A Is MWI Considered Local in Quantum Mechanics?

[PeterDonis]

I found a better reference

Thanks, I'll take a look.

 

[PeterDonis]

your idea that indistinguishability can be reversed

I didn't say indistinguishability could (possibly--I said I wasn't sure) be reversed. I said the swap could (possibly) be reversed.

 

[PeterDonis]

The dip in coincidences of course is due to the photons emerging from a single port, rather than 2

The "dip" is for the case where the photons arrive at exactly the same time at the swap beam splitter. And if they do so, as you say, they both come out the same output port. But if they do that, they can only swap into a limited number of the possible Bell states. In particular, they can't swap into the singlet state, which is the one in which we have one photon coming out each output port of the swap beam splitter. But that is the case that I am interested in doing an MZI analysis for. It is also the only case for which the "event ready" signal occurs in at least one of the entanglement swapping experiments you have referenced in the various threads we have had on that topic.

In other words, unless I'm missing something, the "dip" point, where the photons arrive at the swap beam splitter at exactly the same time, cannot be the only case where a swap occurs, and therefore cannot be the only case in which there is indistinguishability. There must be a finite window of time within which, if both photons arrive at the swap beam splitter, a swap can occur.

 

[DrChinese]

The "dip" is for the case where the photons arrive at exactly the same time at the swap beam splitter. And if they do so, as you say, they both come out the same output port. But if they do that, they can only swap into a limited number of the possible Bell states. In particular, they can't swap into the singlet state, which is the one in which we have one photon coming out each output port of the swap beam splitter. But that is the case that I am interested in doing an MZI analysis for. It is also the only case for which the "event ready" signal occurs in at least one of the entanglement swapping experiments you have referenced in the various threads we have had on that topic.

In other words, unless I'm missing something, the "dip" point, where the photons arrive at the swap beam splitter at exactly the same time, cannot be the only case where a swap occurs, and therefore cannot be the only case in which there is indistinguishability. There must be a finite window of time within which, if both photons arrive at the swap beam splitter, a swap can occur.

I. Mmmm, let's make sure we are saying the same thing. There are 2 completely different Bell states being applied in the experiment:

i) The Bell state for entangled photons 1 & 2, and the Bell state for entangled photons 3 & 4. These are normally chosen to the be same state. That is of course a function of the source, which in this case is PDC. And you are quite correct, for this experiment, the source pairs are created in the singlet state. That being the |ψ−> Bell state. The choice of this for the source Bell state is mostly a matter of convenience. I think we agree here just fine.

ii) The Bell state for entangled photons 1 & 4 applies of course only if a swap occurs. That can be any one of 4 possible Bell states |ψ+>, |ψ−>, |φ+>, |φ->. However, only 2 of the 4 can be detected/differentiated by detector clicks at the BSM (for photons 2 & 3 arriving nearly coincident). The 2 that can be distinguished are |ψ+> and |ψ−>. These 2 allow for 4 fold coincidences. The other 2 Bell states (|φ+>, |φ->) are ignored because they only yield 3 fold coincidences (since Photons 2 & 3 end up in the same detector and only appear as a single click). Only 4 fold coincidences are considered, because the specific detectors that go "click" differentiate between |ψ+> and |ψ−> for Photons 2 & 3. These always occur with one H> click and one V> click. I think we agree here as well.

If those 2 clicks occur in the same output port of the beam splitter (BS), then the resulting Bell state is |ψ+>. If those 2 clicks occur in different output ports of the beam splitter (BS), then the resulting Bell state is |ψ->. At this point, it doesn't matter whether the Bell state chosen for i) is the singlet state or not. What matters is that the 2 identified states are |ψ+> or |ψ−>.

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II. I am sure you are aware of all this, so let's re-examine the HOM dip. This is telling us that the Bell state is not being discriminated, and the singlet output state experiences destructive interference within the beam splitter during an H-O-M test. That is exactly as you said above. You are wanting to present the analysis of the swapping experiment for just that case I guess, although I don't see why. Everything works exactly the same under every interpretation whenever there are 4 fold coincidences. Indistinguishability is a requirement regardless.

The fact that the arrival times for the H-O-M effect to appear is quite narrow means these two photons ARE interacting inside the BS. There is no way to reverse destructive interference. There is no way to distinguish the output photons once indistinguishable - even if you later attempt to check their polarization. Because if you knew (or could know) their polarization going into the BS, they wouldn't be indistinguishable, right?

So let's say we look at the singlet |ψ−> pairs coming out of your MZI. You are basically saying that they are orthogonally polarized (that's the hallmark, right)? That case can be identified, as easily as |ψ+>, which is also orthogonally polarized. So I say everything that applies to one case applies to the other.

I think what you are alluding to that the coincidence window for H-O-M effect (1-10 ps) is much smaller than the window for the swapping experiments (1-10 ns). I'm not sure I understand why those scales are so different, but I would guess it is related to the photon sources being independent and phase linked in some manner.

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III. We still have the following in the swapping scenario for MWI:

a) Some window during which the 2 & 3 photons must overlap at the BS, and be swapped into 1 of 4 possible states. That is not reversible. Coincident with the 4 possible Bell states, there are 2 permutations for each (reflected/transmitted) . There must be MWI branching into 8 worlds here.
b) The subsequent distance to the PBS can be any length. However, in principle a polarization measurement is reversible. Ignoring that, each of the above 8 worlds can be expressed 2 ways in polarization terms. That's 16 worlds by my count.
c) The final distance to the detectors can be any distance as well. I don't see the detectors as being the spot where the branching must occur. Again, that ordering can be arbitrary.

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IV. So I think we end up with a lot of issues for MWI to explain: when/where exactly is branching to occur per MWI? To be fair, QM is not exactly overly specific either - it just says look at the entire context. But QM does not specify observables have specific values at all times, which MWI apparently does make that claim. So when and where do those values change (discontinuously) if not at branching?

 

[PeterDonis]

You are wanting to present the analysis of the swapping experiment for just that case I guess, although I don't see why.

To simplify the analysis, and because, as I mentioned before in at least one entanglement swapping experiment you linked to, that one case was the only one that caused an "event ready" signal to be generated. Adding in the other possible Bell states that the BSM can project into doesn't change anything essential, it just adds more terms to the equations.

I think what you are alluding to that the coincidence window for H-O-M effect (1-10 ps) is much smaller than the window for the swapping experiments (1-10 ns).

Yes, that was my understanding; the HOM "dip" time window is much shorter than the time window required for a swap to occur.

when/where exactly is branching to occur per MWI?

I've already answered this: wherever decoherence occurs. In the scenario under discussion, that happens whenever a photon detector registers a photon, or whenever a time window for a photon to reach a detector expires and the detector does not register a photon (which means the photon escaped into the environment and is lost). So what I was calling "swap" vs. "no swap" is just distinguishing between "one photon is detected in each output arm of the BSM beam splitter" and all the other possible detection/no detection outcomes. Again, that was for the reasons given above.

QM does not specify observables have specific values at all times, which MWI apparently does make that claim.

No, it doesn't. In the MWI, the wave function is all there is. Observables only have specific values in particular branches when the appropriate entanglements are present due to previous unitary interactions (for example, between a photon and a detector).

 

[DrChinese]

1. ...as I mentioned before in at least one entanglement swapping experiment you linked to, that one case was the only one that caused an "event ready" signal to be generated. ...

2. Yes, that was my understanding; the HOM "dip" time window is much shorter than the time window required for a swap to occur.

3. I've already answered this: wherever decoherence occurs. In the scenario under discussion, that happens whenever a photon detector registers a photon, or whenever a time window for a photon to reach a detector expires and the detector does not register a photon (which means the photon escaped into the environment and is lost). So what I was calling "swap" vs. "no swap" is just distinguishing between "one photon is detected in each output arm of the BSM beam splitter" and all the other possible detection/no detection outcomes. Again, that was for the reasons given above.

1. That was in fact the case in some swapping experiments, where a single Bell state is easier to discern than two. The Zeilinger reference did 2 of the 4.

2. I will keep looking for a reference which can suitably explain this. As best I can tell, it is at the beam splitter of the BSM where the tight timing is required for indistinguishability - which certainly makes sense.

3. Just a minor point here: Yes, photons do get lost. But at the BSM, when there is only 1 detector clicking, that almost always means that photons 2 & 3 went to the same detector (same arm, same polarization). Nothing was actually lost. Current detector technology cannot discriminate between 1 photon arriving at a detector versus 2 photons arriving within a few hundred femtoseconds.

But I don't really think what you call "decoherence" matches the MWI concept of branching. Unfortunately, I cannot find enough agreement between sources on MWI to really nail this down one way or the other. I.e when and where does a swap occur?
a. In orthodox QM: the rule is to look to the *entire* context for an answer - and timing/ordering of individual components doesn't matter - ditto for precise location of actions. I might say, for example, that the swap occurs within the Beam Splitter at the point the 2 & 3 photons are irreversibly indistinguishable. However: at that point, which specific Bell State (1 of the 4 possibilities) itself has not yet been determined - and it can't be until the irreversible clicks occur after those photons pass the reversible polarizers. That explanation might be correct, but it is no more specified in standard QM than saying no swap at all occurs at all until the 4 detectors click within the time window (exactly as you say).
b. I don't really think the same can be said for MWI. If MWI is to be meaningful, there should be more detail on precisely when/where branching occurs.PS I think we've arrived at a point where we are close enough in agreement that further effort from you might not be fruitful. Thanks for all your efforts. Of course, I always welcome more discussion from you or anyone.

 

[PeterDonis]

I don't really think what you call "decoherence" matches the MWI concept of branching.

Decoherence is not identical with the MWI concept of branching. It is just a prerequisite for the MWI concept of branching. But branching itself is a separate concept from decoherence. The best simple way I know to describe the connection between the two is that decoherence explains why the branches can't interfere with each other, which is the key fact that allows the MWI to treat each branch as a separate, independent "world".

If MWI is to be meaningful, there should be more detail on precisely when/where branching occurs.

I don't see why. Branching in the MWI is just as "fuzzy" in time as the corresponding concepts are in other interpretations (like "when does collapse occur" in a collapse interpretation, or "when does the swap occur" in your description). The "fuzziness" here is partly due to limitations in our ability to measure and record what happens on very short time scales. But it's also partly due to the inherent arbitrariness in the concepts; we are trying to draw bright lines when the actual physics, as far as we can tell, is continuous. Asking exactly when branching occurs (or collapse, or the swap, etc.) is like asking exactly where on the isthmus of Panama the boundary is between North and South America. Any answer is going to be arbitrary: in the actual geography of the Earth, independent of human ideas, there is no sharp boundary, just a continuous strip of land. Similarly, in the actual physics of the quantum events we are discussing, independent of human ideas, there is no sharp boundary between "not yet branched" and "branched" (or between "not yet collapsed" and "collapsed", or between "not yet swapped" and "swapped"), just a continuous process--at least as far as we can tell with our current knowledge. Maybe at some point we will do experiments on short enough time scales that will explicitly show a sharp boundary--and then we will have to update our theories accordingly.

 

[jbergman]

But in most other interpretations, there also exists something which lives in normal 3+1 dimensional space. If you deny the existence of that 3+1 dimensional space, or at least the existence of anything living in that space, then the locality notion of that 3+1 dimensional space no longer applies to you.

I disagree. For the purposes of this discussion, locality should be framed in terms of the definitions associated Bell's papers. As such, all theories of QM are non-local. We can just consider the unobservable features of MWI as another form of hidden variables.

 

[jbergman]

You say that MWI is alocal because the wave function lives in a higher dimensional abstract space. But that is not specific to this interpretation, it is true for all of them. This just introduces a new word for something different!

Agree, IMO, these are just another form of hidden variables.

 

[PeterDonis]

IMO, these are just another form of hidden variables.

Not really. "Hidden variables", in Bell's formulation, means variables in addition to the ones that appear in standard QM. The wave function appears in standard QM; it's not a hidden variable. And in the MWI, the wave function is literally the only thing there is. So the MWI is not a hidden variable interpretation. It's just a "take the wave function literally in all respects, no matter how extreme and outlandish it turns out to be" interpretation.

 

[gentzen]

I disagree. For the purposes of this discussion, locality should be framed in terms of the definitions associated Bell's papers. As such, all theories of QM are non-local. We can just consider the unobservable features of MWI as another form of hidden variables.

Wait, do you disagree with me, or with Demystifier? I just explained why Demystifier's argument makes sense as an argument, like martinbn interpreted correctly:

Are you saying that @Demystifier says that in MWI the 4 dimensional space-time doesnt exist or nothing exists in it? My understanding of MWI is that there is no such claim, of course i might just not know enough about MWI.

Demystifier's original comment was:

In my view, MWI is neither local nor non-local. It is alocal. See https://arxiv.org/abs/1703.08341

My personal opinion of MWI is that just like Copenhagen, it is not a single interpretation, but different proponents mean quite different interpretations when they say MWI. And those different interpretations contradict each other when you start to dig into details.

 

[Morbert]

For the purposes of this discussion, locality should be framed in terms of the definitions associated Bell's papers. As such, all theories of QM are non-local.

This would leave interesting questions on the table. E.g. Brown and Timpson argue that the non-separability of the wavefunction frees MWI proponents from relying on Bell's local causality as an account of locality, without giving up the idea that correlations in quantum experiments have explanation.

 

[PeterDonis]

Brown and Timpson argue that the non-separability of the wavefunction frees MWI proponents from relying on Bell's local causality as an account of locality, without giving up the idea that correlations in quantum experiments have explanation.

This paper seems questionable to me, at least as far as the MWI is concerned. I find this statement on p. 2:

There is a consistent Lorentz covariant model of quantum phenomena which violates local causality but is local in Bell’s 1964 sense: the Everett picture.

AFAIK there is no accepted relativistic formulation of the MWI (i.e., one that uses quantum field theory instead of non-relativistic QM), so the "Lorentz covariant" claim here is simply false. (It also seems odd on its face, since in a QFT context "local causality" means "operators at spacelike separated events commute", which is true--so Lorentz covariant QFT does not violate "local causality", yet the claim above implies that it does.)

Also, it's not clear to me how even the non-relativistic MWI is "local in Bell's 1964 sense", since that would mean Bell's factorizability condition was met: but the factorizability condition is a condition on the observed correlations, and any QM interpretation violates that condition, since QM's experimental predictions do.

 

[gentzen]

This paper seems questionable to me, at least as far as the MWI is concerned.

Indeed, this paper also seems questionable to me, at least as far as its references to Tim Maudlin are concerned:

For MWI, there are nearly as many schools as there are variants of Copenhagen, and some of those schools are somewhat problematic in their behavior and claims:

 

[jbergman]

Not really. "Hidden variables", in Bell's formulation, means variables in addition to the ones that appear in standard QM. The wave function appears in standard QM; it's not a hidden variable. And in the MWI, the wave function is literally the only thing there is. So the MWI is not a hidden variable interpretation. It's just a "take the wave function literally in all respects, no matter how extreme and outlandish it turns out to be" interpretation.

I think that is too narrow of an interpretation. IMO, the wave function should be considered hidden, hence, it's state is a hidden variable.

 

[PeterDonis]

IMO, the wave function should be considered hidden, hence, it's state is a hidden variable.

On this view, every QM interpretation is a hidden variable interpretation, since QM itself, independent of any interpretation, is a hidden variable theory. Which makes the term "hidden variable" useless, since the whole point of the term was to distinguish between QM interpretations.

 

[Morbert]

AFAIK there is no accepted relativistic formulation of the MWI (i.e., one that uses quantum field theory instead of non-relativistic QM), so the "Lorentz covariant" claim here is simply false.

MWI proponents seem to assume an Everettian interpretation is extendable to relativistic theories. (See e.g. Rubin). Do you have reverences discussing difficulties with extending MWI to relativistic theories?

(It also seems odd on its face, since in a QFT context "local causality" means "operators at spacelike separated events commute", which is true--so Lorentz covariant QFT does not violate "local causality", yet the claim above implies that it does.)

Bell's local causality is different from local commutativity. (See Bell's "La nouvelle cuisine")

 

[PeterDonis]

MWI proponents seem to assume an Everettian interpretation is extendable to relativistic theories. (See e.g. Rubin). Do you have reverences discussing difficulties with extending MWI to relativistic theories?

Wrong question. The question is, do the MWI proponents who "seem to assume" that the MWI is extendable to relativistic theories, have references that actually do that? As far as I can tell, the answer to that is "no". The Rubin paper you reference doesn't give any such extension; it uses nonrelativistic quantum field theory for its computations.

(The Rubin paper is also too quick to dismiss Bell-type nonlocality for the MWI. Bell's original formulation made assumptions about single outcomes, but there are later formulations that don't--they are formulated purely in terms of observed correlations, which applies to the MWI just as much as any other interpretation. In the MWI they become observed correlations in particular branches of the wave function, but they're still there and they still count as "nonlocality" since they violate the relevant inequalities.)

Bell's local causality is different from local commutativity. (See Bell's "La nouvelle cuisine")

Yes, Bell's "local causality" is not the same as relativistic QFT's "local causality". But to just blithely say that relativistic QFT "violates local causality" without even mentioning the different usage of that term in the QFT community vs. the QM interpretation community, does not seem to me to be justified.

 

[Morbert]

Wrong question. The question is, do the MWI proponents who "seem to assume" that the MWI is extendable to relativistic theories, have references that actually do that? As far as I can tell, the answer to that is "no". The Rubin paper you reference doesn't give any such extension; it uses nonrelativistic quantum field theory for its computations.

From the paper:

Indeed, there is a simple line of argument which leads to the conclusion that Everett-interpretation Heisenberg-picture quantum field theory must be local. The dynamical variables of the theory are field operators defined at each point in space, whose dynamical evolution is described by local (Lorentz-invariant, in the relativistic case) differential equations.

I am asking you to clarify your position: Are you saying the relativistic case cannot in fact be constructed, based on some fundamental objection or non-generalizeable character of Everettian interpretations, or are you simply saying you have not seen the relativistic case in literature?

Yes, Bell's "local causality" is not the same as relativistic QFT's "local causality". But to just blithely say that relativistic QFT "violates local causality" without even mentioning the different usage of that term in the QFT community vs. the QM interpretation community, does not seem to me to be justified.

The context is made explicit in the introduction.

 

[PeterDonis]

Are you saying the relativistic case cannot in fact be constructed, based on some fundamental objection or non-generalizeable character of Everettian interpretations, or are you simply saying you have not seen the relativistic case in literature?

The latter.

 

[PeterDonis]

The context is made explicit in the introduction.

I understand perfectly well what the authors are using the term "local causality" to mean. I just think they are being either extraordinarily ignorant or disingenuous by ignoring the other usage of that term in the relativistic QFT community while at the same time making a claim about Lorentz covariant quantum models.

 

[jbergman]

On this view, every QM interpretation is a hidden variable interpretation, since QM itself, independent of any interpretation, is a hidden variable theory. Which makes the term "hidden variable" useless, since the whole point of the term was to distinguish between QM interpretations.

I don't agree with this framing of Bell's work. Bell takes the observables associated with QM as being true. He then shows that no local hidden variable theory can reproduce these observables. These hidden variable theories have no dependence to be augmentations of the standard formalism of QM.

 

[PeterDonis]

I don't agree with this framing of Bell's work.

What you quoted from me has nothing whatever to do with Bell's work. It has to do with your claim about what a "hidden variable theory" is. If you think Bell's definition of what a "hidden variable theory" is was the same as yours, I challenge you to give an explicit reference from his work that supports such a claim. For example, a reference which says that Bell thought the MWI was a hidden variable theory.

(Bell certainly thought the de Broglie-Bohm theory was a hidden variable theory, in fact it was his favorite example of one--a nonlocal one--but dBB is not the MWI. In dBB the hidden variables are the unknown and unknowable particle positions, not the wave function, and AFAIK that was exactly how Bell viewed it.)

Bell takes the observables associated with QM as being true. He then shows that no local hidden variable theory can reproduce these observables. These hidden variable theories have no dependence to be augmentations of the standard formalism of QM.

Nonsense. Bell's formulation of "hidden variable theories" modeled the hidden variables as $\lambda$--which are separate from the wave function, the measurement settings, and the observed results. In other words, his hidden variables precisely are "augmentations of the standard formalism of QM".

 

[jbergman]

What you quoted from me has nothing whatever to do with Bell's work. It has to do with your claim about what a "hidden variable theory" is. If you think Bell's definition of what a "hidden variable theory" is was the same as yours, I challenge you to give an explicit reference from his work that supports such a claim. For example, a reference which says that Bell thought the MWI was a hidden variable theory.

(Bell certainly thought the de Broglie-Bohm theory was a hidden variable theory, in fact it was his favorite example of one--a nonlocal one--but dBB is not the MWI. In dBB the hidden variables are the unknown and unknowable particle positions, not the wave function, and AFAIK that was exactly how Bell viewed it.)Nonsense. Bell's formulation of "hidden variable theories" modeled the hidden variables as $\lambda$--which are separate from the wave function, the measurement settings, and the observed results. In other words, his hidden variables precisely are "augmentations of the standard formalism of QM".

Bell himself states very clearly in, The theory of Local Beables, "Quantum Mechanics is not Locally Causal".

 

[PeterDonis]

Bell himself states very clearly in, The theory of Local Beables, "Quantum Mechanics is not Locally Causal".

Sure. What does that have to do with your claim that the wave function is a hidden variable?

 

[jbergman]

Sure. What does that have to do with your claim that the wave function is a hidden variable?

Here is the comment I made that started this discussion,

I disagree. For the purposes of this discussion, locality should be framed in terms of the definitions associated Bell's papers. As such, all theories of QM are non-local. We can just consider the unobservable features of MWI as another form of hidden variables.

It is quite clear that Bell wouldn't consider MWI as a locally causal theory as he states that QM isn't locally causal.

So really the dispute boils down to whether or not you consider the QM wave function hidden or not. This question isn't of great import, though, to the main point, which is that QM is not embeddable in a local causal theory.

However, I stand by my characterization of the quantum state vector as unobservable and "hidden".

However, I acknowledge that this not the typical usage in this context.

 

[PeterDonis]

Here is the comment I made that started this discussion

That post was in response to @gentzen, not to me. Your post in response to me was:

I think that is too narrow of an interpretation. IMO, the wave function should be considered hidden, hence, it's state is a hidden variable.

And I disagreed with that:

On this view, every QM interpretation is a hidden variable interpretation, since QM itself, independent of any interpretation, is a hidden variable theory. Which makes the term "hidden variable" useless, since the whole point of the term was to distinguish between QM interpretations.

And then you went off on a tangent about Bell's work. Your claim that the wave function should count as a "hidden variable" has nothing to do with Bell's work.

So really the dispute boils down to whether or not you consider the QM wave function hidden or not.

Yes, and I said no, for the reason I gave in the post of mine that I quoted above. If you want to respond to that argument, by all means do so. But the argument I made has nothing to do with Bell's work. It's a simple argument about whether you want the term "hidden variable" to be useful in distinguishing between QM interpretations, or not.

This question isn't of great import, though, to the main point, which is that QM is not embeddable in a local causal theory.

I have not disagreed with that at all, for the definition of "local causal" that you are using (in which a "local causal" theory would not be able to violate the Bell inequalities). As I commented in response to @Morbert, though, I think one needs to be clear that this definition of "local causality" is not the same as the one that is used by the relativistic QFT community; to that community, relativistic QFT is "locally causal", because spacelike separated operators commute.

I stand by my characterization of the quantum state vector as unobservable and "hidden".

It's not a direct observable, no, although quantum tomography can in principle pin it down to any desired degree of accuracy given enough repetitions of a particular preparation procedure.

"Hidden" in the sense of "hidden variable interpretation" would, as I said, make that term useless, since by this definition every QM interpretation is a "hidden variable interpretation" since QM itself is a "hidden variable theory". If you want to use such a definition, I can't stop you, but I don't see the point.

 

[kered rettop]

They're "not an issue" only because the MWI does not contain anything corresponding to what they describe. There are no "mutual influences" or "remote changes" in the MWI. That's because there aren't any in the wave function, and in the MWI, the wave function is all there is.

None of this means the MWI does not have to account for the experimental results. Of course it does, just as any QM interpretation does. It just doesn't do it by appealing to "mutual influences" or "remote changes". It does it by, first, saying that the wave function is all there is; second, saying that there is no collapse, so all of the possibilities contained in the wave function actually exist (meaning that measurements don't have single results--all possible results happen); and third, saying that the wave function is what enforces the correlations such as are observed in Bell inequality violations, entanglement swapping, etc.

Is it not true that the unentangled state is a superposition of Bell states and that entanglement swapping doesn't so much impose entanglement as separate the Bell states (using the tags obtained from the idler photons at D1 and D4)?

 

[kered rettop]

"Hidden" in the sense of "hidden variable interpretation" would, as I said, make that term useless, since by this definition every QM interpretation is a "hidden variable interpretation" since QM itself is a "hidden variable theory". If you want to use such a definition, I can't stop you, but I don't see the point.

I thought "hidden variables" meant "hidden variables added to standard QM". So I'm thinking that you don't need to play devil's advocate, amusing though it may be for us spectators. The definition is simply wrong.

 

[PeterDonis]

Is it not true that the unentangled state is a superposition of Bell states

I don't know what you mean by "the unentangled state". The initially prepared state has photons 1 and 2 entangled, and photons 3 and 4 entangled. After the entanglement swap, photons 1 & 4 are entangled and photons 2 & 3 are entangled. All of the entangled states involved are Bell states (in the simplest version, which is the one I analyzed in detail with math, they are all singlet states). The full 4-photon states are products of two entangled Bell states, not superpositions.

the idler photons

There are no "idler photons" in the experiments discussed.