurement? Or you do a Bayesian update? Is it even possible to do a second measurement on the same ensemble?

 

[vanhees71]

Maybe, I don't understand the question right, because I don't see, where there should be a problem. I just measure the observable I want. If I want to measure one observable at a time $t_1$ and then another on the same system at $t_2$ I just do so. What should I update? Of course, when predicting what's measured at $t_1$ I have to somehow describe what happens to the system due to the interaction with the measurement device at the first measurement. I also don't know what you mean by "Bayesian update". If measurement number 1 is a von Neumann filter measurement, of course I update the state in the usual sense. Also, why shouldn't it be possible to do a second measuerement on the same system within the ensemble? It depends of course what you do to the system with the 1st measurement. If I absorb the photon in the 1st measurement, of course, I cannot measure anything on this very same photon at a later time.

 

[Herman Trivilino]

That makes sense for QFT. But that means NRQM doesn't need collapse because Galilean transformations preserve simultaneity of the experiments.

But Galilean transformations are an approximation valid only when speeds are slow or separation distances between events are not very very large. It's hardly worth using it to form a worldview.

 

[stevendaryl]

OK - that's fair enough - but does assuming a naïve collapse model actually lead one to derive any inconsistent experimental results?

I would say that it doesn't. Instantaneous physical collapse is incompatible with the principles of Special Relativity, but I don't think it is incompatible with any experimental evidence.

 

[stevendaryl]

Maybe, I don't understand the question right, because I don't see, where there should be a problem. I just measure the observable I want. If I want to measure one observable at a time $t_1$ and then another on the same system at $t_2$ I just do so. What should I update?

The wave function used at time t_2. You prepare a system in state |\psi\rangle. At time t_1, the state has evolved to some new state, |\psi(t_1)\rangle = e^{-iHt_1} |\psi \rangle. At this moment, you perform a measurement of observable O and get result \lambda. Then for predictions about measurements at time t_2, you don't use |\psi(t_2)\rangle, you use |\psi'(t_2)\rangle, where |\psi'(t_2)\rangle = e^{-i H (t_2 - t_1)} \Pi_{O,\lambda} e^{-iH t_1} |\psi \rangle, where \Pi_{O,\lambda} is the projection operator that projects onto the subspace of the Hilbert space corresponding to eigenvalue \lambda of operator O. So the "update" being discussed is switching from |\psi(t_1)\rangle = e^{-iHt_1} |\psi\rangle to \Pi_{O,\lambda} |\psi(t_1)\rangle. That update is what people mean by "collapse of the wave function".

 

[vanhees71]

Sure, this describes a von Neumann Filter measurement. This update, however does not mean that there's an action at a distance with a far distant entangled part of the system I've measured.

 

[zonde]

But what is then "collapse" other than that A updates her knowledge due to the achieved polarization measurement of her photon (and the knowledge that it is polarization-entangled before her measurement)? Nothing happens to B's photon, and B still has unpolarized photons. So indeed Gell-Mann is right in his statement that nothing happens to B's photon!

How can we describe "nothing happens to B's photon" in a bit more experimentally accessible way? Would you agree to formulation that "if A has measured it's photon at different angle identical measurement of B's photon would (could) give the same result"?

 

[Ken G]

The reason I can see Gell-Mann's point is that I just don't understand why anyone would want to imagine that measuring one photon does something to the other. There can be reference frames that don't even agree which measurement happened first, nor should it matter-- you have a correlation, it's part of the system. We used to think all the information needed to predict a correlation was either "carried with" each piece of the system independently, or would involve some kind of propagating signal between the parts, but quantum mechanics gave us a well-tested formalism that says that doesn't work, instead correlations are holistic. So why not just accept that correlations are holistic? We've had so many other classical notions that we discarded, like absolute time and space or the idea that two identical preparations could not lead to different outcomes, so from whence comes the need to hang on to the old notion that all information works like attributes "carried with" pieces of a system, coupled with "influences" that propagate between the pieces? We find observationally that correlations are holistic, and a Bell state successfully encodes those holistic correlations, so why tack on some extraneous mechanism for moving those correlations around from place to place like an "influence"? The very idea that a two-photon system can be comprised of two separate photons is already a notion we should look at with suspicion (because of exchange symmetries), so why take it even farther to imagine those two dubious separate photons can do things to each other?

 

[ddd123]

The reason I can see Gell-Mann's point is that I just don't understand why anyone would want to imagine that measuring one photon does something to the other. There can be reference frames that don't even agree which measurement happened first, nor should it matter-- you have a correlation, it's part of the system. We used to think all the information needed to predict a correlation was either "carried with" each piece of the system independently, or would involve some kind of propagating signal between the parts, but quantum mechanics gave us a well-tested formalism that says that doesn't work, instead correlations are holistic. So why not just accept that correlations are holistic? We've had so many other classical notions that we discarded, like absolute time and space or the idea that two identical preparations could not lead to different outcomes, so from whence comes the need to hang on to the old notion that all information works like attributes "carried with" pieces of a system, coupled with "influences" that propagate between the pieces? We find observationally that correlations are holistic, and we have a mathematical formalism for encoding holistic correlations, so why not just accept that? The very idea that a two-photon system can be comprised of two separate photons is already a notion we should look at with suspicion, so why take it even farther to imagine those two dubious separate photons can do things to each other?

I think it is because with the other replacements for the old notions you could still form a picture of what happens inbetween preparation and measurement. Even giving up determinism let's you form a picture, because you just have a different rule for an instantaneous effect. Not having any picture and working in the blind is probably crossing a boundary for how much people are ready to give up. I mean what else could you give up after that if not predictability and thus science itself?

 

[stevendaryl]

Sure, this describes a von Neumann Filter measurement. This update, however does not mean that there's an action at a distance with a far distant entangled part of the system I've measured.

Why doesn't it? The state of the distant component has changed as a result of your measurement.

 

[stevendaryl]

The reason I can see Gell-Mann's point is that I just don't understand why anyone would want to imagine that measuring one photon does something to the other. There can be reference frames that don't even agree which measurement happened first, nor should it matter-- you have a correlation, it's part of the system. We used to think all the information needed to predict a correlation was either "carried with" each piece of the system independently, or would involve some kind of propagating signal between the parts, but quantum mechanics gave us a well-tested formalism that says that doesn't work, instead correlations are holistic.

It seems to me that "holistic" and "nonlocal" might mean the same thing, here.

 

[stevendaryl]

Why doesn't it? The state of the distant component has changed as a result of your measurement.

I understand that the "state" can be interpreted as subjective, rather than objective, but what is there to a particle, other than its state? There is nothing objective, is there? If you assume the existence of something objective (observer-independent) about a particle, and say that the quantum-mechanical state only reflects our information about this, then that's basically a hidden-variables theory, and Bell showed that such a theory has to be nonlocal. If you don't assume that there is anything objective about particles, then it seems like the question of FTL influences is moot. If the distant particle doesn't have any independent reality, then what could it mean to influence it nonlocally?

 

[vanhees71]

Why doesn't it? The state of the distant component has changed as a result of your measurement.

But that interpretation contradicts the locality of the interaction between A's photon and her polarization measurement apparatus. Also if Alice measures something else of her photon after it has passed the polarization filter, say directed to let through H-photons (which with utmost accuracy can indeed be made a v Neumann filter measurement!), all the outcomes of further measurements on her photon are described by associating the polarization state $|H \rangle$ with it. For A it's totally irrelevant what's the state of B's photon, as is for B whatever A does with her photon. The correlations due to the entanglement, which itself is due to the production of the entangled photon pair in the very beginning, can, however be observed by comparing the measurement protocols with accurate timestamps of each single-photon detection event by A and B. From this point of view (the minimal statistical interpretation) there is not need for assuming a collapse at all, and that prevents this interpretation from leading to inconsistency with the very foundations of relativistic QFT!

 

[stevendaryl]

But that interpretation contradicts the locality of the interaction between A's photon and her polarization measurement apparatus.

If you posit, as Von Neumann did, the existence of two kinds of processes: (1) evolution according to Schrodinger's equation (or the equivalent for QFT), and (2) measurements, then nonlocality of the second type doesn't contradict locality for the first type. Of course, that's unsatisfying, because measurements (or observations) surely must be explainable in terms of the quantum mechanics of macroscopic devices, but it seems that any way of making sense of the Born probabilities involves making a distinction between macroscopic and microscopic phenomena. There are no probabilities involved in the evolution of a single electron. There are no probabilities involved in the evolution of two electrons. Probabilities only come into play in the interaction of something large enough to count as an observer, or a measuring device.

 

[zonde]

For A it's totally irrelevant what's the state of B's photon, as is for B whatever A does with her photon.

So you agree with this statement, right? - "If A has measured it's photon at different angle identical measurement of B's photon would (could) give the same result."

 

[stevendaryl]

I don't know how you can say "For A it's totally irrelevant what's the state of B's photon". If Alice knew the state of Bob's photon, then she would know the state of her own photon. So the state of Bob's photon is relevant to Alice.

 

[atyy]

I disagree, and that's also not in accordance with what Peres writes in the here discussed article. Again, you have to distinguish between longranged-correlations ("nonlocality" realized by entanglement also in relativistic QFT) and local interactions (realized by microcausality of the local observables and locality of the interaction Hamiltonian, as also clearly specified by Peres; he gives even a stronger argument, why relativistic QT should be realized as local relativistic QFT, then Weinberg in QT of Fields vol. I!).

Do you disagree with this statement: "Before Alice's measurement the state is $|hh \rangle + |vv \rangle$, and after the measurement the state collapses to $|hh \rangle$ if Alice measures her photon to be horizontal"?

It's the same as what Peres starts his abstract with: " If several interventions performed on a quantum system are localized in mutually space-like regions, they will be recorded as a sequence of “quantum jumps” in one Lorentz frame, and as a different sequence of jumps in another Lorentz frame."

Peres also says in his first sentence of the text: "Quantum measurements [1] are usually considered as quasi-instantaneous processes. In particular, they affect the wave function instantaneously throughout the entire configuration space."

The quantum jumps are obviously nonlocal. If they were local, his whole article would be trivial. It is because they are nonlocal that one has to ask whether that nonlocality can be consistent with locality. The answer is yes, nonlocality can be consistent with locality.

Furthermore Peres writes: "Returning to the Einstein-Podolsky-Rosen conundrum, we must analyze whether it actually involves a genuine quantum nonlocality. Such a claim has led some authors to suggest the possibility of superluminal communication."

Thus for Peres:
wave function collapse: fake quantum nonlocality
superluminal communication: genuine quantum nonlocality
no superluminal communication: genuine quantum locality

So fake quantum nonlocality is consistent with genuine quantum locality.

 

[vanhees71]

I don't know, what you mean by measuring A's photon at different angle. The preparation in the entangled state $|\Psi \rangle=|HV-VH \rangle$ implies that, if A's photon is found to be polarized in an angle $\phi$ (relative to direction $H$), then B's photon will be found in a state perpendicular to it.

Proof: Let $|\phi \rangle=\cos phi |H \rangle + \sin \phi |V \rangle$. Then, if A finds her photon to be polarized in this direction, she adapts her state of the two-photon system to
$$|\Psi_A' \rangle=|\phi \rangle \langle \phi | \otimes 1 |\Psi \rangle=\cos^2 \phi |HV \rangle - \sin^2 \phi |VH \rangle + \cos \phi \sin \phi (|VV \rangle -|HH \rangle) = (\cos \phi |H \rangle + \sin \phi |V \rangle) \otimes (\cos \phi |V \rangle - \sin \phi |H \rangle) .$$
As it turns out, if you consider only those B photons for which A found polarization in direction $|\phi \rangle$, then B will always find polarization in direction $\phi+\pi/2$.

Note that the necessary filtering to figure this out needs the exchange of the measurement protocols between A and B. Both A and B measure just unpolarized photons, i.e., A's photon will go through the $\phi$-polarization filter in 50% of the cases, and in these 50% of the cases B must measure his photon to be $\phi+\pi/2$ polarized. So there is a correlation between the photons but no action at a distance necessary to explain this result. The correlation is due to the preparation of the two-photon state in this entangled state and not due to A's polarization measurement on her single photon.

 

[stevendaryl]

Do you disagree with

Before Alice's measurement the state is $|hh \rangle + |vv \rangle$, and after the measurement the state collapses to $|hh \rangle$ if Alice measures her photon to be horizontal?

It's kind of hard to know what to make of the situation. The situation is this:

1. Initially, Bob's photon is unpolarized.
2. After Alice's measurement, his photon is polarized in some direction, but Bob doesn't know which.

That's a change, of sorts, but it's not a change that makes any difference for Bob. The only "state" of Bob's photon that matters for his measurements is its density matrix. The initial density matrix describes a so-called "improper" mixed state, which is obtained from the two-photon pure state by tracing over Alice's photon. The final density matrix is a proper mixed state, which is obtained by taking a weighted sum of pure-states. So Bob's photon's state went from an improper mixed state to a proper mixed state, which seems like a change, but they are both described by the identical matrix. So from that point of view, nothing has changed for Bob.

 

[stevendaryl]

I don't know, what you mean by measuring A's photon at different angle.

Just a general request about discussions here: If your comment is a direct response to a specific other comment, I would prefer that you either quote some part of the previous comment, or at least the name of the previous commenter.

 

[vanhees71]

Do you disagree with this statement: "Before Alice's measurement the state is $|hh \rangle + |vv \rangle$, and after the measurement the state collapses to $|hh \rangle$ if Alice measures her photon to be horizontal"?

Yes, I disagree with this statement. Correct is: If A's photon passes the h-polarization filter she associates the state $hh \rangle$ to the two photons. However, her measurement has no instantaneous influence on B's photon, i.e., there must not be a collapse if the interpretation should be consistent with the very construction of QED as a local relativistic QFT, and you don't need it!

It's the same as what Peres starts his abstract with: " If several interventions performed on a quantum system are localized in mutually space-like regions, they will be recorded as a sequence of “quantum jumps” in one Lorentz frame, and as a different sequence of jumps in another Lorentz frame."

Well, I'm also against the use of the word quantum jumps, but I guess Peres has the right thing in mind when he states this, and he is right that the temporal sequence for space-like separated "interventions" is frame dependent, which implies that one intervention cannot have a causal influence on the other space-like separated intervention. That's the whole point of our disagreement. In my (and if I understand him right also Peres's) notion of the state as epistemic (particularly the "update" or if you wish to call it with another unsharp word "quantum jump" of the state after a "filtering intervention" as in our example here) there is no tension between causality and relativistic QFT whatsoever, and that's so by construction of the QFT, and Peres's argument in the paper is just another very convincing argument for why (at least) the Hamilton density operator has to commute at spacelike separation of the arguments, i.e., if $(x-y) \cdot (x-y)<0$ (west-coast convention of the metric) you must have $[\hat{\mathcal{H}}(x),\hat{\mathcal{H}}(y)]=0$. Usually one assumes even more, i.e., that any two local operators commute at spacelike separation of their arguments.