# Post-Selection: Pre-existing Correlations or Action At A Distance?

• A
Gold Member

In a recent thread, I argue that Entanglement Swapping + Monogamy Of Entanglement demonstrates "quantum nonlocality" (action at a distance) and "quantum causality" (violation of strict Einsteinian causality). This has the effect of serving as a No-Go for certain QM Interpretations. Any interpretation claiming that swapping can be modeled by statistical post-selection of pre-existing ensembles is necessarily ruled out. Swapping must be an action, a physical event, a projection measurement, an operation, an objective state change, or whatever you want to call it.

Here are a series of references for this thread.

Entanglement Swapping

a. High-fidelity entanglement swapping with fully independent sources (2009)

[We will use their labeling of photons as 1, 2, 3, 4... and the abbreviation BSM as referring to the Bell State Measurement operation on [2 & 3] which leads to the entanglement swap so that [1 & 4] become entangled.]

Entanglement Swapping from Spacetime Separated Sources

b. Characterizing the nonlocal correlations of particles that never interacted (2009)
"Intuitively, it seems that such entanglement swapping experiments exhibit nonlocal effects even stronger than those of usual Bell tests. To make this intuition concrete and to fully grasp the extent of nonlocality in entanglement swapping experiments, it seems appropriate to contrast them with the predictions of local models..."

c. Experimental loophole-free violation of a Bell inequality using entangled electron spins separated by 1.3 km

[Note that entanglement swapping can occur even between combinations of electrons and photons.]

d. Entanglement Between Photons that have Never Coexisted (2012)
"In this work we demonstrate how the time at which quantum measurements are taken and their order, has no effect on the outcome of a quantum mechanical experiment, by entangling two photons that exist at separate times. ... This is a manifestation of the non-locality of quantum mechanics not only in space, but also in time. "

Monogamy Of Entanglement (MOE)

e. Description: https://www.quantiki.org/wiki/monogamy-entanglement

f. Proof: https://arxiv.org/pdf/quant-ph/0604168.pdf

[Conclusion: Given any 3 particles, 2 of which are maximally entangled, the 3rd cannot be entangled with either of the other 2.]

Cascade Entanglement Swapping (6 photons in 3 entangled pairs)

g. Multistage Entanglement Swapping (2008)
"Principle of multistage entanglement swapping: three EPR sources produce pairs of entangled photons 1-2, 3-4 and 5-6. Photon 2 from the inial state and photon 3 from the first ancillary pair are subjected to a joint BSM, and so are photon 4 from the first ancillary and photon 5 from the second acillary pair. The two BSMs project outgoing photons 1 and 6 onto an entangled state. Thus the entanglement of the initial pair is swapped to an entanglement between photons 1 and 6."

h. What is generally agreed

No one seems to be contesting the actual experimental results. No one is contesting that the predictions of QM are correct. Everyone agrees that the ordering of the component measurements can be done in any order, and the results will have no statistical change. Everyone agrees that the spacetime distance of the component measurements can be inside or outside common light cones, and the results will have no statistical change. And there is no requirement that the final entangled [1 & 4] photon pair be indistinguishable (after all their individual sources are known), they could for example have different wavelengths.

Further: we should all agree that because the ordering is irrelevant, there is no need to consider reference frames of any kind. The experiments can always be done so that choice of frame leads to identical experimental conclusions, always in agreement with standard non-relativistic QM. Also, we all agree that there is some observer somewhere who is able to see the results of all the components measurements objectively after they have been completed. This allows us to consider a hypothetical "post-selection" scenario which is part of some QM interpretations. The post-selection being the BSM. So the central question here: can a "Post-Selection = Pre-existing Correlations" (PC) Interpretation be considered viable? Or should we require all Interpretations to follow my conclusion, what might be called "Post-Selection = Action At A Distance" (AAAD). Distance in this case should be considered distance in spacetime, i.e. outside of an Einsteinian lightcone.

1. Entanglement Swapping State Change

We have previously established that the initial quantum state is a Product State of 2 entangled pairs: $$\hat{\rho}=\hat{\rho}_{12} \otimes \hat{\rho}_{34}.$$
We have also established that the final quantum state is a Product State of 2 entangled pairs: $$\hat{\rho}'=\hat{\rho}_{23} \otimes \hat{\rho}_{14}.$$
And I say it should be obvious that: ##\hat{\rho}_{12} \otimes \hat{\rho}_{34} ≠ \hat{\rho}_{23} \otimes \hat{\rho}_{14}## and also ##\hat{\rho}_{12} ≠ \hat{\rho}_{14}##

2. The State Change in PC Interpretations (Pre-existing Correlations)

If you agree with this last conclusion, then we have the following difficulties:

a) Photon [1] could at no time have been maximally entangled with photon [2] and also simultaneously maximally entangled with photon [4]. MOE forbids that.

b) Therefore, "something somewhere" occurred which "quantum causes" (for lack of a better descriptive phrase) the state change of photons [1 & 4] from their initial state to their final entangled state. This despite the fact that [1] is distant from [4] at all times, and the BSM can occur distant to both. What action "caused" the Entanglement Swap? It could *only* have been the BSM performed on photons [2 & 3].

c) The BSM can be performed distant to the [1 & 4] Bell test, leading us to only one possible conclusion: experiments violate Einsteinian locality.

d) They also violate Einsteinian causality, as the BSM can objectively occur either before or after the [1 & 4] is subjected to a Bell test. Because the BSM is a necessary requirement of the Entanglement Swap, it can be considered a "quantum cause".

The only "out" for this last conclusion is a complete rejection of the Monogamy Of Entanglement (MOE), since that is what my argument requires. Yet, MOE is a generally accepted tenet of QM, and is not rejected - as far as I know - by any interpretation. The paradox is therefore that the Pre-existing Correlation Interpretations must reject Monogamy Of Entanglement, but don't and can't.

We have seen one of the problems with reconciling the "Pre-existing Correlations" Interpretations with Monogamy Of Entanglement. Consider an experiment such as g. above where there are 3 entangled photon pairs: [1 & 2], [3 & 4] and [5 & 6]. There are now 2 BSMs (let's call them BSM_23 and BSM_45 - on pairs [2 & 3] and [4 & 5 respectively). Ultimately we perform a Bell test on [1 & 6], which should violate a Bell inequality after BSM_23 and BSM_45 successes. The polarization of photon [1] is tested before BSM_23 is performed. BSM_23 is done before BSM_45, and is also done before the [5 & 6] entangled pair is created.

So: [1] is observed - and ceases to exist - before the [5 & 6] entangled pair is even created. And obviously the swap of entanglement (BSM_45) which comes from identifying which [5 & 6] pairs have suitable statistical correlations to [1] has yet to occur. OK, maybe somehow you can convince yourself that some subensemble of [1] photons and [6] photons will violate a Bell inequality once the [6] photons have been Bell tested.

But we don't even know at this point which [5 & 6] photons are going to be swapped into entanglement with [1] (which no longer even exists). In principle, any entangled pair anywhere in the universe could end up as the [5 & 6] pair. If you had thousands of sources of entangled pairs, they must all in fact contain subensembles that are mutually entangled - where entanglement is mere "Pre-existing Correlations". That conclusion is necessary because there is no act that prevents any entangled pair anywhere in the entire history of the universe from being entangled with the [1] photon we just observed. Because the decision of what photon to swap entanglement with can be made anywhere in the universe, as we already know.

In other words: there can only be a limited number of discrete statistical states that any entangled pair can have, because every stream of entangled pairs has the potential to be swapped with any other entangled pair. This incredible discovery has yet to occur. (Or maybe it just did).
• Stream B contains some pairs that have Pre-existing Correlations with stream A.
• Stream C contains some pairs that have Pre-existing Correlations with stream A, and some with stream B.
• Stream D contains some pairs that have Pre-existing Correlations with stream A, and some with stream B, and some with stream C.
• Stream E contains some pairs that have Pre-existing Correlations with stream A, and some with stream B, and some with stream C, and some with stream D.
• And so on... for all possible entangled particles in the entire universe, without limit. All of which violate MOE.
This result is crazy. But a requirement if you believe in an Interpretation in which Post-Selection = Pre-existing Correlations.

5. The Post-Selection = Action At A Distance Interpretations

Those Interpretations that follow my Action At A Distance template are viable of course, as there are no contradictions. However, this still does not allow one to distinguish between "quantum nonlocal" Interpretations. Nor does it say what mechanism exists that supports the existing experiments as well as quantum theory.

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

I have presented experimental evidence that supports the various swapping permutations. And I have presented support and a proof for Monogamy Of Entanglement. The DrChinese paradox is that Pre-existing correlations between independently prepared entangled photon pairs is ruled out by Monogamy Of Entanglement. That's a hard No-Go for quantum interpretations that feature Pre-existing Correlations (PC) and deny Action At A Distance (AAAD). I have demonstrated objective distant change of state as a result of a BSM. If you don't like the term "action at a distance" then perhaps you might instead use the less controversial term "quantum nonlocality". Same thing.

Does anyone dispute the agreed facts presented above? And if anyone disagrees with my conclusions, can you provide a specific quote from a suitable source that refutes any specific element of my logic? [No self quotes or references to works allowed here, please!]

Cheers,

-DrC

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Nullstein

Gold Member
Consistent histories would dispute b). I.e. Nothing has to happen to photons 1 and 4.

Say the measurement happens at time ##T## by a detector with pointer states ##M_0,M_1,M_2,M_3## each corresponding to one of the possible Bell states of photons 2 and 3, and the photons are prepared in the state ##\Pi_{\psi^-_{12},\psi^-_{34}}## in accordance with the paper. We can write down histories concerning the three times ##0, T-\delta t, T##
$$\Pi_{\psi^-_{12},\psi^-_{34}}(0)\otimes\Pi_{\psi^-_{14}}(T-\delta t)\otimes\Pi_{M_0}(T)$$
$$\Pi_{\psi^-_{12},\psi^-_{34}}(0)\otimes\Pi_{\psi^+_{14}}(T-\delta t)\otimes\Pi_{M_1}(T)$$
$$\Pi_{\psi^-_{12},\psi^-_{34}}(0)\otimes\Pi_{\phi^-_{14}}(T-\delta t)\otimes\Pi_{M_2}(T)$$
$$\Pi_{\psi^-_{12},\psi^-_{34}}(0)\otimes\Pi_{\phi^+_{14}}(T-\delta t)\otimes\Pi_{M_3}(T)$$
These histories decohere, so they are reliable and resolved by the measurement at ##T##. These histories also make reference to the correlation between 1 and 4 prior to measurement, so there is no need to believe the measurement at ##T## caused the correlations. The measurement revealed them.

- Just saw this
[No self quotes or references to works allowed here, please!]
I don't know of any CH paper that deals specifically with entanglement swapping, but since the entanglement is described by the bell states and, according to CH, pure states states have corresponding projectors to subspaces for different properties of the system, this example seems to reduce to the ordinary EPR experiment as far as CH is concerned. (I.e. Asking if the correlation pre-existed in this case is like asking if the polarisation of the unmeasured photon in the original EPR experiment pre-existed)
https://arxiv.org/abs/1007.4281
https://aapt.scitation.org/doi/10.1119/1.14965

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Nullstein
We have also established that the final quantum state is a Product State of 2 entangled pairs: $$\hat{\rho}'=\hat{\rho}_{23} \otimes \hat{\rho}_{14}.$$
And I say it should be obvious that: ##\hat{\rho}_{12} \otimes \hat{\rho}_{34} ≠ \hat{\rho}_{23} \otimes \hat{\rho}_{14}## and also ##\hat{\rho}_{12} ≠ \hat{\rho}_{14}##
This is false or at best incomplete. You have given the post-measurement state of a subensemble. The state of the full ensemble after measurement is given by
##\rho_{\text{after}} = \sum_i P_i (\rho_{12}\otimes\rho_{34}) P_i##
where ##P_i## are projectors onto the Bell state basis at 2&3 (##P_i = \mathbb{1}_1\otimes \tilde P_i\otimes\mathbb{1}##). Just look up "quantum operation" in e.g. Nielsen & Chuang or see equation (8.8) in Isham's "Lectures on Quantum Theory". Your whole misunderstanding derives from confusing the full ensemble with subensembles, as has been explained earlier.

Here is the quote from Isham:

It is then easy to show that ##\rho_{14,\text{after}}=\frac 1 4 \mathbb1\otimes\mathbb1##, i.e. there is no entanglement between 1&4 in the full ensemble.

The whole point of the post-selection argument is that only the full ensemble is physically relevant. Subensembles can be post-selected almost arbitrarily. (If all outcomes occur with some non-zero probability, then it is even possible to get obtain every density matrix by post-selection.) As an example, consider a classical coinflip with probability ##\frac 1 2## for heads and tails. After 1000 coinflips, I have an ensemble of about 500 heads and 500 tails. I can put all heads in a subensemble and I would get a probability distribution of ##1## and ##0##. I can put all heads and only every second tails outcome in a subensemble. Then I would get a distribution of ##\frac 2 3## and ##\frac 1 3##. It is simple to come up with other examples. None of these post-selected distributions represent any physical property of the coin. The coin is always a fair coin, no matter what subensemble you post-select.

In the same way, I can post-select subensembles in an entanglement swapping experiment. It is of zero physical relevance whether there is entanglement between 1&4 in a subensemble, just like it is of zero physical relevance that there is a subensemble of the coinflip where all outcomes are heads. The coin is still a fair coin and the particles 1&4 are still not entangled. There is still only monogamous entanglement between the pairs 1&2 and 3&4 in the state ##\rho_{\text{after}}## of the full ensemble after measurement and no entanglement between 1&4 at all.

(As has been pointed out in the other thread, this is also the position of last years Nobel laureate Anton Zeilinger.)
a) Photon [1] could at no time have been maximally entangled with photon [2] and also simultaneously maximally entangled with photon [4]. MOE forbids that.
When you say "x is entangled with y", you always have to specify with respect to which state. 1 is not entangled with 4 with respect to the full ensemble. There is no entanglement at all between 1&4 in the full ensemble. 1 is entangled with 4 with respect to a post-selected subensemble. Nobody doubts that, but as explained above, it is of zero physical relevance. 1&4 being maximally entangled in a subensemble doesn't forbid that 1&4 are not entangled at all in the full ensemble. If you want to apply MOE, you have to talk about the same state.

Since you never say, which ensemble you are talking about, we can only guess. But it all comes down to the following possibilities:
1. If you say that the state of the full ensemble after measurement has entanglement between 1&4, then you are just plain wrong and contradicting basic undergrad QM.
2. If you say that the state of some subensemble after measurement has maximal entanglement between 1&4 and this entanglement is monogamous (with respect to the subensemble state), then you are right, but it is of no physical relevance at all.
Does anyone dispute the agreed facts presented above? And if anyone disagrees with my conclusions, can you provide a specific quote from a suitable source that refutes any specific element of my logic? [No self quotes or references to works allowed here, please!]
How is anyone supposed to provide a quote from a peer-reviewed source in order to refute a personal and unpublished theory of yourself? I suggest you publish your theory. If you somehow manage to make it through the peer review process, I'm going to publish a rebuttal. Until then, you will have to live with the reference to Nielsen&Chuang or Isham, who explain how to obtain the post-measurement state of the full ensemble.

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lodbrok, DrClaude, vanhees71 and 1 other person
Fra
NIce summary!
What action "caused" the Entanglement Swap? It could *only* have been the BSM performed on photons [2 & 3].

c) The BSM can be performed distant to the [1 & 4] Bell test, leading us to only one possible conclusion: experiments violate Einsteinian locality.
I think it clear that the BSM alone - does not cause any entanglement in any effective ensemble as the physical communication of the results from BSM is required, right? without that "information" the effective final ensemble cant be defined.

So why you feel the need for action at a distance? Do you somehow thing the classical message is not part of the physics?

/Fredrik

vanhees71
Gold Member
Consistent histories would dispute b)
Not surprising, given that CH disputes also Bell-like proofs of nonlocality. Essentially that's because CH denies the existence of a framework independent history, the existence of which Dr. Chinese (and Bell) takes for granted.

DrChinese
Gold Member
2022 Award
NIce summary!

I think it clear that the BSM alone - does not cause any entanglement in any effective ensemble as the physical communication of the results from BSM is required, right? without that "information" the effective final ensemble cant be defined.

So why you feel the need for action at a distance? Do you somehow thing the classical message is not part of the physics?

/Fredrik
Indeed, that's what I try to explain to @DrChinese for years! It's very simple: We have a very well working theory to explain all these experiments with photons or other quantum states of light, called QED, and within this theory by construction there are no actions at a distance. So at least we have one theory, i.e., local (=microcausal) QED that explains all phenomena with an astonishing accuracy, which clearly excludes the possibility of actions at a distance (i.e., causal connections between space-like separated events). So there's no need to claim actions at a distance given all hitherto performed experiments with light (or any other quantum system btw). Only when one has a clear contradiction to the predictions of QED one might be forced to give up the relativistic spacetime description with its specific causality structure but only then!

martinbn and Nullstein
Gold Member
The experiment is reproducing the peculiarities of QM vs CM in the ordinary EPR experiment. If I have an ensemble of Bertleman's socks, I can post-select for a pink sock on the left foot and know that the right foot in this subensemble will not have a pink sock. But the way QM is presented in textbooks, we are tempted to say the act of post-selection via interaction with the left foot induces the colour of the sock on the right foot.

We can post-select by interacting with 2 and 3, and infer a correlation between 1 and 4 in this subensemble. But people will argue over whether the post-selection process induces the correlation in 1 and 4, or just filters it a la classical pairs of Bertleman's socks.

 - fixed typo

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martinbn
Mentor
Here is the quote from Isham
Isham, IIRC, is using Quantum Bayesianism as his interpretation, which is basically a "knowledge" interpretation: the state vector (or density matrix) doesn't tell you anything about the quantum system, only about your knowledge of the quantum system.

Whether or not such interpretations can actually provide a viable explanation of entanglement swapping experimental results is, I think, an open question. Certainly you can't just help yourself to the claim that such an interpretation is the only viable one, which is what the position you are taking amounts to. @DrChinese is not using such an interpretation, and unless you can convince him to adopt one, I don't see how your arguments, or the ones you are referencing, would carry any weight with him, or with anyone who doesn't share your preferred interpretation.

After 1000 coinflips, I have an ensemble of about 500 heads and 500 tails. I can put all heads in a subensemble
But you're picking the subensemble of flips of coin number 1 (the only coin in this scenario) based on knowing the results of the flips of coin number 1. Of course you can get any statistics you like by cherry picking the statistics this way.

However, that's not what's being done to choose the subensemble in the entanglement swapping experiments. A better analogy would be that you're doing 1000 runs, where in each run you flip four coins; then you pick out the subensemble of runs where coins 2 and 3 come up the same, and you find that in that subensemble of runs, coins 1 and 4 also come up the same. In other words, you're finding a correlation between coins 1 and 4 in a subensemble that you picked out without even knowing what the results for coins 1 and 4 were on those runs, and based on something that shouldn't have any correlation whatsoever with coins 1 and 4.

Mentor
people will argue over whether the post-selection process induces the correlation in 1 and 4, or just filters it a la classical pairs of Bertleman's socks
No, there can't be any argument about this, because we already know from Bell's Theorem that "Bertlmann's socks" correlations cannot violate the Bell inequalities, but the correlations we observe in these experiments do. So any claimed "explanation" that makes use of classical "Bertlmann's socks" correlations cannot be correct.

DrChinese
Mentor
How is anyone supposed to provide a quote from a peer-reviewed source in order to refute a personal and unpublished theory of yourself?
This is uncalled for. @DrChinese provided references in the OP. He is describing what he takes to be established by those references. If you don't think his description of what those references say is valid, then critique his description based on the references.

DrChinese
Nullstein
Isham, IIRC, is using Quantum Bayesianism as his interpretation, which is basically a "knowledge" interpretation: the state vector (or density matrix) doesn't tell you anything about the quantum system, only about your knowledge of the quantum system.
Not at all. The book is not about any specific interpretation of quantum mechanics. The initial chapters are about the interpretation-independent math. Everything is derived from the interpretation-independent axioms of QM in their mathematical form. Only the last chapter is about interpretations and it covers all of them with no preference to any specific one. Equation (8.8) is certainly true in all interpretations. Moreover, it's not even true that Isham is a proponent of QBism. And I have also given a reference to Nielsen&Chuang.
Whether or not such interpretations can actually provide a viable explanation of entanglement swapping experimental results is, I think, an open question. Certainly you can't just help yourself to the claim that such an interpretation is the only viable one, which is what the position you are taking amounts to.
On the contrary. I openly admit that a non-local interpretation is also possible. All I'm saying is that it is not necessary. DrChinese is the one who claims that his interpreetation is the only viable one.
@DrChinese is not using such an interpretation, and unless you can convince him to adopt one, I don't see how your arguments, or the ones you are referencing, would carry any weight with him, or with anyone who doesn't share your preferred interpretation.
I don't want to convince him of any interpretation. I want to convince him that his interpretation is not the only viable one.
But you're picking the subensemble of flips of coin number 1 (the only coin in this scenario) based on knowing the results of the flips of coin number 1. Of course you can get any statistics you like by cherry picking the statistics this way.
The point of that example was to illustrate that no conclusion about the physical properties of a system can be drawn by looking at a subensemble. This is invalid reasoning in general.
However, that's not what's being done to choose the subensemble in the entanglement swapping experiments. A better analogy would be that you're doing 1000 runs, where in each run you flip four coins; then you pick out the subensemble of runs where coins 2 and 3 come up the same, and you find that in that subensemble of runs, coins 1 and 4 also come up the same. In other words, you're finding a correlation between coins 1 and 4 in a subensemble that you picked out without even knowing what the results for coins 1 and 4 were on those runs, and based on something that shouldn't have any correlation whatsoever with coins 1 and 4.
If you do something that is illegal in general, you have to argue why it would not be illegal in a specific instance. But in the instance of entanglement swapping, it not just likely invalid to conclude something about the physical system from the statistics of a subensemble. In fact, it is known to be one of the inances where it is definitely illegal to do so. The result at 2&3 is influenced by the preparation of photons 2&3, so it is a common effect. Conditioning on a common effect is well known to induce spurious correlations. This phenomenon is known as Berkson's paradox. So in the instance of entanglement swapping, it is definitely not legal to reason about the full system on the basis of the statistics of the selected subensembles.

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Nullstein
This is uncalled for. @DrChinese provided references in the OP. He is describing what he takes to be established by those references. If you don't think his description of what those references say is valid, then critique his description based on the references.
The idea that one can use monogamy of entanglement in one state (subensemble) to draw conclusions about another state (full ensemble) is certainly not supported by any reference and is his personal theory. (He even calls it the "DrChinese paradox".)

Mentor
I openly admit that a non-local interpretation is also possible.
What do you mean by a "non-local interpretation"?

The usual definition of non-locality is violation of the Bell inequalities, which is not a matter of interpretation: either experiments show them or they don't. Actual experiments do.

Mentor
The idea that one can use monogamy of entanglement in one state (subensemble) to draw conclusions about another state (full ensemble) is certainly not supported by any reference
The references @DrChinese gives certainly seem to me to be saying that the subensemble of runs in which the 2 & 3 BSM gives an "event ready" signal are relevant to monogamy of entanglement.

Your statement about ensembles seems to me to be obviously interpretation dependent, since some interpretations are not ensemble interpretations at all.

DrChinese
Mentor
Equation (8.8) is certainly true in all interpretations.
Mathematically that is the case. But we already know all interpretations agree on the math. The question is what the math "means"; that is where interpretations differ.

DrChinese
Nullstein
What do you mean by a "non-local interpretation"?

The usual definition of non-locality is violation of the Bell inequalities, which is not a matter of interpretation: either experiments show them or they don't. Actual experiments do.
Sorry, I meant to say interpretation with "spooky action at a distance" here (which is what DrChinese advocates).
The references @DrChinese gives certainly seem to me to be saying that the subensemble of runs in which the 2 & 3 BSM gives an "event ready" signal are relevant to monogamy of entanglement.
Nobody denies that there is entanglement between 1&4 in subensembles and that this entanglement is even maximal. What is being denied is that conclusions about the full ensemble can be drawn from this fact. On the contrary, it can be proven to be illegal to draw this conclusion. The math unambiguously says that there is no entanglement between 1&4 in the full ensemble.
Your statement about ensembles seems to me to be obviously interpretation dependent, since some interpretations are not ensemble interpretations at all.
Just because the word "ensemble" pops up, it doesn't mean that is related to the ensemble interpretation. All interpretations agree that the only observable facts about a quantum system are the ensemble predictions. They might provide different mechanisms for how the ensembles come about, but the ensemble predictions are interpretation independent.
Mathematically that is the case. But we already know all interpretations agree on the math. The question is what the math "means"; that is where interpretations differ.
Well, it is a mathematical fact that there is no entanglement between 1&4 in the full ensemble. Everyone has to accept this.

What is open to interpretation is the question: Why is there entanglement in some subensembles? One viable answer is spooky action at a distance. Another viable answer is that the correlation in the subensembles is spurious, introduced by the conditioning on the measurement result.

WernerQH
Why is there entanglement in some subensembles? One viable answer is spooky action at a distance. Another viable answer is that the correlation in the subensembles is spurious, introduced by the conditioning on the measurement result.
My answer would be "causation" along the backward light cones with advanced waves, as they appear for example in the transaction interpretation. I even heard about the so-called Klyshko picture:
the math and physics of two-photon correlations become so intuitive that one can essentially teach it to a dog.

Gold Member
1. Consistent histories would dispute b). I.e. Nothing has to happen to photons 1 and 4.

2. Say the measurement happens at time ##T## by a detector with pointer states ##M_0,M_1,M_2,M_3## each corresponding to one of the possible Bell states of photons 2 and 3, and the photons are prepared in the state ##\Pi_{\psi^-_{12},\psi^-_{34}}## in accordance with the paper. We can write down histories concerning the three times ##0, T-\delta t, T##
$$\Pi_{\psi^-_{12},\psi^-_{34}}(0)\otimes\Pi_{\psi^-_{14}}(T-\delta t)\otimes\Pi_{M_0}(T)$$
$$\Pi_{\psi^-_{12},\psi^-_{34}}(0)\otimes\Pi_{\psi^+_{14}}(T-\delta t)\otimes\Pi_{M_1}(T)$$
$$\Pi_{\psi^-_{12},\psi^-_{34}}(0)\otimes\Pi_{\phi^-_{14}}(T-\delta t)\otimes\Pi_{M_2}(T)$$
$$\Pi_{\psi^-_{12},\psi^-_{34}}(0)\otimes\Pi_{\phi^+_{14}}(T-\delta t)\otimes\Pi_{M_3}(T)$$
These histories decohere, so they are reliable and resolved by the measurement at ##T##. These histories also make reference to the correlation between 1 and 4 prior to measurement, so there is no need to believe the measurement at ##T## caused the correlations. The measurement revealed them.

3. I don't know of any CH paper that deals specifically with entanglement swapping, but since the entanglement is described by the bell states and, according to CH, pure states states have corresponding projectors to subspaces for different properties of the system, this example seems to reduce to the ordinary EPR experiment as far as CH is concerned. (I.e. Asking if the correlation pre-existed in this case is like asking if the polarisation of the unmeasured photon in the original EPR experiment pre-existed)
https://arxiv.org/abs/1007.4281
https://aapt.scitation.org/doi/10.1119/1.14965
Thanks @Morbert for the detail analysis!

1. Consistent Histories preserves relativistic locality. "Objectively real internal properties of an isolated individual system do not change when something is done to another non-interacting system." So in my view, this would be an interpretation I would attack.

I would refer to ##\Pi_{\psi^-_{12},\psi^-_{34}}## as a spatially extended system (which I think is similar to the approach of @vanhees71). So CH locality shields photon [1] from any change performed elsewhere unless the entire system is considered, which has nonlocal extent.

2. That nonlocal extent doesn't really pop out when you look at your 4 histories. The [1] and [4] photons do not share any local space at any time. There cannot be any Bell correlations between them initially.

Further, at some time ##T-\delta t## or ##T## (doesn't matter which) photons [2 & 3] are also out of the local space of both [1] and [4]. I guess what I am saying is that at the time and place of the BSM, according to CH nothing can change elsewhere. Yet certainly the decision to entangle (via BSM) the [1] and [4] photons - out of all the possible photons in the entire universe - must change the statistics for the Bell test on the [1 & 4] photon pairs.

Again, Monogamy Of Entanglement prohibits the kind of association you describe. Each and every [1 & 2] pair starts in the state just as you show (t=0). It is not possible for it to evolve or decohere to any [1 & 4] entangled state where [4] is distant IF you are asserting locality. How would [1] even "know" its new partner is [4], and not [2]? Because it cannot be "partners" with both [2] and [4] simultaneously.

3. Thanks for these references. I will study these papers by Griffiths so I can learn more. (I admit the logic of the CH arguments escapes me at this time.)

While looking for the reference behind a paywall, I found another exposition on CH/DH by Halliwell (it references that Griffiths paper). It says: "This is because the local conservation law (4.1 which is (∂ρ ∂t) + (∇ · j) = 0) permits QV [Quantity in a Volume of space] to change only by redistribution, which is limited by the rate at which the locally conserved quantity can flow out of the volume." [1] and [4] share a conserved observable (spin for example). But they are not local, and never have been. To satisfy the conservation rule, you must look outside the local volume.

Gold Member
1. The state of the full ensemble after measurement is given by
##\rho_{\text{after}} = \sum_i P_i (\rho_{12}\otimes\rho_{34}) P_i##
where ##P_i## are projectors onto the Bell state basis at 2&3 (##P_i = \mathbb{1}_1\otimes \tilde P_i\otimes\mathbb{1}##).

2. ...there is no entanglement between 1&4 in the full ensemble.

3. The whole point of the post-selection argument is that only the full ensemble is physically relevant. Subensembles can be post-selected almost arbitrarily. (If all outcomes occur with some non-zero probability, then it is even possible to get obtain every density matrix by post-selection.) As an example, consider a classical coinflip with probability ##\frac 1 2## for heads and tails. After 1000 coinflips, I have an ensemble of about 500 heads and 500 tails. I can put all heads in a subensemble and I would get a probability distribution of ##1## and ##0##. I can put all heads and only every second tails outcome in a subensemble. Then I would get a distribution of ##\frac 2 3## and ##\frac 1 3##. It is simple to come up with other examples. None of these post-selected distributions represent any physical property of the coin. The coin is always a fair coin, no matter what subensemble you post-select.

In the same way, I can post-select subensembles in an entanglement swapping experiment. It is of zero physical relevance whether there is entanglement between 1&4 in a subensemble, just like it is of zero physical relevance that there is a subensemble of the coinflip where all outcomes are heads. The coin is still a fair coin and the particles 1&4 are still not entangled. There is still only monogamous entanglement between the pairs 1&2 and 3&4 in the state ##\rho_{\text{after}}## of the full ensemble after measurement and no entanglement between 1&4 at all.

4. (As has been pointed out in the other thread, this is also the position of last years Nobel laureate Anton Zeilinger.)

When you say "x is entangled with y", you always have to specify with respect to which state. 1 is not entangled with 4 with respect to the full ensemble. There is no entanglement at all between 1&4 in the full ensemble.

1 is entangled with 4 with respect to a post-selected subensemble. Nobody doubts that, but as explained above, it is of zero physical relevance. 1&4 being maximally entangled in a subensemble doesn't forbid that 1&4 are not entangled at all in the full ensemble. If you want to apply MOE, you have to talk about the same state.

5. Since you never say, which ensemble you are talking about, we can only guess. But it all comes down to the following possibilities:
1. If you say that the state of the full ensemble after measurement has entanglement between 1&4, then you are just plain wrong and contradicting basic undergrad QM.
2. If you say that the state of some subensemble after measurement has maximal entanglement between 1&4 and this entanglement is monogamous (with respect to the subensemble state), then you are right, but it is of no physical relevance at all.

6. How is anyone supposed to provide a quote from a peer-reviewed source in order to refute a personal and unpublished theory of yourself?
1. The state you present as "after measurement" is incorrect as presented. You could call the [1 & 2] pairs entangled after the BSM in the subset of cases where there are no indistinguishable [3] photons present in the BSM. But where there is a successful BSM, no [1 & 2] pairs remain entangled. MOE.

2. Correcting your statement: "...there is no entanglement between any 1&4 pair in the initial full ensemble." MOE.

3. Why you bring up classical probabilities and classical subensembles, I don't understand. This is the quantum world. Bell inequality violations are not possible in a classical example, there are plenty of papers that say the same. No, you cannot pick your own subensembles at random and get the quantum statistics. Only the successful BSM pairs have that characteristic.

4. The Zeilinger team says in the reference: "A successful entanglement swapping procedure will result in photons 1 and 4 being entangled, although they never interacted with each other. This is done by performing a Bell-state measurement [BSM] on particles 2 and 3, i.e. by projecting them on one of the four Bell states. ... We confirm successful entanglement swapping by testing the entanglement of the previously uncorrelated photons 1 and 4."

I am saying the same thing.

5. I identify the ensembles clearly.
i) The subensemble of [1 & 4] pairs is entangled upon successful BSM, and only upon such. I never say otherwise.
ii) Glad you agree those same pairs are monogamously and maximally entangled. The relevance: They were previously monogamously and maximally entangled with other partners. What changed? The swap, that's what changes things.

6. My conclusion is only about certain interpretations, and this is a place to discuss same. Each individual element I have presented is standard QM, no interpretation needed. If any of those elements seem wrong to you, please point them out.

Mentor
I meant to say interpretation with "spooky action at a distance" here
Even that is still vague, but yes, the particular interpretation @DrChinese is adopting assigns a particular meaning to violations of the Bell inequalities, and experiments like entanglement swapping, that other interpretations do not.

Since this is an interpretational issue, however, please bear in mind that there is no way to resolve it, since we all agree on the experimental results. (The general guidelines for this subforum discuss this.) The best we can do is for everyone to state as clearly as possible what their preferred interpretation entails.

Gold Member
I think it clear that the BSM alone - does not cause any entanglement in any effective ensemble as the physical communication of the results from BSM is required, right? without that "information" the effective final ensemble cant be defined.

So why you feel the need for action at a distance? Do you somehow thing the classical message is not part of the physics?

/Fredrik

The swap is a nonlocal action (AAAD). Yes, you need the classical communication to put all the pieces together, But that doesn't mean the classical component is in any way involved in the quantum piece (which violates c). You could snail mail the results at one place to another, right?

The quantum nonlocality in my examples center around the BSM "quantum causing" the change in state of a remote subensemble of [1 & 4] pairs. I can't imagine how you could model it differently if all [1] and [4] photons were previously monogamously and maximally bound to other partners.

Again: certainly you agree about monogamy of entanglement, right?

Mentor
it is a mathematical fact that there is no entanglement between 1&4 in the full ensemble.
The mathematical fact is that the full ensemble has a particular density matrix. But whether that means "no entanglement between 1 & 4" depends on how you interpret the density matrix. In the interpretation @DrChinese is adopting, entanglement is not a property of an ensemble at all, it's a property of individual quantum systems. So looking at a density matrix that averages over an ensemble of systems does not and cannot tell you whether particular individual systems in individual runs are entangled.

DrChinese
Mentor
a density matrix that averages over an ensemble of systems
Note, btw, that the density matrix for the "full ensemble" is a mixture of four different subensembles, one for each of the 4 possible Bell states. As has been discussed in other threads on this topic, only one of those 4 Bell states for 2 & 3 is detected as an "event ready" signal from the BSM on 2 & 3. Each of the four subensembles has 2 & 3 entangled, and 1 & 4 entangled, but in different entangled states, and the average of all four gives the density matrix for the full ensemble. (Actually, in the real experiments only a small fraction of runs even meet the entry criteria for the BSM, so the density matrix describing the real experiment is a mixture of what I described above and a density matrix where nothing at all happens because the 2 & 3 photons don't arrive at the BSM beam splitter in a narrow enough time window.)

Gold Member
No, there can't be any argument about this, because we already know from Bell's Theorem that "Bertlmann's socks" correlations cannot violate the Bell inequalities, but the correlations we observe in these experiments do. So any claimed "explanation" that makes use of classical "Bertlmann's socks" correlations cannot be correct.
Yes, there can be arguments about this. Dropping statistical independence or employing a single framework rule let us interpret measurement as selecting a subensemble of runs with pre-existing properties, without invoking spooky action at a distance a la measurements on Bertleman's socks.

Nullstein
1. The state you present as "after measurement" is incorrect as presented. You could call the [1 & 2] pairs entangled after the BSM in the subset of cases where there are no indistinguishable [3] photons present in the BSM. But where there is a successful BSM, no [1 & 2] pairs remain entangled. MOE.
You are in contradiction with established math. The state I have given is absolutely correct. I have given you two references to standard textbooks where the formula is written the exact way I quoted where it is also explained that this is the state of the system if no selection has been performed. So you are basing your argument on an invalid premise. And you can't just shout "MOE" and think that this somehow disproves a theorem. As I explained, your understanding of MOE is fundamentally flawed, because you are confusing different ensembles with each other.
2. Correcting your statement: "...there is no entanglement between any 1&4 pair in the initial full ensemble." MOE.
No, the correct statement is that there is no entanglement in the 1&4 system in both the initial and the final state, as long as no subensemble has been selected. This is a proven mathematical and interpretation-independent fact. You are in contradiction with published standard textbook knowledge.
3. Why you bring up classical probabilities and classical subensembles, I don't understand. This is the quantum world. Bell inequality violations are not possible in a classical example, there are plenty of papers that say the same. No, you cannot pick your own subensembles at random and get the quantum statistics. Only the successful BSM pairs have that characteristic.
I bring up classical probabilities, because they make for a simple example so everyone can understand that drawing conclusions about the full ensemble from the statistics of subensembles is invalid reasoning. It doesn't suddenly become valid reasoning if we go quantum. If anything, it becomes even more invalid. Classical probability theory is contained in quantum theory as a special case after all.
4. The Zeilinger team says in the reference: "A successful entanglement swapping procedure will result in photons 1 and 4 being entangled, although they never interacted with each other. This is done by performing a Bell-state measurement [BSM] on particles 2 and 3, i.e. by projecting them on one of the four Bell states. ... We confirm successful entanglement swapping by testing the entanglement of the previously uncorrelated photons 1 and 4."

I am saying the same thing.
No you are saying that there is an action at a distance. Zeilinger is not saying anything like that and it certainly isn't implied by your quote. Moreover, in the previous thread, a quote of Zeilinger has been presented to you, where he says specifically that a knowledge based interpretation of the quantum state is viable and that no spooky action at a distance is required in such an interpretation. You are in direct contradiction with this claim.
5. I identify the ensembles clearly.
i) The subensemble of [1 & 4] pairs is entangled upon successful BSM, and only upon such. I never say otherwise.
ii) Glad you agree those same pairs are monogamously and maximally entangled. The relevance: They were previously monogamously and maximally entangled with other partners. What changed? The swap, that's what changes things.
You are comparing apples and oranges again. The system does not become entangled by the BSM. There is no entanglement between 1&4 in the full ensemble before and after measurement, so nothing did physically change about the 1&4 pair. The entanglement in the subensemble can be interpreted as correlation that has been introduced by the conditioning.
6. My conclusion is only about certain interpretations, and this is a place to discuss same. Each individual element I have presented is standard QM, no interpretation needed. If any of those elements seem wrong to you, please point them out.
No, you are in stark contradiction with standard QM as I have shown above. The state of the full ensemble after measurement is given by the formula in the textbooks that I have quoted. You apparently deny this. Moreover, you claim that a knowledge based interpretation is not viable, which is a general statement about all interpretations. As I explained, your claim is wrong and it is also in contradiction with the opinion of the recent Nobel laureate Anton Zeilinger.

Nullstein
Since this is an interpretational issue, however, please bear in mind that there is no way to resolve it, since we all agree on the experimental results. (The general guidelines for this subforum discuss this.) The best we can do is for everyone to state as clearly as possible what their preferred interpretation entails.
Well, I'm not claiming that a certain interpretation is impossible. I agree that this cannot be resolved. I explain that DrChinese is wrong when he claims that a certain interpretation is not viable. This is certainly resolvable.

The mathematical fact is that the full ensemble has a particular density matrix. But whether that means "no entanglement between 1 & 4" depends on how you interpret the density matrix.
Entanglement has a mathematical definition that can be checked. The full ensemble of the 1&4 system after measurement is in a product state, so it is by definition not entangled.

Gold Member
1. Well, it is a mathematical fact that there is no entanglement between 1&4 in the full ensemble. Everyone has to accept this.

2. What is open to interpretation is the question: Why is there entanglement in some subensembles? One viable answer is spooky action at a distance. Another viable answer is that the correlation in the subensembles is spurious, introduced by the conditioning on the measurement result.
1. We agree.

2. In an "event ready" variation of this experiment, somewhat similar as presented in my reference c, you can have a situation such as this (starting with the usual [1 & 2] and [3 & 4] entangled pairs):

a) Ilya receives a [2] photon and a [3] photon and performs a BSM. When a successful Psi+ BSM occurs, he sends a classical signal to Jean, who is distant.

b) Jean receives a [1] photon and a [4] photon and stores them without observing their polarization. This can occur before or after a), but the storage of these photons occurs outside the light cone of Ilya's BSM.

The BSM cannot, according to the "Pre-existing Correlations" viewpoint, change the state of either [1] photon or the [4] photon that Jean is storing. No Bell measurement is performed yet on [1] and [4].

c) Jean freely and randomly chooses a common polarization setting to measure the [1] and [4] photons.

d) Jean gets the signal from Ilya saying that the swap has been performed, indicating the stored [1] photon and [4] photon are currently entangled. Jean retrieves the [1] photon and [4] photon, and measures them at the chosen angle.

Each and every single pair will be perfectly polarization correlated for any choice of angle setting. There are no spurious correlations. You could also say there are no subensembles and no statistical considerations.
The cause is simple, a successful BSM occurred and led Ilya to press the "event ready" button (which sent the signal to Jean). The BSM remotely changes the [4] photon from being entangled with [3], to being entangled with [1]. That change is physical, and occurs faster than c.

Gold Member
1. ...where the formula is written the exact way I quoted where it is also explained that this is the state of the system if no selection has been performed. ... No, the correct statement is that there is no entanglement in the 1&4 system in both the initial and the final state, as long as no subensemble has been selected. This is a proven mathematical and interpretation-independent fact. You are in contradiction with published standard textbook knowledge.

2. Zeilinger is not saying anything like that and it certainly isn't implied by your quote. Moreover, in the previous thread, a quote of Zeilinger has been presented to you, where he says specifically that a knowledge based interpretation of the quantum state is viable and that no spooky action at a distance is required in such an interpretation. You are in direct contradiction with this claim.

3. The system does not become entangled by the BSM. There is no entanglement between 1&4 in the full ensemble before and after measurement, so nothing did physically change about the 1&4 pair. The entanglement in the subensemble can be interpreted as correlation that has been introduced by the conditioning.

1. I am not disputing this, never have. If there are no BSMs, then there are no swaps. So there is no [1 & 4] entanglement. None. Not in any subensemble, not in any single pair. Zero.

2. Nobel prize winning Zeilinger might be in contradiction with Nobel prize winning Zeilinger, I can't say. I am simply quoting from his published experimental work with other team members. I'd say that's a fair quote though, coming within a referenced paper that this thread is based on. He has plenty of other papers in his incredible body of work saying virtually the same. So it was a surprise to me when someone said he followed an information-type interpretation. Hey, from time to time, Bell hinted that he was a Bohmian. So I guess these greats also have the same troubles with interpretations as us mortals.

3. Go back to your 1. where we agree there is no entanglement if no BSM. Iff there is a BSM, then there is [1 & 4] entanglement. And it occurs remotely. Cause and effect, but without consideration of direction in time.

Gold Member
Further, at some time ##T-\delta t## or ##T## (doesn't matter which) photons [2 & 3] are also out of the local space of both [1] and [4]. I guess what I am saying is that at the time and place of the BSM, according to CH nothing can change elsewhere. Yet certainly the decision to entangle (via BSM) the [1] and [4] photons - out of all the possible photons in the entire universe - must change the statistics for the Bell test on the [1 & 4] photon pairs.
The bit in bold is the crux of the matter. CH says the BSM resolves one of the four histories I presented, and this resolution gives us information about what to expect with future measurements. If we perform a different measurement, or even no measurement, we cannot construct the subensembles we are interested in. I.e. We cannot identify runs where a particular Bell property like ##\psi^-_{14}## obtain. Consider this analogy:

Say you have two friends, Alice and Bob, with spacelike-separated feet. Alice is standing on Bob's left, so that Alice's right foot is next to Bob's left foot, but Alice's left foot is far away from Bob's right foot. Each friend is wearing a pair of odd-coloured socks, but you don't know any other information. So learning the colour of Alice's left sock will not tell you anything about Bob's right sock.

Now say I perform a measurement of Alice's right sock and Bob's left sock (i.e. the socks that are close together), and I tell you one of two possible outcomes: "The socks are the same colour" and "The socks are not the same colour". Once you learn the outcome of this measurement, you suddenly know the relation between Alice's left sock and Bob's right sock. Now, looking at the colour of Alice's left sock will tell you the colour of Bob's right sock, even though there has been no interaction (say Alice and Bob are lightyears tall).

In summary: The original ensemble exhibits no correlation between the distant socks, but the measurement has partitioned the ensemble into subensembles each exhibiting a correlation between non-interacting, spacelike separated socks.

If the socks are quantum variables, this simple picture breaks down because of noncommuting variables, but CH lets you recover this simple reasoning so long as we operate within a framework of histories that decohere. This is the elegance of CH.

Again, Monogamy Of Entanglement prohibits the kind of association you describe. Each and every [1 & 2] pair starts in the state just as you show (t=0). It is not possible for it to evolve or decohere to any [1 & 4] entangled state where [4] is distant IF you are asserting locality. How would [1] even "know" its new partner is [4], and not [2]? Because it cannot be "partners" with both [2] and [4] simultaneously.
Staying with the CH formalism, what MoE forbids are histories containing operators like ##\Pi_{\psi^-_{12},\psi^-_{14}}## because these are not projectors to any Hilbert subspace, and so are not physical. But it does not forbid ##\Pi_{\psi^-_{12}}## and ##\Pi_{\psi^-_{14}}## in the same history so long as i) they are asserted at different times and ii) the histories decohere.

Nullstein
1. We agree.
Apparently we don't. In your previous post you say the opposite and your current post is also in contradiction with that. You claim that the full ensemble after measurement is given by ##\rho_{23}\otimes\rho_{14}##. This is provably wrong.
2. In an "event ready" variation of this experiment, somewhat similar as presented in my reference c, you can have a situation such as this (starting with the usual [1 & 2] and [3 & 4] entangled pairs):

a) Ilya receives a [2] photon and a [3] photon and performs a BSM. When a successful Psi+ BSM occurs, he sends a classical signal to Jean, who is distant.

b) Jean receives a [1] photon and a [4] photon and stores them without observing their polarization. This can occur before or after a), but the storage of these photons occurs outside the light cone of Ilya's BSM.

The BSM cannot, according to the "Pre-existing Correlations" viewpoint, change the state of either [1] photon or the [4] photon that Jean is storing. No Bell measurement is performed yet on [1] and [4].

c) Jean freely and randomly chooses a common polarization setting to measure the [1] and [4] photons.

d) Jean gets the signal from Ilya saying that the swap has been performed, indicating the stored [1] photon and [4] photon are currently entangled. Jean retrieves the [1] photon and [4] photon, and measures them at the chosen angle.

Each and every single pair will be perfectly polarization correlated for any choice of angle setting. There are no spurious correlations. You could also say there are no subensembles and no statistical considerations.
The cause is simple, a successful BSM occurred and led Ilya to press the "event ready" button (which sent the signal to Jean). The BSM remotely changes the [4] photon from being entangled with [3], to being entangled with [1]. That change is physical, and occurs faster than c.
Your reference c is a standard entanglement swapping experiment where subensembles are taken based on the BSM measurement. In entanglement swapping, entanglement appears always only in subensembles. So the argument that no conclusion about the full ensemble based on statistics of subensembles can be drawn, still holds.

Nullstein
1. I am not disputing this, never have. If there are no BSMs, then there are no swaps. So there is no [1 & 4] entanglement. None. Not in any subensemble, not in any single pair. Zero.
There are BSM's though. The state of the full ensemble after the BSM is given by equation (8.8) in Isham. Do you agree or not?
2. Nobel prize winning Zeilinger might be in contradiction with Nobel prize winning Zeilinger, I can't say. I am simply quoting from his published experimental work with other team members. I'd say that's a fair quote though, coming within a referenced paper that this thread is based on. He has plenty of other papers in his incredible body of work saying virtually the same. So it was a surprise to me when someone said he followed an information-type interpretation. Hey, from time to time, Bell hinted that he was a Bohmian. So I guess these greats also have the same troubles with interpretations as us mortals.
Well, your quote just doesn't imply what you believe it implies. Zeilinger makes a much weaker statement than you do, so he is not contradicting himself.
3. Go back to your 1. where we agree there is no entanglement if no BSM. Iff there is a BSM, then there is [1 & 4] entanglement. And it occurs remotely. Cause and effect, but without consideration of direction in time.
In 1., I explained that there is no entanglement between 1&4 in the full ensemble even after the BSM. This is a mathematical theorem and it seems like you are contradicting it. It is actually hard to tell what your claim is, since sometimes you claim to agree but then again you say something completely in disagreement with it.

Mentor
Entanglement has a mathematical definition that can be checked.
Yes, but which state you apply that mathematical definition to is interpretation dependent.

On a non-ensemble interpretation, where the 4-photon system for each individual run is described by its own state, the 1 & 4 photons are entangled on the individual runs, by the same mathematical definition you are using. You are dismissing that because those individual runs are part of a "subensemble", but that just emphasizes the fact that you are using an ensemble interpretation where @DrChinese is not.

Mentor
spooky action at a distance a la measurements on Bertleman's socks.
Measurements on Bertlmann's socks cannot show any spooky action at a distance, because such measurements cannot violate the Bell inequalities.

I haven't read the papers you referenced yet, so I don't know whether they use the Bertlmann's socks metaphor or not; but you should not be using it in this thread since it is irrelevant to what we are discussing.

DrChinese