Quantum mechanics is not weird (locality and non-locality weirdness)

[rubi]

I consider Bertlmann's socks to be an illustration of exactly the opposite conclusion. Bertlmann's socks is an example of an apparent nonlocality that disappears when you are given more information about local conditions. That's the whole point of the search for hidden variables: You have a correlation that is apparently nonlocal, but you can eliminate the nonlocality if you can find local information that make the same predictions as the nonlocal information.

The correlation doesn't suddenly become local once you found a hidden variable that explains it. It is still a non-local correlation, the numbers haven't changed. Non-locality of correlations is a relation between numbers. The point is that the correlation can be explained by a common cause in the past. The socks were separated in a local region and then sent to Alice and Bob at a slower speed than the speed of light. Thus neither the discovery of the red sock causes the blue sock to be blue nor does the discovery of the blue sock cause the red sock to be red. Instead, there is a common cause in the past.

If you can't find such variables, then you're stuck with a nonlocal theory.

Only under the assumption that the world is classical (i.e. realistic). If the world is classical, the common cause explanation is not possible and thus the discovery of the spin-up particle would have caused the other particle to have spin-down and the other way around: The proof of Bell's inequality assumes realism (modeling on a combined probability space) and locality. Thus a violation of Bell's inequality implies that not both realism and locality are true (by the logical principle of contraposition), which means that realism or locality or both are false. If we know that the world is realistic, must conclude that the locality assumption in the proof must have been false. However, if we don't assume a realistic world (for example a quantum mechanical world), then we can't infer that the locality assumption is false. Neither can we infer that it is true. We just don't know it.

That's only because nobody really knows what "causation" means.

Well, in the case of a Bell-test experiment, we can explain the non-local correlations if we accept that the world is quantum mechanical: The entangled quantum particles had been created locally in the past and then sent to Alice and Bob respectively at a speed lower than the speed of light. This is a perfectly causal explanation as soon as we accept that the world is quantum mechanical and quantum objects just happen to exist. If we hadn't entangled the particles in the past, before we sent them to Alice and Bob respectively, we wouldn't have seen the perfect correlations in the Bell-test experiment. A classical physicist couldn't accept such an explanation. But for a quantum physicist, who accepts that the world behaves quantum mechanically, it is not problematic. It's only the classical thinking that makes it seem weird.

That's a huge difference. In the socks case, we know that our nonlocal description is incomplete, that a complete description of Alice's and Bob's situation is local. So the apparent nonlocality is due to our ignorance about the state of the system. The nonlocality is in principle eliminable. That's not the case with QM.

I agree that it is a huge difference and the non-local correlation of quantum objects are very different from the non-local correlations of classical objects. But I don't agree that this implies that quantum mechanics is non-local, because for non-locality, it's the cause that matters, not the correlation. Quantum mechanics is not complete, but that doesn't imply that non-local correlations can't be explained causally.

In the one case, the most complete description of reality is local. In the other case, it's not.

In the other case, it's also local, because I can also give a common cause explanation for the non-local correlations. It just involves quantum objects rather than classical objects. The only thing you need to accept is that using quantum objects as if they did really happen to exist in the universe is a valid way of reasoning. And given the success of quantum mechanics, I don't find this assumption unreasonable.

If you know everything there is to know about the local region near Bob, you can never do better than 50% probability of spin-up, 50% probability of spin-down.

That's right. Quantum mechanics is not complete. But completeness isn't required for locality. It's an entirely different concept.

If, in addition, you know that Alice measured spin-up at angle $\vec{A}$, then you can predict with certainty that Bob will measure spin-down along that axis.

That's also right. Non-local correlations exist, just as they do in classical theories (see Bertlmann's socks). In both cases, they can be explained by common cause explanations in the past, although in the quantum case, you must resort to quantum reasoning.

So the quantum facts are nonlocal; the best prediction for what Bob will measure may involve facts about the situation far away from Bob.

The facts are non-local, just as in the classical case. The explanation is local (i.e. causal) in both cases.

Are you suggesting that it is possible to predict Bob's results using some kind of quantum local variables that fail to belong to a combined probability space?

No, we can neither predict Alice's nor Bob's results. We can however predict the non-local correlations and we can explain them causally. No FTL is involved. The reason for the correlation lies in the past light cone of both Alice and Bob.

 

[stevendaryl]

The correlation doesn't suddenly become local once you found a hidden variable that explains it.

[edit] It's not about whether it's "explainable". It's about whether the result is predictable, given local information. It is, in the Bertlemann's socks case, and it is not in the EPR case.

The point of Bertlemann's socks is that the nonlocality is due to our ignorance of the complete state of affairs. So it's a subjective kind of nonlocality.

Well, in the case of a Bell-test experiment, we can explain the non-local correlations if we accept that the world is quantum mechanical:

My issue is not explaining correlations, it's about whether the information needed to predict Bob's future results is nonlocal, or not. Does there exist local information that allows you to predict that Bob will definitely measure spin-down along axis $\vec{A}$? No, that information does not exist. Does there exist nonlocal information that allows you to predict it? Yes, knowing that Alice obtained along that axis allows you to predict that Bob will measure spin-down.

In the Bertlemann's socks case, the information needed to predict the color of the sock exists locally. You just don't happen to know it. In the quantum case, the information doesn't exist locally.

 

[stevendaryl]

No, we can neither predict Alice's nor Bob's results.

That's not true. After Alice measures spin-up along axis $\vec{A}$, she can predict that Bob will measure spin-down along axis $\vec{A}$. So we certainly can predict Bob's results, in certain cases.

 

[stevendaryl]

In the other case, it's also local, because I can also give a common cause explanation for the non-local correlations.

I think you keep getting off onto "explanation", when I'm trying to talk about predictions. There is no local information that will allow you to predict Bob's result. But there is nonlocal information that allows you to predict Bob's result. That's the sense in which quantum predictions are nonlocal.

 

[DrChinese]

Well, in the case of a Bell-test experiment, we can explain the non-local correlations if we accept that the world is quantum mechanical: The entangled quantum particles had been created locally in the past and then sent to Alice and Bob respectively at a speed lower than the speed of light. This is a perfectly causal explanation as soon as we accept that the world is quantum mechanical and quantum objects just happen to exist. If we hadn't entangled the particles in the past, before we sent them to Alice and Bob respectively, we wouldn't have seen the perfect correlations in the Bell-test experiment. ... We can however predict the non-local correlations and we can explain them causally. No FTL is involved. The reason for the correlation lies in the past light cone of both Alice and Bob.

Ah, sorry, this is not factually correct. I say this not even considering the general Bell Theorem issues that others have pointed out.

You can entangle, and get perfect correlations, from particles that have never existed in any common light cone. There is no common cause.

http://arxiv.org/abs/1209.4191

"The role of the timing and order of quantum measurements is not just a fundamental question of quantum mechanics, but also a puzzling one. Any part of a quantum system that has finished evolving, can be measured immediately or saved for later, without affecting the final results, regardless of the continued evolution of the rest of the system. In addition, the non-locality of quantum mechanics, as manifested by entanglement, does not apply only to particles with spatial separation, but also with temporal separation. Here we demonstrate these principles by generating and fully characterizing an entangled pair of photons that never coexisted. Using entanglement swapping between two temporally separated photon pairs we entangle one photon from the first pair with another photon from the second pair. The first photon was detected even before the other was created. The observed quantum correlations manifest the non-locality of quantum mechanics in spacetime."

And

http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.80.3891

We experimentally entangle freely propagating particles that never physically interacted with one another or which have never been dynamically coupled by any other means. This demonstrates that quantum entanglement requires the entangled particles neither to come from a common source nor to have interacted in the past. In our experiment we take two pairs of polarization entangled photons and subject one photon from each pair to a Bell-state measurement. This results in projecting the other two outgoing photons into an entangled state.

 

[rubi]

[edit] It's not about whether it's "explainable". It's about whether the result is predictable, given local information. It is, in the Bertlemann's socks case, and it is not in the EPR case.

No, it's not about predictability. It's only about causation. A theory is local if the cause for each event lies in the past lightcone of that event, rather than a spacelike separated region. That's the definition from relativity. The question is: Did Alice cause Bob's particle to be spin-down, when she measured hers to be spin-up, or would Bob have measured spin-down anyway, even if Alice hadn't measured first? This possibility is certainly consistent with QM. In the first case, Alice would influence Bob's particle non-locally. In the other case, there is no causal relation between the events. QM is silent on this issue. And we can't go back in time and try it.

The point of Bertlemann's socks is that the nonlocality is due to our ignorance of the complete state of affairs. So it's a subjective kind of nonlocality.

Yes, that is correct. But that doesn't imply that all other kinds of non-locality require non-local causal relationships.

My issue is not explaining correlations, it's about whether the information needed to predict Bob's future results is nonlocal, or not. Does there exist local information that allows you to predict that Bob will definitely measure spin-down along axis $\vec{A}$? No, that information does not exist. Does there exist nonlocal information that allows you to predict it? Yes, knowing that Alice obtained along that axis allows you to predict that Bob will measure spin-down.

As I said, predictability is not important. You only require that all events in spacetime that are in a cause-and-effect relationships can be connected by causal curves (i.e. timelike or lightlike curves).

That's not true. After Alice measures spin-up along axis $\vec{A}$, she can predict that Bob will measure spin-down along axis $\vec{A}$. So we certainly can predict Bob's results, in certain cases.

Once we know the result of one measurement, we can predict the other one, just as in Bertlmann's socks. The question is, whether us measuring the first particle/sock caused the second particle/sock to have the value that we can measure.

I think you keep getting off onto "explanation", when I'm trying to talk about predictions. There is no local information that will allow you to predict Bob's result. But there is nonlocal information that allows you to predict Bob's result. That's the sense in which quantum predictions are nonlocal.

As I said, the only relevant question is whether there are spacelike cause-and-effect relationships or not.

 

[rubi]

Ah, sorry, this is not factually correct. I say this not even considering the general Bell Theorem issues that others have pointed out.

You can entangle, and get perfect correlations, from particles that have never existed in any common light cone. There is no common cause.

http://arxiv.org/abs/1209.4191

"The role of the timing and order of quantum measurements is not just a fundamental question of quantum mechanics, but also a puzzling one. Any part of a quantum system that has finished evolving, can be measured immediately or saved for later, without affecting the final results, regardless of the continued evolution of the rest of the system. In addition, the non-locality of quantum mechanics, as manifested by entanglement, does not apply only to particles with spatial separation, but also with temporal separation. Here we demonstrate these principles by generating and fully characterizing an entangled pair of photons that never coexisted. Using entanglement swapping between two temporally separated photon pairs we entangle one photon from the first pair with another photon from the second pair. The first photon was detected even before the other was created. The observed quantum correlations manifest the non-locality of quantum mechanics in spacetime."

And

http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.80.3891

We experimentally entangle freely propagating particles that never physically interacted with one another or which have never been dynamically coupled by any other means. This demonstrates that quantum entanglement requires the entangled particles neither to come from a common source nor to have interacted in the past. In our experiment we take two pairs of polarization entangled photons and subject one photon from each pair to a Bell-state measurement. This results in projecting the other two outgoing photons into an entangled state.

Interesting. I'll read the articles, but I won't have time to reply before sunday evening. Maybe there is another causal explanation for these experiments that just isn't as simple as the one I proposed earlier. If not, I'll admit that I am wrong!

 

[stevendaryl]

No, it's not about predictability. It's only about causation

Maybe that's what you're talking about, but it isn't what I was talking about. I'm talking about predictability. That's exactly the issue, ever since E, P and R wrote their original paper:

If, without in any way disturbing a system, we can predict with certainty...the value of a physical quantity, then there exists an element of reality corresponding to this physical quantity.

In QM, such "elements of reality" are nonlocal.

 

[stevendaryl]

As I said, the only relevant question is whether there are spacelike cause-and-effect relationships or not.

It is not the only relevant question. It might be the only question you care about, but it is not what I'm talking about when I describe QM as nonlocal.

There are two different concepts: signal locality and "Bell" locality (or whatever you it should be called). Everyone agrees that QM satisfies signal locality. But it's not local, in the Bell sense.

 

[zonde]

Or in other words: Bell's factorization criterion requires that all local observables that are needed to predict the local future can be modeled on a combined probability space.

All predictions of QM are made on the same probability space. And it does not matter that in order to arrive at these predictions you have to use different probability spaces.

 

[rubi]

It is not the only relevant question. It might be the only question you care about, but it is not what I'm talking about when I describe QM as nonlocal.

There are two different concepts: signal locality and "Bell" locality (or whatever you it should be called). Everyone agrees that QM satisfies signal locality. But it's not local, in the Bell sense.

I'm arguing that Bell's criterion does not adequately capture the notion of locality in a quantum world. QM violates Bell's criterion, we agree here, but calling it "locality" criterion is a misnomer, because it only captures locality in realistic scenarios.

All predictions of QM are made on the same probability space. And it does not matter that in order to arrive at these predictions you have to use different probability spaces.

That's false. Quantum mechanics does not have a state, in which a particle has both a definite $S_x$ and a definite $S_y$ value. However, a product space has such an event in its $\sigma$-algebra.

I'll answer the remaining posts on sunday.

 

[stevendaryl]

Let me introduce a hypothetical device to explore this concept of nonlocality.

Suppose you have a device with a button on it, and when you push the button, a light on the device glows some color: Red, green or yellow. The color is completely random, except that any two devices, anywhere in the universe always show the same sequence of colors.

I think that people would assume that either the sequence of colors is predetermined, or that the devices are somehow signalling each other. But obviously, if there is no way to use the devices to signal FTL.

 

[zonde]

As I said, predictability is not important. You only require that all events in spacetime that are in a cause-and-effect relationships can be connected by causal curves (i.e. timelike or lightlike curves).

Cause is a fuzzy concept. You can't define experimental criteria based on it.

 

[stevendaryl]

I'm arguing that Bell's criterion does not adequately capture the notion of locality in a quantum world. QM violates Bell's criterion, we agree here, but calling it "locality" criterion is a misnomer, because it only captures locality in realistic scenarios.

Well, I disagree. QM provides a nonlocal description of the results of experiments, in the sense that knowledge about conditions in one region of the universe allows you to make predictions about results in a far-distant region of the universe. We don't have to say anything about "realism" in order to come to that conclusion.

 

[rubi]

Well, I disagree. QM provides a nonlocal description of the results of experiments, in the sense that knowledge about conditions in one region of the universe allows you to make predictions about results in a far-distant region of the universe. We don't have to say anything about "realism" in order to come to that conclusion.

I had already adressed this using nothing but pure logic:

Only under the assumption that the world is classical (i.e. realistic). If the world is classical, the common cause explanation is not possible and thus the discovery of the spin-up particle would have caused the other particle to have spin-down and the other way around: The proof of Bell's inequality assumes realism (modeling on a combined probability space) and locality. Thus a violation of Bell's inequality implies that not both realism and locality are true (by the logical principle of contraposition), which means that realism or locality or both are false. If we know that the world is realistic, must conclude that the locality assumption in the proof must have been false. However, if we don't assume a realistic world (for example a quantum mechanical world), then we can't infer that the locality assumption is false. Neither can we infer that it is true. We just don't know it.

Ok, now I'm really gone till sunday. Bye!

 

[stevendaryl]

I had already adressed this using nothing but pure logic:

Since my criterion for "nonlocal" has nothing to do with common causes or realism, I don't see how that response is at all relevant.

 

[DrChinese]

Interesting. I'll read the articles, but I won't have time to reply before sunday evening. Maybe there is another causal explanation for these experiments that just isn't as simple as the one I proposed earlier. If not, I'll admit that I am wrong!

Take your time. One can always rationalize any position, but the fact is that photons can be entangled (via swapping) that have never interacted in any way, are from independent sources, etc. And in fact you can decide to entangle them AFTER they no longer exist. Pretty hard to make the case for a common cause in that.

And you still would not be making sense even if these experiments didn't exist. Because if you make the case for a common cause you are asserting realism (hidden variables); and we know (due to Bell) that hidden variables cannot be local. It doesn't make sense for you to assert otherwise, as this is generally accepted science (in fact see my tag line, from the Wiki page on Bell's Theorem).

In fact, reading your comments more closely: you are arguing for rejection of classical reality and acceptance of locality. Fine, that is a viable position. But in that quantum world, there is no causal explanation as you imply.

 

[jk22]

If we apply literally EPR's criterion, then the elements of reality should exist only when measuring at same angles for EPRB. But does this not already implies non locality, since how would the elements of reality "know" the state of both detectors to know if they have to exist or not ?

On this view Bell did not show that Einstein was wrong, since Bell suppose elements of reality (lambda) at any angles, whereas EPR would reply, except for same angle, there is no evidence that elements of reality exist.

 

[zonde]

Well, in the case of a Bell-test experiment, we can explain the non-local correlations if we accept that the world is quantum mechanical: The entangled quantum particles had been created locally in the past and then sent to Alice and Bob respectively at a speed lower than the speed of light. This is a perfectly causal explanation as soon as we accept that the world is quantum mechanical and quantum objects just happen to exist. If we hadn't entangled the particles in the past, before we sent them to Alice and Bob respectively, we wouldn't have seen the perfect correlations in the Bell-test experiment. A classical physicist couldn't accept such an explanation. But for a quantum physicist, who accepts that the world behaves quantum mechanically, it is not problematic. It's only the classical thinking that makes it seem weird.

The statement "if we accept that the world is quantum mechanical" is very problematic. First, it might mean a lot of different things related to different interpretations. Second predictions of a theory should be formulated in terms that are independent of theory. So if we analyze just predictions there should be no need to accept anything from the theory.
The second part about causal explanation taken alone is clearly not enough. We can see it by using reductio ad absurdum type argument:
Let's imagine that we are presented with evidence that monozygotic twins can relay messages faster than light. So if we can trace these twins back to the same zygote we can claim that there is no FTL phenomena?
Obviously we have to include into analysis relied messages and correlations between them. In case of entanglement it's measurement angles that we have to include into consideration.

 

[strangerep]

Ah, sorry, this is not factually correct. [...]

You can entangle, and get perfect correlations, from particles that have never existed in any common light cone. There is no common cause.

http://arxiv.org/abs/1209.4191
[...]
http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.80.3891

As well as a "like" of your post, I just wanted to give an explicit "thank you" for emphasizing that. I had not appreciated how strongly such experiments challenge (some) interpretations of physics.

 

[DrChinese]

As well as a "like" of your post, I just wanted to give an explicit "thank you" for emphasizing that. I had not appreciated how strongly such experiments challenge (some) interpretations of physics.

Thanks for your kind comment!

This class of experiments is very difficult for many QM interpretations - regardless of what one's favorites are. Because they don't fit naturally with either the Many Worlds or the Bohmian Mechanics groups. That doesn't stop those groups from claiming they are not ruled out, but again there is nothing natural about how they address this. MW says there is a splitting of worlds upon observation (its signature feature), but clearly that doesn't help much when entanglement is performed AFTER the splitting of worlds. And BM says there are non-local guide waves (its signature feature), which seemingly fails to explain why a photon that no longer exists is entangled with one that exists now - but is not entangled with anything else.

Again, not trying to turn this thread into another battle of interpretations. We can do that in a new thread if that is desired by my many MW and BM friends here ... :)

 

[jk22]

Is it that the only remaining were then the copenhagen ?

 

[atyy]

Thanks for your kind comment!

This class of experiments is very difficult for many QM interpretations - regardless of what one's favorites are. Because they don't fit naturally with either the Many Worlds or the Bohmian Mechanics groups. That doesn't stop those groups from claiming they are not ruled out, but again there is nothing natural about how they address this. MW says there is a splitting of worlds upon observation (its signature feature), but clearly that doesn't help much when entanglement is performed AFTER the splitting of worlds. And BM says there are non-local guide waves (its signature feature), which seemingly fails to explain why a photon that no longer exists is entangled with one that exists now - but is not entangled with anything else.

Again, not trying to turn this thread into another battle of interpretations. We can do that in a new thread if that is desired by my many MW and BM friends here ... :)

I'm not sure the word "entanglement" is used in a standard way in those papers.

I labelled the following statements from http://arxiv.org/abs/1102.1490 as "A" and "B".

"A: We also demonstrate that this two-photon signal may violate Bell's inequalities in the Clauser and Horne (CH74) formulation [12].

B: In view of the above statement this implies that entanglement among two photons may exist even though the two photons do not overlap in time."

In fact, B does not follow from A, if one uses a standard meaning of entanglement. A is what is observed, so as long as an interpretation can explain A, it is fine.

 

[stevendaryl]

I'm not sure the word "entanglement" is used in a standard way in those papers.

I always had a simplistic view of entanglement: A two-particle state is entangled if its state cannot be written as a product. But that doesn't actually make sense, because the Fermi and Bose statistics forces the state to be symmetrized in a way that it can't be a simple product. So I'm not sure what the mathematical definition of entangled state ought to be.

 

[kith]

I'm not sure the word "entanglement" is used in a standard way in those papers.

That's also my sentiment. I didn't have a look at the papers yet but I don't see how the standard definition of entanglement (inseparable states of a tensor product space) can be applied to photons which didn't coexist. On the other hand, violations of Bell-type inequalities without entanglement are maybe even more interesting.

 

[A. Neumaier]

So I'm not sure what the mathematical definition of entangled state ought to be.

It is precisely what you say. Entanglement is a mathematical property that makes only sense between distinguishable particles. They are typically distinguished by their preparation (label the particles by the beam in which they are at the beginning) before they get entangled.

Indistinguishable particles in a multiparticle state have no identity - they don't have a true particle existence since the physical Hilbert space for them has no position operator for one particle! This is why it is much more natural to describe them by fields, which give naturally rise to indistinguishable particle states as anonymous excitations.

If you want to treat indistinguishable particles in a 2-particle state them as two particles with an identity you need to describe them in an unphysical bigger Hilbert space of distinguished particles. There they will be automatically entangled, and remain so if the interaction is physical, since they will remain indistinguishable.

Thus forcing realistic quantum physics into a particle picture creates weirdness almost from the start.

 

[zonde]

On the other hand, violations of Bell-type inequalities without entanglement are maybe even more interesting.

I find it interesting that experiments that test Bell inequalities with efficient detection (with fair sampling loophole closed) actually use non-maximally entangled states as they allow violation of Bell inequalities at lower efficiency.
By itself it does not mean anything but it sort of suggests that entanglement might be just special case of some other more fundamental phenomena (say interference).

 

[DrChinese]

That's also my sentiment. I didn't have a look at the papers yet but I don't see how the standard definition of entanglement (inseparable states of a tensor product space) can be applied to photons which didn't coexist.

Sure they are inseparable, no different in any way mathematically than any other entangled system if drawn up appropriately. The question is: what is the physical meaning of an entangled system with components that do not co-exist? That is certainly no "weirder" (see thread name) than when the entangled components are not co-located (i.e. not local to each other). Keep in mind that standard QM does not favor one over the other (non-local vs. non-contemporaneous entanglement) in any manner. It does not favor entanglement before detection over entanglement after detection either. All of these are equivalently entangled, and you cannot signal with any of the variations.

 

[atyy]

That's also my sentiment. I didn't have a look at the papers yet but I don't see how the standard definition of entanglement (inseparable states of a tensor product space) can be applied to photons which didn't coexist. On the other hand, violations of Bell-type inequalities without entanglement are maybe even more interesting.

http://arxiv.org/abs/1209.4191 Eq 2 and 3.

It's just entanglement swapping. Nothing very mysterious. What is observed is "Entanglement swapping creates correlations between the first and last photons non-locally not only in space, but also in time. Quantum correlations are only observed a posteriori, after the measurement of all photons is completed."

So as long as one can explain the correlations, that's fine. I'm sure there should be a notation (probably using second quantization to fully allow creation and destruction of photons) that will allow even the quantum mechanics to be put into non-mysterious English.

 

[atyy]

I always had a simplistic view of entanglement: A two-particle state is entangled if its state cannot be written as a product. But that doesn't actually make sense, because the Fermi and Bose statistics forces the state to be symmetrized in a way that it can't be a simple product. So I'm not sure what the mathematical definition of entangled state ought to be.

But in this case, one can use the simplistic view, since identical particles need not be involved.

 

[vanhees71]

It is precisely what you say. Entanglement is a mathematical property that makes only sense between distinguishable particles. They are typically distinguished by their preparation (label the particles by the beam in which they are at the beginning) before they get entangled.

Now it's indeed very confusing (not to say weird). Usually the experiments on entanglement are done with photons, which are indistinguishable bosons. Using parametric down conversion they prepare, e.g., the singlet state
$$|\Psi \rangle =\frac{1}{\sqrt{2}} [|\phi_A, \phi_B \rangle -|\phi_B,\phi_A \rangle] \otimes [|1,-1 \rangle-|-1,1 \rangle],$$
where I've factorized the states in a spatial and a helicity (in one arbitrarily given direction) part. It's a symmetrized state as it must be; $|\phi_A \rangle$ denotes a state that refers to a single-photon "wave packet" moving in A's direction. The photons are indistinguishable as it must be. What's entangled are the polarizations, i.e., if A finds $+1$, B finds $-1$ and vice versa. You can't say who measures which individual photon. You can only say that A measures a photon and its polarization state as well as B at the location of their experimental setups (polarizer+photon detector).

I think it's very clear, if you write the state in this complete way, including the spatial (or momentum) part of the states, that the photons are indistinguishable, particularly in this case. You can't say, which individual photon has which helicity. It doesn't even make any sense to try so, because of the very preparation discussed here.

Also you don't need many-body states to have entanglement. A nice example is the Stern-Gerlach experiment which can be seen as an apparatus preparing single-particle states, where position and spin are entangled. In the above notation this single-particle state would read as follows
$$|\psi \rangle=c_1 |\phi_1 \rangle \otimes |+1/2 \rangle + c_2 |\phi_2 \rangle \otimes |-1/2 \rangle,$$
where $|\phi_j \rangle$ refers to wave packets that peak in FAPP well separated regions of space. Then the particle has a spin component +1/2 if found at location 1 and -1/2 if found in region 2.

Indistinguishable particles in a multiparticle state have no identity - they don't have a true particle existence since the physical Hilbert space for them has no position operator for one particle! This is why it is much more natural to describe them by fields, which give naturally rise to indistinguishable particle states as anonymous excitations.

If you want to treat indistinguishable particles in a 2-particle state them as two particles with an identity you need to describe them in an unphysical bigger Hilbert space of distinguished particles. There they will be automatically entangled, and remain so if the interaction is physical, since they will remain indistinguishable.

Thus forcing realistic quantum physics into a particle picture creates weirdness almost from the start.

Well, in non-relativistic QT, where you have a fixed number of particles you can describe everything in terms of appropriate symmetrized or antisymmetrized wave functions. There's no need for QFT, although of course you can use QFT in this case either, and creation and annihilation operators are just more convenient to handle than the cumbersome (anti)symmetrized wave functions of the "1st-quantization formalism".

 

[A. Neumaier]

it's indeed very confusing (not to say weird)

Indeed, it is,. Many experiments about quantum foundations are confusing and hence weird by choice if their language.
Much of it t makes sense only by being sloppy enough. This sloppiness is enough to widely open the gate for all sorts
of weirdness to enter, and for rationality to leave.

Entanglement is something very useful for discussing quantum information theory, where one deals from the start with true tensor products, and exploits superpositions to get computational advantages for cryptographic security or faster algorithms.

it is misplaced for the description of 2-photon states. The two photons in a 2-photon state exist only figuratively. In no sense covered by the formal side of QED, a 2-photon state contains two single photons since photons are intrinsic relativistic objects and there are no associated operators in photon Fock space. The fact that the photon number operator has a discrete spectrum doesn't make single photons existent in a 2-photon state. If it did, we'd also have angular particles and angular antiparticles describing quantum states of high angular momentum.

 

[vanhees71]

This I don't understand either. The usual definition of an $N$-photon state is that it is an eigenstate of the total-photon-number operator of eigenvalue $N$. Then you have two photons by definition. All this, of course, refers to non-interacting photons, because there is no clear definition of a photon number for interacting quantum fields.

 

[stevendaryl]

Now it's indeed very confusing (not to say weird). Usually the experiments on entanglement are done with photons, which are indistinguishable bosons. Using parametric down conversion they prepare, e.g., the singlet state
$$|\Psi \rangle =\frac{1}{\sqrt{2}} [|\phi_A, \phi_B \rangle -|\phi_B,\phi_A \rangle] \otimes [|1,-1 \rangle-|-1,1 \rangle],$$
where I've factorized the states in a spatial and a helicity (in one arbitrarily given direction) part.

So the more sophisticated view is that the interesting situation is not when particles that are entangled (identical particles are always entangled, by my definition), but when specific attributes of the particles (spin or helicity or whatever) are entangled.

 

[A. Neumaier]

The usual definition of an N-photon state is that it is an eigenstate of the total-photon-number operator of eigenvalue N. Then you have two photons by definition.

The first is correct, the second only holds figuratively. For you cannot point to a single property (apart from mass 0 and spin 0, which are nondynamical) that any of the two photons whose existence you assert has. By definition you can conclude only that you have something called a 2-photon state.

Calling something (by analogy to the nonrelativistic case) a photon-number operator doesn't bring photons into existence, just as renaming the angular momentum operator ''angular particle number operator'' doesn't bring angular particles into existence.

I am taking the QFT formalism seriously as a valid description of nature, but not the talk about it, which is largely historical and to some extent inappropriate. It is a similar issue as your fight against the notion of ''second quantization''.