Entanglement in QM - Understanding Instant Correlation & Randomness

[GiuseppeP]

Hello,

I have red on the web about the entanglement, but there is one thing that it is not clear to me: to explain it, why we cannot just say that randomness and correlation between the two entangled particles is happening at instant of their creation and from that time on they stay in that condition until the entanglement is broken?

Thank you.

 

[zonde]

I can offer you to look at this informal proof why this can't be the case:
https://www.physicsforums.com/threads/a-simple-proof-of-bells-theorem.417173/#post-2817138

 

[DrChinese]

why we cannot just say that randomness and correlation between the two entangled particles is happening at instant of their creation and from that time on they stay in that condition until the entanglement is broken?

As zonde mentions: Your assumption leads to a contradiction with observed results and QM's predictions (which match observations). This was discovered by Bell (1964) and is expressed by Bell's Theorem. The short version is that entangled pairs are correlated at many angles in a fashion that no predetermined data set could support.

The only meaningful loophole is that there is faster than light communication between members of the pair. The result is that you either deny the existence of predetermined values at all possible measurement setting (there are no hidden variables); or you deny that the speed of light c is the limit for causal influences. Your choice between these 2 options...

 

[Jilang]

Would it preclude the possibility of one real and one random variable also?

 

[jk22]

What do you mean by real variable and random variable ?

Anyhow Bell's theorem has nothing against global variable in the sense $$AB (a,b,\lambda)$$

There is an argument by https://arxiv.org/abs/quant-ph/0006014 saying the Chsh operator should contain a different variable for each pair ie $$A (a,l1)B (b,l1)-A (a,l2)B (b,l2)... $$

But if you think this argument comes back to admit the pairs are independent. Calculation of the probabilities with the local model gives but the same as bell's result the difference is that there is a variance namely $$|<Chsh>|_{lhv}=2\pm \sqrt {3}/2$$

Whereas $$|<Chsh>|_{qm}=2\sqrt {2}\pm 1/\sqrt {2} $$

 

[Jilang]

I am thinking of a fixed flagpole with a spinning flag, where for entangled pairs the flagpoles are in opposite directions and the spinning direction is opposite for each. The fixed flagpole is the real variable, the spinning flag direction wrt the pole, the variable one.

 

[jk22]

This is exactly the local model used which is A=sgn (a.l) : the result is the signum of the projection of the flag on the measurement direction.

This function howere has nothing 'transcendent' since one would imagine very quick varying one in the neighborhood of one given lambda

 

[DrChinese]

I am thinking of a fixed flagpole with a spinning flag, where for entangled pairs the flagpoles are in opposite directions and the spinning direction is opposite for each. The fixed flagpole is the real variable, the spinning flag direction wrt the pole, the variable one.

Jilang, certainly by now you know that model wouldn't work to enable agreement with observation.

 

[Jilang]

This is exactly the local model used which is A=sgn (a.l) : the result is the signum of the projection of the flag on the measurement direction.

This function howere has nothing 'transcendent' since one would imagine very quick varying one in the neighborhood of one given lambda

That's what's bothering me. It might be "quick-varying"' but will still hit the plane of observation from a given direction.

 

[Jilang]

Jilang, certainly by now you know that model wouldn't work to enable agreement with observation.

No, I have only seen proofs that rule out predetermined existing variables. I am struggling to see though how a composite of one real and one random variable wouldn't work.

 

[DrChinese]

No, I have only seen proofs that rule out predetermined existing variables. I am struggling to see though how a composite of one real and one random variable wouldn't work.

Either: a) there is no difference between predetermined and "random" variables functionally, in which case the outcome is the same as Bell; or b) the random variable acquires its value independently at a later time. If so, and the random variable is local, then you cannot reproduce perfect correlations. So you are back to where you started.

 

[vanhees71]

As is well-known in this forum I disagree, and I'd answer the question in #1 in the positive. The entangled state is created in the very beginning and stays an entangled state as long as there's no disturbance that changes it to a "disentangled" state.

To be specific, let's take the standard experiment with a pair of polarization-entangled photons. They are created by interaction of a laser field with a birefringent crystal by a process called parametric down conversion. Then the corresponding two-photon state, with a polarization part given by
$$|\Psi \rangle=\frac{1}{\sqrt{2}} (|HV \rangle-|VH \rangle).$$
The single photons have maximally uncertain polarization, i.e.,
$$\hat{\rho}_A=\mathrm{Tr}_B |\Psi \rangle \langle \Psi|=\frac{1}{2} \hat{1}=\hat{\rho}_B,$$
i.e., A and B measure just perfectly unpolarized photons. Nevertheless there is a 100% correlation between measurements, i.e., whenever A measures her photon to be H polarized Bob necessarily finds his photon V polarized and vice versa.

The correlation described by the entanglement is there for the whole time, from the creation of the two photons until to their detection much later. There is for sure no faster-than-light interaction triggered by A's measurement with B's photon and vice versa since this is ruled out by the very construction of QED as a local microcausal relativistic QFT. There are very long threads about this in this forum, and we don't need to repeat all the arguments against this view, but for me this is still the only view that is consistent with both the theoretical foundations of QED and observations.

 

[PeterDonis]

why we cannot just say that randomness and correlation between the two entangled particles is happening at instant of their creation and from that time on they stay in that condition until the entanglement is broken?

We can; in fact that's what we do say. What we don't say is that "correlation between two entangled particles" means that the results of all possible measurements that could be made on them are determined at the instant of their creation. That sort of model is ruled out by experiment, as zonde and DrChinese have pointed out. But, as vanhees71 has pointed out, that does not mean the entanglement itself is ruled out; it isn't. You just have to be clear on what entanglement means.

The short version is that entangled pairs are correlated at many angles in a fashion that no predetermined data set could support.

I don't think the OP was necessarily assuming that "entanglement" means "predetermined data set".

I'd answer the question in #1 in the positive.

So would I, because I don't think that the OP necessarily had in mind a local hidden variable model to begin with.

 

[DrChinese]

I don't think the OP was necessarily assuming that "entanglement" means "predetermined data set".

You and vanhees71 probably read that as he meant it. I just assumed he was referencing some form of a hidden variable model. My bad.

The effect we call "entanglement" does not have a clear, well-defined start and end point. In the OP's example, there looks to be a simple explanation of the starting point for the entanglement - it's when the particle pair is created. (Not so clear when the end point occurs though.) On the other hand, there are entanglement examples (swapping comes to mind) where the start point of entanglement is not so well defined. You can have entangled particles that have never co-existed. To me (and not everyone would agree), that shows how difficult it is to reduce such effects to everyday language.

 

[vanhees71]

Well, for entanglement swapping you need entangled systems, you have prepared as such. I've no example, where some entangled system is not somehow prepared with some local process, i.e., where there's no causal connection between the entangled parts of a system in the past.

 

[jk22]

... or you deny that the speed of light c is the limit for causal influences. Your choice between these 2 options...

Note that if you accept supraluminal influences then information has to be sent from one to the other and that not only the result has to but the measurement configuration too which makes this solution to seem quite improbable since the variable would have to carry a huge information like a real number and not only a bit.

Functionally B (A(a,l),b) is not sufficient but it has to be B (A (a,l),a,b) which makes a slight difference.

 

[DrChinese]

Well, for entanglement swapping you need entangled systems, you have prepared as such. I've no example, where some entangled system is not somehow prepared with some local process, i.e., where there's no causal connection between the entangled parts of a system in the past.

To entangle particles A & D, there must be "something" they must have interacted with in some common past individually. But A & D never need to have co-existed in a common light cone; so they have never interacted with each other, nor has there been any possible signal between them intermediated by that "something". The decision to entangle A & D can occur after they no longer exist.

So this blurs the idea that entanglement begins at point T1 and ends at point T2.

 

[vanhees71]

Give an example with a complete preparation/measurement procedure, where this is done. Maybe then I can point out on this example what I mean.

 

[jk22]

The decision to entangle A & D can occur after they no longer exist.

So this blurs the idea that entanglement begins at point T1 and ends at point T2.

Is this similar to the before before experiment where Suarez deduced that the correlation comes from 'outside' spacetime ?

 

[DrChinese]

Give an example with a complete preparation/measurement procedure, where this is done. Maybe then I can point out on this example what I mean.

This is the effect I am referring to. A-D in my example correspond to their photons 1-4. When does the entanglement between photons 1&4 start, and when does it end? Certainly you can define it however you like, but no answer is very convincing.
https://arxiv.org/abs/1209.4191
Entanglement between photons that have never co-existed.
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.

 

[jk22]

The fact that entanglement is represented in an euclidean 4 dimensional space for Bell's state is maybe important since it proves the existence of a fourth space dimension, that has nothing todo with time ?

 

[vanhees71]

I remember to have seen this before. However, I don't see, where there is any problem. If you read the introduction, it's clear how the entanglement of photons 1 and 4 are coming about. There's nowhere anything which is not completely understandable with standard QED, there are non nonlocal interactions necessary (of course not, since everything is fully explanable with QED). The entanglement of photons 1 and 4 is just due to the described preparation procedure.

 

[rubi]

I remember having explained this already some years ago. The photons 1 and 4 aren't really entangled. Their statistics will be completely uncorrelated. In order to get the entangled statistics, one needs to postselect photons based on data that was collected in the region where photons 2 and 3 meet, so the probability distributions that feature dependence require non-local information.

There is an fully classical analoous situation. If instead of emitting pairs of entangled photons, classical pairs light pulses were emitted with either red or blue color, then the light pulses 1 and 4 were still completely uncorrelated. However if you postselect based on data of photons 2 and 3 (for example you select only those events such that the light pulses 2 and 3 have the same color), then pulses 1 and 4 will show perfect correlations, even though they have never coexisted. It's just a Monte Carlo procedure to generate a certain probability distribution. You just discard all events that don't fit your desired probability distribution.

 

[DrChinese]

I remember to have seen this before. However, I don't see, where there is any problem. If you read the introduction, it's clear how the entanglement of photons 1 and 4 are coming about. There's nowhere anything which is not completely understandable with standard QED, there are non nonlocal interactions necessary (of course not, since everything is fully explanable with QED). The entanglement of photons 1 and 4 is just due to the described preparation procedure.

I never said otherwise, yes it's standard QM. The OP question involved a reference to when entanglement began. The strange time sequencing of this experiment points out some of the oddities of QM and causality. I don't see there is any particular point where entanglement began, or where it ended, other than whatever you want to define it to be. QM is quiet, by any reasonable method.

 

[DrChinese]

I remember having explained this already some years ago. The photons 1 and 4 aren't really entangled. Their statistics will be completely uncorrelated. In order to get the entangled statistics, one needs to postselect photons based on data that was collected in the region where photons 2 and 3 meet, so the probability distributions that feature dependence require non-local information.

Sorry rubi, I must disagree on this point. 1 & 4 are absolutely entangled, and will show perfect correlations that indicate the same. If they were not entangled, which can be performed after the fact (or any time), that would not be the case. There is a form of post-selection done when they are entangled, that's true. You need to know that a Bell state was registered, and which one. But the effect itself is unambiguous, and you will not find generally accepted explanations to the contrary.

Here is another authoritative reference, very similar setup in some respects. They also performed a variation in which the entanglement was made to occur after detection.

https://arxiv.org/abs/quant-ph/0201134
Quantum teleportation strikingly underlines the peculiar features of the quantum world. We present an experimental proof of its quantum nature, teleporting an entangled photon with such high quality that the nonlocal quantum correlations with its original partner photon are preserved. This procedure is also known as entanglement swapping. The nonlocality is confirmed by observing a violation of Bell's inequality by 4.5 standard deviations. Thus, by demonstrating quantum nonlocality for photons that never interacted our results directly confirm the quantum nature of teleportation.

 

[DrChinese]

There is an fully classical analoous situation. If instead of emitting pairs of entangled photons, classical pairs light pulses were emitted with either red or blue color, then the light pulses 1 and 4 were still completely uncorrelated. However if you postselect based on data of photons 2 and 3 (for example you select only those events such that the light pulses 2 and 3 have the same color), then pulses 1 and 4 will show perfect correlations, even though they have never coexisted. It's just a Monte Carlo procedure to generate a certain probability distribution. You just discard all events that don't fit your desired probability distribution.

And just to address your above point: obviously classical scenarios do NOT generate Bell inequality violations. The Zeilinger experiment violated one by 4.5 SD. The example above is just a variation of Bertlmann' socks, so I won't go any farther.

 

[rubi]

Sorry rubi, I must disagree on this point. 1 & 4 are absolutely entangled, and will show perfect correlations that indicate the same. If they were not entangled, which can be performed after the fact (or any time), that would not be the case. There is a form of post-selection done when they are entangled, that's true. You need to know that a Bell state was registered, and which one. But the effect itself is unambiguous, and you will not find generally accepted explanations to the contrary.

If you take all detected events into account, you will find completely uncorrelated statistics. The point is that you can emulate entangled statistics by taking only those events into account that produce the entangled statistics. I agree that this emulated statistics will violate Bell's inequality and show perfect correlations. But you only get the effect if you perform a post-selection based on non-local data. Alice (photon 1) and Bob (photon 4) can't decide which events to keep and which to discard without having information from Charlie (photons 2 & 3), so the post-selected probability distributions for Alice and Bob are obtained from data localized in overlapping space-time regions (Charlie). Only data obtained only from spacelike separated regions can be a test for locality. While your post-selected statistics will undoubtedly show quantum correlations, it's not a test for locality.

Here is another authoritative reference, very similar setup in some respects. They also performed a variation in which the entanglement was made to occur after detection.

https://arxiv.org/abs/quant-ph/0201134
Quantum teleportation strikingly underlines the peculiar features of the quantum world. We present an experimental proof of its quantum nature, teleporting an entangled photon with such high quality that the nonlocal quantum correlations with its original partner photon are preserved. This procedure is also known as entanglement swapping. The nonlocality is confirmed by observing a violation of Bell's inequality by 4.5 standard deviations. Thus, by demonstrating quantum nonlocality for photons that never interacted our results directly confirm the quantum nature of teleportation.

I don't deny that the photons have never interacted. All I'm saying is that the post-selected probability distributions are computed from data obtained from overlapping space-time regions.

And just to address your above point: obviously classical scenarios do NOT generate Bell inequality violations. The Zeilinger experiment violated one by 4.5 SD. The example above is just a variation of Bertlmann' socks, so I won't go any farther.

I agree that this scenario will not generate Bell inequality violations. The point of the example is to show that it is not surprising that photons that have never interacted can be correlated if you perform a post-selection based on non-local data. The post-selected probability distributions in my example will undoubtedly show perfect (classical) correlations, even though there is no common cause in their intersecting past light cones. This is not surprising for a classical physicist and it should not be surprising for a quantum physicist either. Even though my example shows perfect classical correlations over spacelike distances without a common cause in the past light cone, it doesn't imply that classical Maxwell electrodynamics is non-local.

 

[DrChinese]

If you take all detected events into account, you will find completely uncorrelated statistics. The point is that you can emulate entangled statistics by taking only those events into account that produce the entangled statistics. I agree that this emulated statistics will violate Bell's inequality and show perfect correlations. But you only get the effect if you perform a post-selection based on non-local data. Alice (photon 1) and Bob (photon 4) can't decide which events to keep and which to discard without having information from Charlie (photons 2 & 3), so the post-selected probability distributions for Alice and Bob are obtained from data localized in overlapping space-time regions (Charlie). Only data obtained only from spacelike separated regions can be a test for locality. While your post-selected statistics will undoubtedly show quantum correlations, it's not a test for locality.

I don't know why you would make this assertion. First, it completely goes against the science. Second, the test performed on photons 2 & 3 does NOT check for the angle of polarization as is done at 1 & 4. The 2 & 3 Bell State Analyzer can be oriented at any angle, it matters not. If there was nothing non-local going on, then this could not occur.

And I don't know why you "diss" an experiment just because there is post selection. Entanglement is a rare event even using standard PDC. So you have to look for the signature of entanglement. That means you may include some cases where there is no entanglement by mistake. That hurts the stats, not helps them.

It is called entanglement-swapping for a reason, and this is one of the core technologies for quantum communication protocols. Perhaps you have a citation that supports your belief that photons 1 & 4 are not entangled?

 

[rubi]

I don't know why you would make this assertion. First, it completely goes against the science. Second, the test performed on photons 2 & 3 does NOT check for the angle of polarization as is done at 1 & 4. The 2 & 3 Bell State Analyzer can be oriented at any angle, it matters not. If there was nothing non-local going on, then this could not occur.

I don't deny anything that is written in the article. I'm just telling you that the post-selected probability distributions that show the quantum correlations require data from Charlie, so they are obtained from overlapping space-time regions. I don't see how you can deny this.

And I don't know why you "diss" an experiment just because there is post selection. Entanglement is a rare event even using standard PDC. So you have to look for the signature of entanglement. That means you may include some cases where there is no entanglement by mistake. That hurts the stats, not helps them.

I don't "diss" the experiment. The experiment is great and it shows that you can obtain quantum statistics over spacelike separated regions. Post-selection is a perfectly admissible procedure. All I'm saying is that this experiment is not a threat to the locality of quantum electrodynamics, just like my example above is not a threat to the locality of classical electrodynamics and the authors of the article don't claim that it is. I'm in 100% agreement with the authors of that article.

It is called entanglement-swapping for a reason, and this is one of the core technologies for quantum communication protocols.

Nothing wrong with that.

Perhaps you have a citation that supports your belief that photons 1 & 4 are not entangled?

You can read in any quantum mechanics textbook that the definition of entanglement is that the state vector cannot be written as a tensor product state. Since photons 1 and 4 never coexist, there is no reference frame in which the state vector contains both photon 1 and 4, neither as a tensor product nor as a non-trivial linear combination of tensor products. Hence, there is no state vector of photons 1 and 4 that meets the textbook definition of entanglement.
--
DrChinese, unlike some other people in this forum, I don't believe that it is your intention to spread misinformation about QM, but I really think you have misunderstood something essential about that paper and the intention of the authors. I have tried as good as I can to clarify this misunderstanding and I don't think I can explain it any better. I don't want to spend another 20 pages discussing this, so maybe we can just agree to disagree. Maybe @vanhees71 wants to continue the discussion and knows a simpler way to explain what I mean.

 

[DrChinese]

You can read in any quantum mechanics textbook that the definition of entanglement is that the state vector cannot be written as a tensor product state. Since photons 1 and 4 never coexist, there is no reference frame in which the state vector contains both photon 1 and 4, neither as a tensor product nor as a non-trivial linear combination of tensor products. Hence, there is no state vector of photons 1 and 4 that meets the textbook definition of entanglement.

DrChinese, unlike some other people in this forum, I don't believe that it is your intention to spread misinformation about QM, but I really think you have misunderstood something essential about that paper and the intention of the authors. I have tried as good as I can to clarify this misunderstanding and I don't think I can explain it any better. I don't want to spend another 20 pages discussing this, so maybe we can just agree to disagree.

As to the "textbook" definition of entanglement, I fail to see how the system of photons 1&4 are in a product state. Obviously they share a state.

So I guess that is where we leave it. I stand by the characterization of photons 1&4 being entangled (the title is after all "Entanglement Between Photons that have Never Coexisted"). Such entanglement is both non-local ("the nonlocality is confirmed by observing a violation of Bell's inequality by 4.5 standard deviations. ") and non-temporal ("the observed quantum correlations manifest the non-locality of quantum mechanics in spacetime") exactly as claimed in the cited articles. Obviously at some level, the descriptions used by Zeilinger et al do not match yours. You have yet to present any reference to support your position other than to agree with the articles but disagree with me - which seems odd.

Vanhees71 and I have had a constructive private discussion in the past about entanglement - especially single particle entanglement. If either you or he think there is any point to continuing, that's fine. Any 2 particles (or a stream of same) that exhibit perfect spin correlations at any angle are entangled. There is no classical effect that can reproduce this for quantum states. Unless you have something more specific that says otherwise, I don't know what there is to discuss.

 

[rubi]

As to the "textbook" definition of entanglement, I fail to see how the system of photons 1&4 are in a product state. Obviously they share a state.

So if you fail to see how photons 1&4 are in a product state, then you should agree that their state doesn't meet the textbook definition of an entangled state, so they aren't entangled by the standard definition.

So I guess that is where we leave it.

The problem is that you misunderstand everything I write:

I stand by the characterization of photons 1&4 being entangled (the title is after all "Entanglement Between Photons that have Never Coexisted"). Such entanglement is both non-local ("the nonlocality is confirmed by observing a violation of Bell's inequality by 4.5 standard deviations. ") and non-temporal ("the observed quantum correlations manifest the non-locality of quantum mechanics in spacetime") exactly as claimed in the cited articles.

The authors use a non-textbook definition of entanglement. What they mean is just that their experiment reproduces the statistics of entangled particles. I never denied that the correlations in their experiment are non-local and non-temporal. However, this is completely unproblematic in a local theory. The correlations in my above example are also non-local and non-temporal, even though classical Maxwell electrodynamics is a perfectly local theory. There is no contradiction between your claim and my claim.

Obviously at some level, the descriptions used by Zeilinger et al do not match yours. You have yet to present any reference to support your position other than to agree with the articles but disagree with me - which seems odd.

I have told you that my definition of entanglement agrees with the textbook definition. I thought it was enough to point you to the textbooks, but if you want a specific one, you can check out "Quantum Theory, Concepts and Methods" by Peres.

Any 2 particles (or a stream of same) that exhibit perfect spin correlations at any angle are entangled.

This does not apply to your photons 1 and 4. Photons 1 and 4 show absolutely no correlation at any angle. In order to get the correlations, you need to select a subsequence of events based on measurements of photons 2 and 4, so you need data from overlapping spacetime regions. You seem to ignore this point entirely, so it doesn't make sense to continue the discussion, since this is my main point. The subset of events that show quantum correlations must be collected from data from overlapping spacetime regions, otherwise you will see no correlation.

By the way, I'm not saying anything that can't be found in textbooks. In fact, I'm defending the mainstream position.

 

[PeterDonis]

So if you fail to see how photons 1&4 are in a product state, then you should agree that their state doesn't meet the textbook definition of an entangled state, so they aren't entangled by the standard definition.

I'm confused. Isn't the textbook definition of entanglement that the state can't be written as a product state (but only as the sum of multiple product states)? So if photons 1 and 4 are not in a product state, doesn't that mean they do meet the textbook definition of an entangled state?

 

[DrChinese]

The authors use a non-textbook definition of entanglement.

As I said, you are rejecting the work of some of the top physicists in the field. And it is a bit lame to reference an entire textbook in support of your position. If there is something specific being referenced in support of a position, the norm is to cite that as I have. On the other hand, since you cited the book by Peres written in 1995, I wonder what he had to say in 2000 relevant to the experiment I cited by Zeilinger et al. Hmmm, perhaps you would look at page 5 of that.

"As shown in Fig. 3, the observed fidelity of the entanglement of photon [1] and photon [4] matches the fidelity in the non-delayed case within experimental errors. Therefore, this result indicate that the time ordering of the detection events has no influence on the results and strengthens the argument of A. Peres [4, published in 2000]: this paradox does not arise if the correctness of quantum mechanics is firmly believed."

The reason that the entangled state stats are reproduced in the cited papers is that the photons (1&4) are entangled. In fact, you could set it up so you could predict with certainty the outcome of each and every photon 4 polarization outcome, for example, and they (1&4) still will never have interacted in the past nor existed in a common light cone. And in conjunction with the OP question: it will not be possible to identify the point in time they became entangled, nor precisely at which point they ceased to be entangled. And yet all of this is standard QM.

So if you want to cling to your textbook definition of entanglement, I will simply say that it is outdated - and the author of your citation has agreed in a subsequent paper referenced by Zeilinger et al. Seriously, the state of the art on these experiments has moved a long way in the past 5, 10, 20 years. Entanglement comes in many exotic forms, and traditional notions of locality and causality - such as you may adhere to - do not serve to adequately describe what is going on. You can entangle particles that have never interacted after detection, before detection, within a common light cone, or fully non-locally. For another example of the technique I have cited (swapping), check out this important 2015 result:

https://arxiv.org/abs/1508.05949
"We employ an event-ready scheme that enables the generation of high-fidelity entanglement between distant electron spins. ... Our experiment realizes the first Bell test that simultaneously addresses both the detection loophole and the locality loophole." Photons [2&3] are used as critical components for the entanglement of the electrons, which serve in the same role as photons 1&4 in my examples.

 

[rubi]

I'm confused. Isn't the textbook definition of entanglement that the state can't be written as a product state (but only as the sum of multiple product states)? So if photons 1 and 4 are not in a product state, doesn't that mean they do meet the textbook definition of an entangled state?

For entanglement, you need a tensor product of individual system and a state that can't be written as a tensor product within this space. However, that doesn't apply in this case, since at no point in time (for any observer), the two photons 1 and 4 exist simultaneously. If we want to put in mathematically, we have to describe this experiment in a Fock space. The states of the system during its time evolution are (very roughly, up to phases and normalization and we really need density matrices):

1. $\psi_1\otimes\psi_2 + \psi_2\otimes\psi_1$
2. $\psi_2$
3. $\psi_2\otimes(\psi_3\otimes\psi_4+\psi_4\otimes\psi_3)$
4. $\psi_4$

But at no time, for no observer, the system is in a state $(\psi_1\otimes\psi_4+\psi_4\otimes\psi_1)\otimes\psi_{\text{something else}}$, since at no time, photons 1 and 4 coexist.
After post-selection, we will find statistics that matches the statistics of a hypothetical state $\psi_1\otimes\psi_4+\psi_4\otimes\psi_1$ for coexisting photons 1 and 4. However, it's only the statistics that matches this state. The system itself is never in this state.

 

[DrChinese]

I'm confused. Isn't the textbook definition of entanglement that the state can't be written as a product state (but only as the sum of multiple product states)? So if photons 1 and 4 are not in a product state, doesn't that mean they do meet the textbook definition of an entangled state?