# I Confused by nonlocal models and relativity

#### vanhees71

Gold Member
Then why put in the extra word "local"? Particularly when it causes interminable arguments in threads like this, because the word "nonlocal" is commonly used to describe correlations that violate the Bell inequalities?
The problem is that this terminology is used all the time. It's hopeless to change terminology which is established for nearly 100 years. That makes the debates about the foundations so difficult.

#### vanhees71

Gold Member
1. Finally!

2. Agreed.

3. No, and once again, please give a reference other than yourself. I have never seen such a [*] description in any entanglement swapping paper [**]. If the entanglement swap is not executed at 2 & 3, as I indicated above, the far separated 1 & 4 will not be entangled - contradicting your purely local [***] characterization.

Once 1 & 4 are entangled, they are mirrors of each other and have quantum properties that far exceed what 2 otherwise independently created particles could ever have. Of course, they are now part of a quantum system and are no longer separable. You could never construct such a system of 2 particles otherwise as you could never know that many non-commuting observables. Zeilinger says, of the 2 entangled particles that never interact:

"The aim is that the distant experimenter Bob [looking at photon 4] obtains an exact replica of that particle. It is evident that no measurement whatsoever Alice [looking at photon 1] might perform on the particle could reveal all necessary information to enable Bob to reconstruct its state."

That doesn't happen because of post-selection! That should be obvious. There are NO such 2 particles that can be independently created and later post selected (they must be entangled, and cannot be entangled with any other particle). That is strictly forbidden by the HUP. The mirror particles (the ones projected into the singlet state) have the mirroring values for non-commuting p and q, for example. They must be physically connected as part of a single, non-separable, quantum system. It's non-local in spatial extent.

4. The humorous element is that you completely ignore everything Zeilinger actually says, and then recast his viewpoint to match yours. Zeilinger does not claim to have an understanding of the mechanism by which quantum nonlocality operates (and I don't either). But he would never refer to QM as being local causal, and I certainly can't recall a respected paper of the last 20 years using any terminology similar to yours.

And please, despite our going back and forth on the subject, I hope you would not take our discussion as anything other than friendly.

*Let's just say "contrived".
*** Seriously, it's 2019 my friend. Quantum non-locality was generally accepted some time ago.
For sure, I think this is a completely friendly discussion, and I think all there is is a misunderstanding between our points of view due to different understanding of unsharply defined words.

The most puzzling thing for me is our disagreement about 3. I also don't see that my viewpoint is any different from what's written in the paper by Zeilinger et al which we discuss here.

Again ad 3). In the paper they emphasize themselves that in order to get a preparation of a subensemble, where the pair 1&4 is entangled is the sufficient locality of the manipulations/measurements on the pair 2&3. The Bell measurement of 2&3 must be made such that it's happening in a sufficiently small space-time region to be within the temporal and spatial coherence lengths of the photons. Thus I agree with both you and the authors of the paper that the entanglement swapping won't work if you don't match this constraint.

The photons 1&4 have never causally interacting and are nevertheless entangled after the entanglement-swapping procedure. We also agree on this, and I never denied that quantum states can describe "non-local correlations through entanglement". I emphasize however, that it describes "correlations" not "causal connections". As I said, I think our mutual misunderstanding is due to the unsharply defined meaning of these words.

Finally, I don't believe that physics is about understanding some "mechanism". Whatever that might be. Quantum theory as any physical theory describes (admittedly in a quite formal way) phenomena, and all there is to understand is this formalism and how it's applied to phenomena. The very fact that Zeilinger et al can invent all their beautiful experiments which finally all confirm QT shows that they have very well understood QT. There's nothing more to understand! Just take the formalism seriously, and this formalism is in some sense a miracle, making the long-range correlations described by entanglement and the implied "inseparability" on the one hand consistent with Einstein causality through the microcausality constraint of relativistic QFT on the other hand!

That doesn't happen because of post-selection! That should be obvious. There are NO such 2 particles that can be independently created and later post selected (they must be entangled, and cannot be entangled with any other particle). That is strictly forbidden by the HUP.
Why's it obvious that entanglement can't be created via a quantum post-selection procedure?

Imagine that you've already performed some Bell tests on 1&4 and you have the results in front of you. With those results you should be able to classically select pairs so that those results seem to violate a Bell inequality. However, because you've measured them, at this point the entanglement correlations are broken.

I think what's happening in this experiment is almost the same thing but unlike with the classical example, it doesn't reveal the measurement results for 1&4, so they can maintain their entanglement. Initially, even without any interaction, there is a sub-ensemble of 1&4 which is entangled along a chosen direction. Maybe it's correct to say every pair of identical particles is entangled along a random direction. Then you use the quantum post-selection procedure on pairs 2&3 to identify an appropriate sub-ensemble of 1&4.

#### vanhees71

Gold Member
Exactly!

In other words: You can post-select the subensemble 100 years after the experiment was done provided you have a sufficiently complete measurement protocol. There's nothing mysterious in this as long as you don't insist on the collapse interpretation in the sense of a causal action at a distance as some Copenhagen flavored interpretations insist upon but interpret entanglement as a description of very strong correlations between parts of a quantum system.

#### microsansfil

*For example: The entanglement swap can occur such that the entangled photons never co-exist. So obviously the entanglement did not occur at their creation - they were not created together (as you proposed). Variations on entanglement swapping wreck attempts to describe in terms of locality and/or (micro)causality.

Entanglement Between Photons That Have Never Co-existed
When photon 1 is measured in a certain basis the correlation with photon 2 is break.
Photon 2 is delayed until a second pair (3–4) is created and photons 2 and 3 are projected onto the Bell basis.

When photon 2 is projected, with photon 3, onto bell basis (Bell measurement), will the entanglement swapping still be realized ? Since photon 2 has lost its correlation with photon 1.

/Patrick

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#### vanhees71

Gold Member
Then it won't be realized. That's the whole point of the discussion! To get entanglement swapping it is necessary to measure "locally (in space and time!) enough" to be within the coherence lengths of the photons. If you perform the measurments on 2&3 at a time where 2 is no longer entangled with 1 and/or 3 no longer entangled with 4, all you get are uncorrelated photons 1&4, no matter what you meaure on photons 2&3.

#### DrChinese

Gold Member
Why's it obvious that entanglement can't be created via a quantum post-selection procedure?
Ha, perhaps not obvious because of my poor description.

They can't be created by post-selection! Post-selection (as an explanation for what is occurring) implies that these 2 photons - 1 & 4 - are occasionally and randomly entangled (with each other); and then post-selection simply reveals that entanglement. That's a reasonable hypothesis, but here's why that is absolutely impossible.

A. You can create 2 photons that have the same polarization (at a specific angle), for example light from a laser, but a test for entanglement will fail on those. There are a number of mechanisms for creating entangled pairs; here, pair 1 & 2 are entangled via PDC and 3 & 4 are as well. In fact they are maximally entangled. You may recall there is something called "monogamy of entanglement". That says that if photon 1 is maximally entangled with photon 2, then it CANNOT be entangled with photon 4 or with any other particle in the universe.

B. Now you run into our problem: what happened to cause 1 & 4 to become entangled - meaning that ANY observation on one tells you the matching value of an observable on the other? That state for 1 & 4 only occurs IF they were entangled, there is no other quantum state that yields the same results. So what could possibly have occurred to cause this state? "Something" must have happened to change all that!

C. I say (and this simply follows the rules of entanglement swapping, see the cited paper for more detail):
- Photons 1 & 4 were entangled as a result of a decision to cast them into a Bell State (by action on photons 2 & 3), else they weren't entangled at all.
- The mechanism for entangling 1 & 4 is to bring 2 & 3 together in the Bell State Analyzer (BSA) and perform a Bell State Measurement (BSM) on them. Some of the 2 & 3 pairs will randomly be cast into a Bell State which indicates that 1 & 4 are now entangled.
- Such only happens if 2 & 3 are completely indistinguishable. In the experiment, that occurs when they arrive at the BSA within a very small time window, perhaps within 5-10 picoseconds. They must also show up in different arms (different PBS) of the BSA, after passing through a common Beam Splitter (BS), as that indicates a Bell State.

D. If the vanhees hypothesis were correct (local causality, no physical collapse, his minimal interpretation where observations reveal pre-existing attributes):

- Photons 2 & 3 are speeding towards the BSA and whatever happens (or doesn't happen) at the BSA cannot change photons 1 & 4 because they are spacelike separated from the BSA.
- Photons 1 & 4 are NOT yet entangled with each other, because they are entangled with 2 & 3 respectively.
- The 2 & 3 photons enter the BSA around the same time (within a very small window), but they generally cannot interfere or otherwise interact with each other. So presumably, whether they are going to register as meeting the conditions for a Bell State (psi+ or psi-) has already been determined and the BSA reveals the appropriate detector clicks to so indicate.

E. And now we have big problems with no good answers (although you can try):
i) How (and when) did photon 1 lose its entangled connection to photon 2? Presumably that occurs when photon 2 is measured, correct? But then it no longer exists!
ii) How (and when) did photon 1 gain an entangled connection to photon 4? After all, we have postulated it was previously entangled with photon 2 and photon 2 alone (due to monogamy). And nothing is changing for its state, as it is too far away.
iii) More importantly, why would photon 4 - out of all the particles in the universe - suddenly have this unique and monogamous connection to photon 1? (Remember: attributes such as momentum (frequency), polarization, etc. are all part of the entanglement.)

I would hope these problems would convince you of the futility of the hypothesis. There is no narrative described in any paper I have read that remotely matches the vanhees hypothesis (I've asked and asked without any success for a reference). But if E. didn't convince you...

F. Furthermore: If you accept the vanhees hypothesis, then the detector clicks at the BSA are revealing pre-existing attributes of the 2 & 3 pair which is to be used to identify Bell States. And critically, photons 2 & 3 do not interact in any way, as mentioned above. So if we were to delay photon 2 from arriving at the BSA by (say) 100 ps, that should not in any way affect the detector click outcomes at the BSA other than to have one of the clicks occur about 100 ps later than the other. Here's how it might look for the relative detection time stamps, and I am making up simple time stamps for purposes of illustration:

No delay:
Detector 1V: 1000000ps (photon 1)
Detector 2H: 1000003ps (either photon 2 or 3, not sure which)
Detector 3H: 1000004ps (either photon 2 or 3, not sure which)
Detector 4V: 1000002ps (photon 4)

Add in a 100ps delay to the photon 2 path to make it distinguishable from photon 3:
Detector 1V: 1000000ps (photon 1)
Detector 2H: 1000003ps (either photon 2 or 3, not sure which)
Detector 3H: 1000104ps (this must be photon 2, which now makes the 2H detector click due to photon 3)
Detector 4V: 1000002ps (photon 4)

You can see that this should NOT identify different 2 & 3 pairs if the BSA is revealing pre-existing properties of those photons. The only difference is that 2 is easily identified. To put this in perspective: 4-fold coincidences similar to the above only occur every few seconds on the average. That would place the next occurrence perhaps 1000000000000+ ps later than my example. Not much chance to get confused about which clicks belong together.

Of course: if 2 & 3 are distinguishable, the rule is that there is no entanglement swap. So 1 & 4 are not entangled, and now demostrate Product State statistics which are clearly different than Entangled State statistics. But we postulated their polarizations were simply being revealed, and nothing occurring at the BSA changed anything. But actually: choosing to add a delay to the photon 2 path does change the 1 & 4 statistics, and that occurs both non-locally and without any causal direction in time. QED.

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#### PeterDonis

Mentor
The problem is that this terminology is used all the time.
Really? Everybody uses "local" to mean "microcausal"?

That certainly isn't true of Bell's original paper or the copious literature it has spawned; in that literature, "local" means what Bell defined in his original paper (that the joint probability function factorizes). That definition has nothing whatever to do with causality, or with QFT.

#### vanhees71

Gold Member
Ha, perhaps not obvious because of my poor description.

They can't be created by post-selection! Post-selection (as an explanation for what is occurring) implies that these 2 photons - 1 & 4 - are occasionally and randomly entangled (with each other); and then post-selection simply reveals that entanglement. That's a reasonable hypothesis, but here's why that is absolutely impossible.
So you say the cited paper by Zeilinger et al is fake? I don't believe that!

Again: Before the described selection by the (local) manipulations of photons 2&3 the photons 1&4 were not entangled because the initial state the four photons are prepared in are factorizing in the following way
$$|\Psi_{1234} \rangle = |\Psi_{12} \rangle \otimes |\Psi_{34} \rangle.$$
The two-photon states $|\Psi_{12} \rangle$ and $|\Psi_{34} \rangle$ are maximally entangled (Bell states) but the pair 1&4 are not. This state now describes and ensemble of equally prepared four photons, and this preparation is described by this state and nothing else.

Now if you do the described manipulations on photons 2&3, however, and select (or post-select, which doesn't matter at all) an subensemble, i.e., after choosing only those photon pairs 1&4, for which the pair 2&3 was found in one of the maximally entangled Bell states, then this subensemble of photon pairs 1&4 is described by a maximally entangled Bell state $|\Psi_{14} \rangle$.

The subensemble can be chosen whenever you like after the experiment is done and an appropriate measurement protocol is taken. The possibility of postselection due to entanglement is more explicit in the famous quantum-erasure experiments a la Wheeler, now realized several times with photons (e.g., Kim et al or Walborn et al).

#### vanhees71

Gold Member
Really? Everybody uses "local" to mean "microcausal"?

That certainly isn't true of Bell's original paper or the copious literature it has spawned; in that literature, "local" means what Bell defined in his original paper (that the joint probability function factorizes). That definition has nothing whatever to do with causality, or with QFT.
That's precisely what I mean. Bell talks about correlations and (in)separability, while in QFT locality has a different meaning, namely locality of interactions and thus microcausality. That's what's confusing, and that's why one has to read each paper in context. This confusion is why some people think Bell's non-locality would contradict the QFT-meaning of locality, but that's NOT the case!

The most accurate experiments are done with entangled-photon state, and if anything gets relativistic it's photons. The experiments are analyzed with standard QED, which is a microcausal QFT, and as long as there's no contradiction between experiment and QED I believe that relativistic QFT is consistent with both locality of interactions (microcausality and thus Einstein causality) and the fact that there are non-local correlations described by entanglement and inseparability.

What's not so clear to me is whether Bell was convinced that QT (no matter whether relativistic QFT or non-relativistic QM) is wrong and there's some local hidden-variable theory being right and QT being wrong or not. Before Aspect that could well have been possible, but after his experiments and the much more accurate ones possible since then, I think that's out of the question.

#### PeterDonis

Mentor
in QFT locality has a different meaning, namely locality of interactions and thus microcausality
"Microcausality" in QFT means operators at spacelike separated events commute. That in itself does not mean the same thing as "locality of interactions". You could have as many nonlocal interactions as you like, as long as the outcomes of spacelike separated measurements do not depend on the order in which they are performed.

#### vanhees71

Gold Member
Then define what you mean by "locality of interactions". The very reason to use the constraint of microcausality in fact is to have local interactions only, i.e., you build the interaction Lagrangians as integrals over local operators obeying the microcausality contraint. This is at least sufficient for locality. I'm not sure whether it's also necessary, but the Standard model, including QED, is using the microcausality constraint anyway.

#### PeterDonis

Mentor
Then define what you mean by "locality of interactions".
I didn't introduce the term so I feel no need to define it.

I already gave my definition of "locality": it's the one Bell used in his paper.

The very reason to use the constraint of microcausality in fact is to have local interactions only, i.e., you build the interaction Lagrangians as integrals over local operators obeying the microcausality contraint.
But those operators also have to obey a nonlocal constraint: they have to commute with other operators at spacelike separated events.

#### vanhees71

Gold Member
This is of course fulfilled either, because you build the local observables from quantum fields obeying bosonic or fermionic commutation or anticommutation relations, respectively. Then the observables built up from them are all obeying the microcausality constraints.

Again, the only point I want to make is that you have to clearly specify whether you mean locality in the sense of interactions/microcausality or in Bell's sense, where it means separability.

I really don't understand what you are after.

#### DrChinese

Gold Member
So you say the cited paper by Zeilinger et al is fake?
I don't get this, where do I say the paper I cited is fake? I realize that English may not be your first language, but this is a bit extreme.

Post selection identifies the 4-fold events to be analyzed. What I said is that the post selection process itself does NOT create the entanglement. Zeilinger says nothing different, and if he does, feel free to point that out.

And I again call for a narrative from ANY entanglement swapping experiment that describes the swapping action as you do. Specifically: your opinion that it is local and causal, nothing happening at the BSA that causes the swap.

Entanglement swapping IS due to the Bell state cast which non-locally affects photons 1 & 4 (changing Product state stats to Entangled state stats), and the entire process lacks any semblance of a causal direction. Ordering has no bearing on the outcomes.

#### PeterDonis

Mentor
I really don't understand what you are after.
You were arguing with @DrChinese about whether QFT is "local". You were actually in violent agreement with him, because he was using "local" the way I am using it (Bell's definition), but you were using "local" to mean "microcausal", and both are true; QFT is microcausal, but it violates the Bell inequalities. If you had just left out the word "local", there would have been no argument.

#### DrChinese

Gold Member
1. Now if you do the described manipulations on photons 2&3, however, and select (or post-select, which doesn't matter at all) an subensemble, i.e., after choosing only those photon pairs 1&4, for which the pair 2&3 was found in one of the maximally entangled Bell states, then this subensemble of photon pairs 1&4 is described by a maximally entangled Bell state $|\Psi_{14} \rangle$.

2. The possibility of postselection due to entanglement is more explicit in the famous quantum-erasure experiments a la Wheeler, now realized several times with photons (e.g., Kim et al or Walborn et al).
1. There are no papers that say that 4-fold coincidences simply "reveal" entanglement properties of 1 & 4 that were pre-existing. They all say that decision to project them into a Bell state - or not - is what is responsible for 1 & 4 entanglement. The below quote from the cited paper is a good example that confirms my description and rejects yours:

"We confirm successful entanglement swapping by testing the entanglement of the previously uncorrelated photons 1 and 4."

That is diametrically the opposite of what you describe, which is that 1 & 4 are correlated if 2 & 3 click a certain way. They aren't, the swap must occur for that to happen. Please note that the action is called "entanglement swapping" because photon 1's entangled partner changes from photon 2 to photon 4. Nowhere is the process of projection, casting, swapping referred to in terms that imply pre-existing relationships are being "revealed".

And if you can address my example F. in my post #57, that would be on target. Ditto for explaining why entanglement monogamy doesn't prevent 1 & 4 from being entangled while 1 & 2 are. I look forward to that.

2. Try referencing something a little more modern... and relevant. We are not discussing quantum erasers in this thread. We are discussing entanglement swapping using independent PDC sources.

Again, please QUOTE a suitable citation for your position - or retract it. If you are correct, and this is standard QFT, such quotes must be everywhere. I would let this drop (since I know I won't change your opinion regardless of what I cite) but there are plenty of others who are following who are drawing a completely wrong picture of things because of what you are saying. People are drawn to local causality, but the whole point of entanglement swapping experiments is to show in the strongest of terms that position is not tenable in any form.

And if you can address my example F. in my post #57, that would be on target. Ditto for explaining why entanglement monogamy doesn't prevent 1 & 4 from being entangled while 1 & 2 are. I look forward to that.
This is also something I would like to know. At the very least, it should be possible for 1 to be entangled with 2, while (1&2) is entangled with (3&4), but that doesn't seem like this case. Maybe monogamy of entanglement is only about "known" entanglement; after all, having two qubits entangled along random directions isn't a useful resource.
People are drawn to local causality, but the whole point of entanglement swapping experiments is to show in the strongest of terms that position is not tenable in any form.
Regardless of the outcome of this discussion, this really hasn't been shown in a convincing manner. Some MWI and anti-realist proponents say QM has been shown to be locally causal. Not everyone agrees those proofs are convincing, but the opposite also hasn't been generally accepted either. If entanglement swapping provides a clear argument against local causality, it should be better explained.

#### DrChinese

Gold Member
1. This is also something I would like to know. At the very least, it should be possible for 1 to be entangled with 2, while (1&2) is entangled with (3&4), but that doesn't seem like this case. Maybe monogamy of entanglement is only about "known" entanglement; after all, having two qubits entangled along random directions isn't a useful resource.

2. Regardless of the outcome of this discussion, this really hasn't been shown in a convincing manner. Some MWI and anti-realist proponents say QM has been shown to be locally causal. Not everyone agrees those proofs are convincing, but the opposite also hasn't been generally accepted either.
1. No, absolutely not possible if 1 & 2 are maximally entangled (which they are when exiting the PDC crystal). You can entangle N number of quantum particles (no specific limit), but they will not be maximally entangled in that case. In fact, PDC occasionally produces 4 entangled photons - but again, no 2 are maximally entangled.

2. Post Bell, local realism has been roundly excluded along with most variations of same (depending on your particular exact definition, of course: local causality, local determinism). And yet many physicists are completely unaware of swapping, and how far the swapping experiments have come. There's no one that "needs" to be convinced, but what I am saying is a direct reflection of the mainstream. As a note: Because I am a Science Advisor, I try to label any non-standard opinion I hold as such. Of course, like vanhees71, I still think my personal opinions are correct too.

#### microsansfil

Then it won't be realized.
The document has been published at PHYSICAL REVIEW LETTERS

Here it is write :

To prove it, the researchers first used a laser to cause entanglement between a pair of photons, P1, P2. They then measured the polarization of P1, which was immediately followed by the entangling of another pair of photons, P3, P4. This was followed by measuring P2 and P3 simultaneously and causing them to become entangled with one another—a process known as projective measurement. Then, P4 was measured. Measuring P1 caused its demise of course—before P4 was born—but the measurement of P4 showed that it had become entangled with P1 nevertheless, if only for a very short period of time.
No one said anything?

/Patrick

#### vanhees71

Gold Member
I don't get this, where do I say the paper I cited is fake? I realize that English may not be your first language, but this is a bit extreme.

Post selection identifies the 4-fold events to be analyzed. What I said is that the post selection process itself does NOT create the entanglement. Zeilinger says nothing different, and if he does, feel free to point that out.

And I again call for a narrative from ANY entanglement swapping experiment that describes the swapping action as you do. Specifically: your opinion that it is local and causal, nothing happening at the BSA that causes the swap.

Entanglement swapping IS due to the Bell state cast which non-locally affects photons 1 & 4 (changing Product state stats to Entangled state stats), and the entire process lacks any semblance of a causal direction. Ordering has no bearing on the outcomes.
Maybe it's my lack of English that we are running in circles. First of all let me again stress that one cannot describe ANY entanglement-swapping experiment at once, but one has to concentrate carefully in any specific case. So let's discuss the experiment by Zeilinger again in detail, because that's what we are pondering for most of the time in this thread by now. You started with another one, where also time delays are involved, and we can analyze this too again. Also in this experiment there's nothing I can find contradicting my very standard minimal interpretation of local relativistic QFT point of view either (though I don't agree with some jargon in this paper indicating some "retrocausality argument" and at the same time stating this interpretation doesn't contradict standard relativistic QFT, which is for me a contradiction in itself).

So let's first concentrate again on the paper by Pan, Zeilinger et al, i.e.,

Since it seems not to be available from free legal sources, I'll try to summarize it completely. In quotation marks I write literal quotes from the paper, which I may also comment to make my terminology clear again.

First of all let's quote the abstract:

"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."

Maybe my English is too bad, but for me it's clearly stated that at the end of some manipulations two "particles" (though the authors use photons, but that's usual jargon and not too critical) ARE (sic!) entangled though they have never interacted nor are from a common source.

Now let's describe the experiment in mathematically clear ways (taken also from the paper). The authors create two polarization entangled photon pairs in the following state
$$|\psi_{1234} \rangle = \frac{1}{2} (|H_1 V_2 \rangle-|V_1 H_2 \rangle) \otimes (|H_3 V_4 \rangle-|V_3 H_4 \rangle).$$
Note that here the labels 1...4 can be interpreted to indicate the momentum part of the state which translates by geometry to the places where the various interactions with optical elements like beam splitters, polarizers, and photo detectors etc.

Defining the complete CONS of Bell states of photon pairs (with momenta labeled by a,b)
$$|\phi_{ab}^{\pm} \rangle = \frac{1}{\sqrt{2}} (|H_a H_b \rangle \pm |V_a V_b \rangle),\\ |\psi_{ab}^{\pm} \rangle = \frac{1}{\sqrt{2}} (|H_a V_b \rangle \pm |V_a H_b \rangle),$$
there's a typo in the paper concerning Eq. (3) which should read
$$|\psi_{1234} \rangle = \frac{1}{2} (|\psi_{14}^+\rangle \otimes \psi_{23}^+ \rangle - |\psi_{14}^- \rangle \otimes |psi_{23}^- \rangle - |\phi_{14}^+ \rangle \otimes \phi_{23}^+ \rangle + |\phi_{14}^{-}\rangle \otimes |\phi_{23}^{-} \rangle). \qquad (3^{\text{corrected}})$$
This of course doesn't change anything concerning the outcome of the measurement, i.e., projecting the pair 2&3 to the Bell state $|\psi_{23}^- \rangle$ leads necessarily to a preparation of the pair 1&4. Note that this projection only acts on photons 2&3, i.e., the corresponding projection operator on the four-photon state is described by
$$\hat{P}_{\psi_23^-}= |\psi_{23}^- \rangle \langle \psi_{23}| \otimes \hat{1}_{14},$$
i.e. NOTHING (sic!) happens to the pair 1&4.
Applying this projector to the corrected Eq. (3) above we get
$$\frac{1}{2} |\psi_{23}^- \rangle \otimes \psi_{14}^- \rangle. \qquqad (\text{Projector})$$
This implies, by the way, that only 1/4 of the four photons are left in the ensemble after this SELECTION.

To verify that pair 1&4 in this partial ensemble is indeed in the expected entangled state the authors verify by doing the coincidence measurements on polarization states of the photons in the pair 1&4 and they precisely verify the expectation.

The authors emphasize all the very points I'm making, particularly that you have to measure coincently the four photons:

Concerning the measurement of the pair 2&3 enabling the wanted projection:

"One, therefore, has to guarantee good spatial and temporal overlap at
the beam splitter and, above all, one has to erase all kinds
of path information for photon 2 and for photon 3."

Then the authors describe how they achieved this goal (here and in the following I do not explain this in detail; if necessary, I can try to do also this, but it's all standard textbook stuff concerning standard optical elements like half-wave plates, polarizers, (coincidence) photo detectors).

Concerning the measurement on the pair 1&4 for state verification after the projection

"According to the entanglement swapping scheme, upon
projection of photons 2 and 3 into the $|\psi_{23}^- \rangle$ state, photons
1 and 4 should be projected into the $|\psi_{14}^- \rangle$ state. To
verify that this entangled state is obtained, we have to
analyze the polarization correlations between photons 1
and 4 conditioned on coincidences between the detectors
of the Bell-state analyzer."

and

Thus, indeed the authors state that by this procedure of coincidence measurments, i.e., the projection of the pair 2&3 to the said Bell state necessarily leads to entanglement of the pair 1&4 in the corresponding partial ensemble (which is the case in 1/4 of the cases, neglecting real-world inaccuracies of the equipment), and I agree with them. Note that the projection acts only on the photons in the pair 2&3 and NOT on those in the pair 1&4, as shown in Eq. (Projector). On pair 1&4 nothing at all is done in the analysis, as indicated by the unit operator in the second factor of the Kronecker product in Eq. (Projector), and this is so provided the locality of interactions is as described by standard QED based on the microcausality constraint, and thus the experiment indeed precisely verifies the predictions of QED (note that in the description above the authors as well as I never have used anything contradicting standard QED). Nowhere is a causal action at a distance which would be violating the very principles of relativity and also standard QED!

#### vanhees71

Gold Member
1. No, absolutely not possible if 1 & 2 are maximally entangled (which they are when exiting the PDC crystal). You can entangle N number of quantum particles (no specific limit), but they will not be maximally entangled in that case. In fact, PDC occasionally produces 4 entangled photons - but again, no 2 are maximally entangled.

2. Post Bell, local realism has been roundly excluded along with most variations of same (depending on your particular exact definition, of course: local causality, local determinism). And yet many physicists are completely unaware of swapping, and how far the swapping experiments have come. There's no one that "needs" to be convinced, but what I am saying is a direct reflection of the mainstream. As a note: Because I am a Science Advisor, I try to label any non-standard opinion I hold as such. Of course, like vanhees71, I still think my personal opinions are correct too.
Integerstingly, here I fully agree with @DrChinese ;-)). So now again it seems we share the same (pretty conservative!) view on the meaning of the QT (and particularly relativistic QFT) formalism.

Yes, and I claim as well that my point of view is the standard one. Maybe the discrepancies between my view and @DrChinese 's is also due to the fact that we are working in different communities. I don't know on what he is working. I'm in theoretical high-energy heavy-ion research, using equilibrium and non-equilibrium relativistic QFT. Quantum optics is only my hobby interest on another fascinating subject, and for me what quantum optics has shown over the recent years is confirming more and more the standard minimal interpretation of relativistic QFT. There's no need for additional interpretational elements like MWI (which cannot be checked anyway since all the parallel universes predicted are by construction unobservable), state collapse contradicting microcausality and locality of standard relativistic QFT, Bohmian mechanics (which is not convincingly generalized to standard relativistic QFT)...

#### Mentz114

Gold Member
1. No, absolutely not possible if 1 & 2 are maximally entangled (which they are when exiting the PDC crystal). You can entangle N number of quantum particles (no specific limit), but they will not be maximally entangled in that case. In fact, PDC occasionally produces 4 entangled photons - but again, no 2 are maximally entangled.

2. Post Bell, local realism has been roundly excluded along with most variations of same (depending on your particular exact definition, of course: local causality, local determinism). And yet many physicists are completely unaware of swapping, and how far the swapping experiments have come. There's no one that "needs" to be convinced, but what I am saying is a direct reflection of the mainstream. As a note: Because I am a Science Advisor, I try to label any non-standard opinion I hold as such. Of course, like vanhees71, I still think my personal opinions are correct too.
I have to say that the discussion between yourself and prof @vanhees71 is the most informative about EPRB ever !
I just want to clear up one point about the swapping experiment. You said or showed that adding a certain time delay to a path did not affect the statistics. Is this true of any delay that does not cause disentanglement ?
It seems to me that the one explanation for the outcomes is that between preparation and disentanglement everything happens at once . At the time of preparation the settings at A and B already exist. Nothing needs to communicate and no information goes backwards in time. This reminds me of Feynmans QED description.

#### DrChinese

Gold Member
1. First of all let me again stress that one cannot describe ANY entanglement-swapping experiment at once, but one has to concentrate carefully in any specific case. So let's discuss the experiment by Zeilinger again in detail, ...

2. Concerning the measurement on the pair 1&4 for state verification after the projection

"According to the entanglement swapping scheme, upon
projection of photons 2 and 3 into the $|\psi_{23}^- \rangle$ state, photons
1 and 4 should be projected into the $|\psi_{14}^- \rangle$ state. To
verify that this entangled state is obtained, we have to
analyze the polarization correlations between photons 1
and 4 conditioned on coincidences between the detectors
of the Bell-state analyzer."

and

View attachment 246043

Thus, indeed the authors state that by this procedure of coincidence measurments, i.e., the projection of the pair 2&3 to the said Bell state necessarily leads to entanglement of the pair 1&4 in the corresponding partial ensemble (which is the case in 1/4 of the cases, neglecting real-world inaccuracies of the equipment), and I agree with them. Note that the projection acts only on the photons in the pair 2&3 and NOT on those in the pair 1&4, as shown in Eq. (Projector).

3. On pair 1&4 nothing at all is done in the analysis, as indicated by the unit operator in the second factor of the Kronecker product in Eq. (Projector), and this is so provided the locality of interactions is as described by standard QED based on the microcausality constraint, and thus the experiment indeed precisely verifies the predictions of QED (note that in the description above the authors as well as I never have used anything contradicting standard QED).

4. Nowhere is a causal action at a distance which would be violating the very principles of relativity and also standard QED!
1. It makes it difficult for myself and readers to follow a paper behind a paywall, but I see that the paper I cited did not have the equation you presented. Here's an freely available similar one from Zeilinger et al which does:

Experimental Nonlocality Proof of Quantum Teleportation and Entanglement Swapping

2. Now, you failed to mention a key point of formula (3) in this reference: This does NOT describe the state prior to photons 2 & 3 arriving together at the beamsplitter. So your "conclusion" that "nothing changes" for 1 & 4 after selection is tainted by this omission. If entanglement swapping does not happen (for example 2 & 3 are not brought together), there is no entanglement between 1 & 4 (and that has nothing to do with post selection).

3. There is no microcausality constraint. And certainly none mentioned in any entanglement swapping paper. Again, you are not quoting the authors here, and this is one of the things I object to. Please don't put your words in their mouth. What the authors say in no way supports your position.

4. And now you are mangling my position as well: I don't assert there are causes which propagate FTL, although it is possible as Bohmian Mechanics is a viable interpretation. What I am saying is that the action narrative for entanglement swapping does not follow any clear causal description, precisely because the detection ordering is not relevant, and because the detectors are spacelike separated. I don't know any more than anyone else what is occurring under the quantum covers.

------------------

I notice you keep ignoring key points I am making:
a) Entanglement monogamy prevents photon 1 from being entangled with both 2 and 4 as you claim. It's one or the other, and I will challenge you to provide a reference to the contrary. That alone makes your description not viable.
b) My counter-example F from post #57. As mentioned previously, this counter-example is not theoretical: it is done as part of the tuning process for every entanglement swapping experiment. It must be performed, as this is how the 2 & 3 streams are calibrated so the the swap is enabled. It starts off with a timing difference and no swaps. As calibration improves, swaps occur and that continues until an optimal performance level is achieved. They simply don't publish those results because it is a null result (because 1 & 4 won't be entangled because no swap occurred).

#### DrChinese

Gold Member
1. I have to say that the discussion between yourself and prof @vanhees71 is the most informative about EPRB ever !

2. I just want to clear up one point about the swapping experiment. You said or showed that adding a certain time delay to a path did not affect the statistics. Is this true of any delay that does not cause disentanglement ?

3. It seems to me that the one explanation for the outcomes is that between preparation and disentanglement everything happens at once . At the time of preparation the settings at A and B already exist. Nothing needs to communicate and no information goes backwards in time. This reminds me of Feynmans QED description.
1. Yay!

2. Adding delay to change the ordering (sequence) does not change the statistics. The following can occur in any order:

a. Detection of photon 1.
b. Detection of photon 4.
c. Projection of the photons 2 & 3 into a Bell state via co-arrival at the beam splitter.
d. Creation of photons 1 & 2 (must precede a. and c. though).
e. Creation of photons 3 & 4 (must precede b. and c. though).

3. I don't know if it makes sense to refer to "everything happening at once" in a normal temporal sense. Precisely because there is no required order other than that photons must be created before they are detected, and photons 2 & 3 must be created prior to projection.

My "narrative" to describe entanglement swapping is as follows:

When photons 1 & 2 are created, they form an entangled system "X" which grows to have spatio-temporal extent. When photons 3 & 4 are created, they too form an entangled system "Y" which grows to have spatio-temporal extent. As elements of quantum systems X and Y intersect at the beam splitter, they split into 2 new systems that are likewise entangled, but consisting of different pairing of the photons. After the *beamsplitter* portion of the BSA, ALL 4 PHOTONS ARE STILL ENTANGLED: 1 & 4, and 2 & 3. And in some experimental versions, the 2 & 3 pair is in the singlet state and therefore otherwise has the same characteristics as the 1 & 4 pair. Both sets now exhibit perfect correlations.

a. What can't be described in this narrative is the nature of how systems with spatio-temporal extent "collapse", if indeed there is something that can be called collapse. Because of Bell: this implies that "something" changes non-locally, and it certainly appears that it is NOT the revealing of quantum properties that had preexisting values. Because entangled particles lack well-defined preexisting values until observation (again per Bell, and this particular characteristic appears one way or another in all interpretations).

b. When can it be said that 1 & 4 become entangled? They need not ever have been in causal contact, don't need to exist at the same time, don't need to even exist when they became entangled. And because of entanglement monogamy, they cannot remain entangled (as they were previously) with their birth twins.

c. Returning to the OP: special relativity does NOT in any way figure in, constrain, or otherwise involve itself in the quantum description. In fact, SR can be even considered time symmetric (just to add to the confusion).

"Confused by nonlocal models and relativity"

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