A Hidden Assumptions in Bell's Theorem?

[Simple question]

But why do that when the particles in ch1 and ch4 are not correlated originally?

Can you explain what correlation you are talking about ? What is the quantum property that two unrelated particles are supposed to share/be-correlated before measurement ? Are you invoking super determinism, and that any random pair of particle in the universe are entangled (because of the BB), you just have to look at them the right way ?

Because they want to identify the subset of particles in ch1 that are correlated with co-propagating particles in ch4. It follows that if a particle in ch4 is correlated with a particle in ch1, then its entangled sibling in ch3 will be correlated with the sibling of the other particle in ch2.

Therefore by simply doing a BSM measurement between particles in ch1 & ch4, you can detect the subset of particles in ch1 & ch4 that are correlated and use this information to post-select subsets of ch2 and ch3 which would show the same correlation.

Indeed. And I haven't seen anybody describe this otherwise.

This is the basis of entanglement swapping.

Again, true. Some unrelated series of particle pairs that had no business to be untangled in the first place can be "assigned" an entanglement, by a process taking place in a completely disjoint (space-like) region.
I think we call it "swapped" because the other pair entanglement is destroyed (or the particle themselves) to preserve monogamy.

When you describe it as though an experiment was done on exactly 4 particles, two of which interacted with each other and the other two which never interacted gained entanglement, it is wrong and very misleading.

I don't think Dr Chinese did that... at all. There a many (but finite) pairs of pair that are processed.

What I fail to understand is that it is obvious that 100% of swapping can never been achieved, but I haven't found the lower bound in the paper, nor how to compute it.

 

[DrChinese]

Coincidence counting is a heralding mechanism. Particle pairs which are matched by coincidence counting are entangled in this case. A BSM does not transfer anything between particle pairs, it simply heralds that their properties are correlated.

In entanglement-swapping experiments, you have a stream of entangled pairs [1 & 2] and another stream of entangled particles [3 & 4]. You use the interaction between the pairs from the [2 & 3] to select subsets of [1 & 4] that would be correlated. The entanglement is transferred by the post-selection of the [1&4] stream using information from the [2 & 3] interaction and the reason it works is that 1 is already correlated with 2 and 3 with 4.

From: https://www.nature.com/articles/srep09333 (NOTE: in this paper, the BSM is done on 1 & 4 instead of 2 & 3)Note that they are detecting an entangled state between ch1 & ch4. But why do that when the particles in ch1 and ch4 are not correlated originally? Because they want to identify the subset of particles in ch1 that are correlated with co-propagating particles in ch4. It follows that if a particle in ch4 is correlated with a particle in ch1, then its entangled sibling in ch3 will be correlated with the sibling of the other particle in ch2.

Therefore by simply doing a BSM measurement between particles in ch1 & ch4, you can detect the subset of particles in ch1 & ch4 that are correlated and use this information to post-select subsets of ch2 and ch3 which would show the same correlation.

This is the basis of entanglement swapping.

When you describe it as though an experiment was done on exactly 4 particles, two of which interacted with each other and the other two which never interacted gained entanglement, it is wrong and very misleading.

There are no experimental references, and certainly not the one you presented, which treat swapping as post-selection rather than a quantum operation. From your reference (which is just another confirming swapping experiment with different labeling, no better or worse than those already cited, so I don't know why you are distracting us with its inclusion):

"The detection of an entangled state in ch1 and ch4 heralds the existence of entanglement in ch2 and ch3, which originally have no correlation." (All: note that in this reference, channels labeled [2 & 3] perform the same role as [1 & 4] in the other references. These are the photons that a Bell test are performed on.)

As I get tired of saying: the monogamy of entanglement (you can start another thread if you doubt this and want to debate it, but it is orthodox QM) prevents the kind of correlations you think exist between photons prior to a swap. There is no such thing, you have made this up to preserve your opinion. Swapping is a quantum operation, and if you can find an experiment that says otherwise, please... quote. And save us all some time by a verbatim quote, not a vague/useless reference requiring reading of an entire paper.

In conclusion: Swapping is an experiment done on [sets of] exactly 4 particles, two of which interacted with each other and the other [distant] two which never interacted [but still] gained [maximal] entanglement. This is orthodox science, and sorry: it is you who is misleading. I guess you are free to imagine your own interpretation, but please, even here is not the place to debate such a point. Even my esteemed colleague @vanhees71 - who views the situation quite differently than I - recognizes the need for a successful Bell State Measurement as a requirement for the swap.

 

[DrChinese]

What I fail to understand is that it is obvious that 100% of swapping can never been achieved, but I haven't found the lower bound in the paper, nor how to compute it.

Good question.

The rate of successful swaps compared to all entangled pairs generated is indeed very low. From source A, the rate might be 1 in 100 million; and the same in source B. So roughly, there might be 1 swap out of (100 million)^2 pairs. You might get 10 per seconds, to 1 in 10 minutes; obviously this varies widely based on laser strength, etc.

However: these experiments set a narrow time window for the successful Bell State Measurement (BSM). Any and all events that match the stated criteria are used. There is no scientific issue with this; as when the BSM is successful, there will be maximum entanglement for the other distant photons which have never been in a common light cone.

If a sufficient number of the included pairs are *not* actually entangled, then the CHSH S value might drop below 2. That doesn't happen, as is shown in many papers. Vice versa: if some of the entangled pairs are excluded, then the S value won't be high enough above 2 to violate a Bell inequality in a convincing manner. Either way, you can't "accidentally" violate a Bell inequality in this kind of test (assuming a reasonable sample size).

 

[PeterDonis]

the entanglement of (14) for the subensemble which you get by projecting on a Bell state of (23) is already present in this entanglement of the original pairs.

This doesn't make sense. The original pairs are (12) and (34). Each of those pairs are entangled, but the overall 4-photon state is a product state of the two pairs. So where in the original state is there any entanglement between (14)? There can't be. And if there isn't any entanglement between (14) in the original state--which there can't be--how can any entanglement between (14) in the final results be explained using the original state? It's no answer to say "subensemble" because there aren't even any subensembles of the original state that have entanglement between (14). You can't "project out" subensembles that don't exist in the first place.

 

[PeterDonis]

conditioning on a common effect can make correlations appear that are not at all induced by a cause-and-effect relationship. This phenomenon is commonly known as Berkson's paradox.

But here we're talking about perfect correlation or anti-correlation. Berkson's paradox can't account for that.

To take Berkson's original example as it is described in the Wikipedia page on Berkson's paradox [1], if we have two diseases that are uncorrelated in the general population, they can appear to be negatively correlated in a particular subpopulation, say hospitalized patients. So, for example, if Alice is in the hospital for disease A, she is less likely to have disease B than a member of the general population, and if Bob is in the hospital for disease B, he is less likely to have disease A than a member of the general population. So if we just sample Alices and Bobs from hospitals, we might be led to believe that disease A has some negative causal impact on the chance of getting disease B, and vice versa, when actually it doesn't if we look at the whole population. And we could similarly find subpopulations that showed a spurious positive correlation between diseases A and B.

But now, to paraphrase @DrChinese's description of what is going on in entanglement swapping experiments: suppose Alice and Charlie are "prepared" so that they both have disease A and not B, and Bob and Donna are "prepared" so that they both have disease B and not A. Alice and Bob each go off on their own and never meet each other. But Charlie and Donna meet and decide to get a disease test, and they find that they now both have disease B and not A; and Alice and Bob each decide at some point to get a disease test, and both find that they have disease A and not B. There's no way to account for that using Berkson's paradox. There's no way to somehow "pick subensembles" to accomplish it, because there are no subensembles of the starting ensemble in which Bob has disease A and not B, and Charlie has disease B and not A.

[1] https://en.wikipedia.org/wiki/Berkson's_paradox

 

[Nullstein]

I am tired of quoting Zeilinger, Weinberg, and authors of well known experiments who all say the same thing in one way or another: ...the non-locality of quantum mechanics, as manifested by entanglement, does not apply only to particles with spatial separation, but also with temporal separation." This is the standard viewpoint of the scientists designing and performing the experiments, in complete opposition to your viewpoint.

I am also tired of you quoting them as if it would support your point. I am in full agreement with the quoted sentence and in general in full agreement with the cited scientists. It is you, who is in disagreement with them, because you use your personal definition of "non-locality". The quoted scientist use the term "non-locality" to refer to the presence of Bell-violating correlations, nothing more. You use it in a much stronger way, implying that Bell-violating correlations somehow require the existence of non-local cause-and-effect relationships. This is not true. Some interpretations explain the correlations that way, but it is not a necessary consequence. Moreover, respected scientist usually want nothing to do with commitments to particular interpretations, because there is nothing to be gained from it. It's is not experimentally testable anyway and typically just hurts ones reputation.

There aren't any suitable papers on swapping where they say anything like you do: "of course, quantum teleportation across time and space always respects c". By definition (since it is called teleportation), it never respects c.

I don't say anything like this, you just made it up. I'm saying there is no evidence for a non-local cause-and-effect relationship. Correlation doesn't imply causation.

I remain confident no quote will be forthcoming.

I have already given you citations in earlier threads, but you didn't bother to read or understand them anyway. The argument I'm making is a standard argument in causal inference and is explained in all gory detail e.g. in "Causality" by Pearl.

1. Correlation may not always assure us there is causation... but that is exactly what violations of Bell's Inequality (by photons 1 & 4) tells us! That's the whole point!! The "cause" of such a violation - keep in mind it is not a classical cause, but one that follows quantum mechanical rules (which transcend the usual spacetime restrictions) - is the overall context.

No that's not what Bell violation tell us. Bell violations tell us that the correlations can't be explained by a classical common cause in the past. But this is not the only possible local explanation, especially in entanglement swapping, where a perfectly reasonable explanation is possible.

And the "cause" of the entanglement swap (again, not a classical cause as time order is NOT a factor) is the Bell State Measurement (BSM) on photons 2 & 3.

That is your personal interpretation that is not advocated by most physicists. If you disagree, you could just provide a citation by a respectable scientist making a commitment to such a causal interpretation. Please note: The appearance of the term "non-local" doesn't indicate such a commitment.

In QM, a complete measurement context involves elements that defy normal past-to-future ordering (classical = cause must precede effect), and defy restrictions imposed by light cones (locality=respects c). There aren't any generally accepted papers being written by the community that say otherwise.

There aren't any generally accepted papers being written by the community that make a commitment to either way, because it is interpretation dependent and untestable.

2. You need to read what you wrote again. You say the post-selection on 2 & 3 places distant 1 & 4 into an entangled state, which would be an example of spooky action at a distance if true.

No, that's not what I'm saying. I'm saying that the subensemble is described by a state that doesn't factorize and is thus entangled by definition. I'm not saying that the post-selection actively "places" anything into an entangled state.

And then you say a common cause is excluded by Bell. I quite agree! What you have actually done is demonstrate that without the 2 & 3 swap, 1 & 4 would not be entangled.

No, I haven't. The Bell state measurement does not affect the 1&4 pair. The full ensemble of 1&4 is not entangled. The subensembles are entangled, but it is a well understood statistical fallacy to conclude that there is a cause-and-effect relationship because of the conditioning on an effect.

That is correct sir!! In the quantum world, the swap "causes" the entanglement (where "cause" means cause in the quantum sense, which is not classical).

Again, there is no evidence for this. This is completely interpretation dependent and there is currently no known way to test it even in principle.

We all know the swap is a condition for 1 & 4 entanglement.

That's not even true. There are entangled subensembles even if no operation is conducted at 2&3. This is a mathematical necessity, because the full ensemble of the 1&4 system is not altered by the BSM.

You say: "Therefore, the measurement result is a common effect." Hand-waving at its best, sorry, but this is not a valid deduction.

There is no hand waving here, its really trivial that the initial preparation of the state influences the result of the measurement, there can hardly be a debate about this.

3. Berkson's paradox is a red herring. We are talking about actual experiments in which there are perfect correlations between photons 1 & 4, which have never interacted - and at the same time violate Bell inequalities. No classical example will match this scenario. And folks who quote this paradox are grasping at straws.

It's completely irrelevant that the correlations are perfect. Berkson's paradox arises whenever there is conditioning on a common effect. There is no further qualification.

And just to address your ridiculous assertion that the full 1 & 4 ensemble contains a uniform distribution ("...a uniform distrubution can always be decomposed into entangled subensembles...") which includes pairs that were entangled without the performance of a swap: Because 1 & 2 are maximally entangled, monogamy of entanglement prevents 1 & 4 from also being maximally entangled. You basically made this argument up on your own, and I thought we had previously dispelled you from this viewpoint in this forum - even considering the latitude allowed here.

There can not be a debate about whether the full ensemble of the 1&4 subsystem is uniformly distributed. This is a mathematical fact and if you cannot calculate it yourself, you have to do your homework first.
$\mathrm{Tr}_{23}(\rho_{12}\otimes\rho_{34}) = \mathrm{Tr}_{2}(\rho_{12})\otimes\mathrm{Tr}_{3}(\rho_{34}) = \frac{\mathbb 1_2}{2}\otimes\frac{\mathbb 1_2}{2}$
If you disagree, then please point out which equality sign you disagree with.

 

[Nullstein]

@Nullstein, "No, I don't argue that the measurement didn't do anything physical. I argue that it does something physical only to the 2&3 photon pair, but does nothing physical to the 1&4 photon pair".

It seems to me that you do. I am quite sure you don't question that entanglement between two particles IS physical AND change "density matrix of all P1&P2" of such ensembles of monogamously entangled pairs (compared to "classical" entanglement)

I don't know your definition of "physical". The correlations can of course be found in the lab, if that's what you mean by it. I don't understand the rest of the sentence.

Now are you saying that the full ensemble of pair 1&4 have the same "density matrix" of that of non-entangled pair ?
If yes, it means that what(or when)ever happens at 2&3, one would never be able to successfully entangled 100% (or really even more than the classical mechanic limit) of pairs, because it would select the whole (or above classical/ detectable=>FLT communication) sub-ensemble, an this is a contradiction.

The full ensemble of the 1&4 pair has the same density matrix before and after the Bell state measurement at 2&3. This follows from a very simple standard calculation. Yes, it is true that entanglement swapping can not be used to entangle 100% of the events. No, this is not in contradiction to anything.

 

[PeterDonis]

@Nullstein, as has been commented in previous threads, you are saying entanglement is a property of ensembles, not of individual pairs of photons. But that means your arguments can only possibly be valid if one adopts an ensemble interpretation of QM. @DrChinese, as far as I can tell, is not using an ensemble interpretation, so one would not expect your arguments to carry any weight with him even if we assume for the sake of argument that they would carry weight with someone who did adopt an ensemble interpretation.

 

[Nullstein]

This doesn't make sense. The original pairs are (12) and (34). Each of those pairs are entangled, but the overall 4-photon state is a product state of the two pairs. So where in the original state is there any entanglement between (14)? There can't be. And if there isn't any entanglement between (14) in the original state--which there can't be--how can any entanglement between (14) in the final results be explained using the original state? It's no answer to say "subensemble" because there aren't even any subensembles of the original state that have entanglement between (14). You can't "project out" subensembles that don't exist in the first place.

There is no entanglement between 1&4 in the full ensemble, not even after the BSM, so if there are entangled subensembles after the BSM, they must have been there before the BSM. It's very easy to perform the decomposition. If you have an ensemble of coinflips, then 50% of them will be heads and 50% will be tails. You can just sort them into two subensembles that are no longer uniformly distributed. Sorting the uniformly distributed ensemble of the 1&4 system into buckets that match the statistics of the entangled Bell states is only insignificantly harder.

 

[PeterDonis]

The argument I'm making is a standard argument in causal inference and is explained in all gory detail e.g. in "Causality" by Pearl.

Are there any references in the causal inference literature that specifically discuss the case of quantum measurements on entangled particles that violate the Bell inequalities?

 

[Nullstein]

@Nullstein, as has been commented in previous threads, you are saying entanglement is a property of ensembles, not of individual pairs of photons. But that means your arguments can only possibly be valid if one adopts an ensemble interpretation of QM.

But that doesn't follow. The statistical predictions of QM are the same in every interpretation of QM and all predictions of QM are statistical. (One may argue about things like selection rules, but that's not important here.) It may or may not be true that individual pairs of photons are entangled (whatever that means in practice), but it doesn't matter, since it would have no observable consequences other than statistical. Lab experiments only collect statistics and compare it to the statistical predictions of QM. I'm making statistical assertions out of the necessity that only statistical predictions are experimentally accessible and not because I'm advocate of the ensemble interpretation. (In fact, I'm not. I am in the shut up and calculate camp.)

 

[Nullstein]

Are there any references in the causal inference literature that specifically discuss the case of quantum measurements on entangled particles that violate the Bell inequalities?

Yes, Pearl himself has several publications on that and e.g. Spekkens and co-workers publish on the topic quite regularly.

 

[PeterDonis]

It may or may not be true that individual pairs of photons are entangled (whatever that means in practice), but it doesn't matter, since it would have no observable consequences other than statistical.

No, not all observable consequences are "only statistical". In the case of the entanglements observed in these experiments, as I have already mentioned, we are looking at perfect correlation or anti-correlation of each individual pair. And an obvious further improvement in all such experimental protocols is to find more ways to set up scenarios in which the outcomes are like this: all or nothing instead of a statistical test.

(Roger Penrose, in one of his "Mind" books, can't remember which, proposed a scenario involving three particles in which, instead of just having statistical violations of Bell inequalities as we do in normal EPR experiments, you can, if QM is correct, observe outcomes that are literally impossible on any model that satisfies the assumptions of Bell's theorem. I don't know if that scenario ever got published as an actual paper, but IIRC similar suggestions have appeared in the literature for more definitive tests that aren't open to statistical loopholes. I don't see any reason why scenarios of that kind could not be adapted to show swapping of such outcomes.)

 

[PeterDonis]

Yes, Pearl himself has several publications on that and e.g. Spekkens and co-workers publish on the topic quite regularly.

If you have any specific references I would be interested in seeing them.

 

[Nullstein]

No, not all observable consequences are "only statistical". In the case of the entanglements observed in these experiments, as I have already mentioned, we are looking at perfect correlation or anti-correlation of each individual pair. And an obvious further improvement in all such experimental protocols is to find more ways to set up scenarios in which the outcomes are like this: all or nothing instead of a statistical test.

Yes, but already the word "correlation" implies that we are talking about statistical predictions. In order to observe correlations, you need to perform lots of measurements and calculate the covariances. A single measurement doesn't suffice to prove perfect correlation. The prediction "perfect correlation" is statistical in nature.

(Roger Penrose, in one of his "Mind" books, can't remember which, proposed a scenario involving three particles in which, instead of just having statistical violations of Bell inequalities as we do in normal EPR experiments, you can, if QM is correct, observe outcomes that are literally impossible on any model that satisfies the assumptions of Bell's theorem. I don't know if that scenario ever got published as an actual paper, but IIRC similar suggestions have appeared in the literature for more definitive tests that aren't open to statistical loopholes. I don't see any reason why scenarios of that kind could not be adapted to show swapping of such outcomes.)

Sounds like the GHZ experiment. But also here, one needs many runs to test the predictions. Among the axioms of QM, the only one that makes quantitative predictions about measurements in the lab, is the Born rule, which is probabilistic. Sometimes the probability distribution is sharply peaked and the Born rule predictions certainty. But it is still a probability distribution, so the prediction can only be tested by collecting statistics.

 

[PeterDonis]

A single measurement doesn't suffice to prove perfect correlation.

If you don't like my usage of "perfect correlation" I can use a different term. But the kind of result I am talking about is not statistical: it is a result that is literally impossible for any model that satisfies the assumptions of Bell's theorem. Such a result can be observed in a single run of an experiment.

Sounds like the GHZ experiment.

That's similar but not quite the same as the proposal of Penrose that I was referring to. I'll have to see if I can find a more specific reference for the latter.

 

[PeterDonis]

Sounds like the GHZ experiment. But also here, one needs many runs to test the predictions.

Can't one observe (++-) instead of (+++) for the three photon polarization measurement outcomes as a result in a single run?

 

[Demystifier]

I argue that it does something physical only to the 2&3 photon pair,

OK, what physical does it do to them?

 

[vanhees71]

This doesn't make sense. The original pairs are (12) and (34). Each of those pairs are entangled, but the overall 4-photon state is a product state of the two pairs. So where in the original state is there any entanglement between (14)? There can't be. And if there isn't any entanglement between (14) in the original state--which there can't be--how can any entanglement between (14) in the final results be explained using the original state? It's no answer to say "subensemble" because there aren't even any subensembles of the original state that have entanglement between (14). You can't "project out" subensembles that don't exist in the first place.

Yes, the total ensemble is in a pure state with the state ket of the form
$$|\Psi_{12} \rangle \otimes |\Psi_{34} \rangle,$$
where $|\Psi_{12} \rangle$ and $|\Psi_{34} \rangle$ are Bell states by preparation.

Now projecting the two photons (23) to another Bell state, you get a subensemble, where (14) are entangled. That's the entire point of "entanglement swapping". The subensemble is selected by the projection measurement to a Bell state of the two photons (23)! Of course you are right in saying that this subensemble must indeed be prepared by doing this projective measurement.

 

[vanhees71]

@Nullstein, as has been commented in previous threads, you are saying entanglement is a property of ensembles, not of individual pairs of photons. But that means your arguments can only possibly be valid if one adopts an ensemble interpretation of QM. @DrChinese, as far as I can tell, is not using an ensemble interpretation, so one would not expect your arguments to carry any weight with him even if we assume for the sake of argument that they would carry weight with someone who did adopt an ensemble interpretation.

No matter which interpretation you choose, the statistical properties of the measurements discussed here, are uniquely described by Q(F)T, and in practice you can test statistical inferences only by preparing an ensemble and doing these measurements. In this sense the probabilistic predictions of QT (or any other statistical description) refer to ensembles, no matter which other sophisticated metaphysical interpretation you follow.

 

[vanhees71]

There is no entanglement between 1&4 in the full ensemble, not even after the BSM, so if there are entangled subensembles after the BSM, they must have been there before the BSM. It's very easy to perform the decomposition. If you have an ensemble of coinflips, then 50% of them will be heads and 50% will be tails. You can just sort them into two subensembles that are no longer uniformly distributed. Sorting the uniformly distributed ensemble of the 1&4 system into buckets that match the statistics of the entangled Bell states is only insignificantly harder.

It's even shown in the very short and to-the-point PRL by Zeilinger et al:

https://web.physics.ucsb.edu/~quopt/swap.pdf

It's a nice exercise with manipulations in the bra-ket formulation, suitable for a problem in the QM1 lecture!

 

[martinbn]

@DrChinese Let me ask you some direct questions.

1. In which sense do you use the term non-locality? Is it meant as violations of Bell's inequality or something else. If it is only in the sense of violations of Bell's inequalities, then why do you need to talk about entanglement swapping? The violations can be demonstrated without swapping. If it is something else, can you clarify what it is? And do you claim that your references (Zeilinger, Weinberg and so on) use it also that way?2. Consider a pair of two systems A and B in an entangled state. Then there are two standard facts, that have been mentioned and can be found in the literature. The density matrix of B has all the information about the statistics of the possible outcomes of measurements on B. Any measurement on A doesn't change the density matrix of B. The question is: do you dispute any of this? If yes, can you explain why. If not, then why do you say that the statement "Measurement on A does have an effect on B. Or the cause of a result at B is the measurement on A." is interpretation independent? Actually you say it is a proven fact.

 

[Simple question]

The quoted scientist use the term "non-locality" to refer to the presence of Bell-violating correlations, nothing more.

As does DrChinese. All this is a-causal, only you are invoking "causation" and using prefered "interpretation". As shown in your next sentence:

You use it in a much stronger way, implying that Bell-violating correlations somehow require the existence of non-local cause-and-effect relationships.

Strikeout mine. This is your mistake only.

I'm saying there is no evidence for a non-local cause-and-effect relationship. Correlation doesn't imply causation.

Correct. But if we stick to experimental facts, the SMB at 2&3 is "spooklily" sniffing those correlation about a big chunk of a space-like region (1&4 detection's events). That is why anyone can use "non-local" in a perfectly simple and SR compliant way.

The Bell state measurement does not affect the 1&4 pair.

That's your way to "interpret" things. I think everybody is silent on this, because there is simply no standard understanding on how nature does this (an stunningly, in a-causal way). Nor does exist any theory that can explain this using locality, as proved by Bells (if those assumptions are right)

The full ensemble of 1&4 is not entangled.

Correct. So why are they running those experiments ?

The subensembles are entangled, but it is a well understood statistical fallacy to conclude that there is a cause-and-effect relationship because of the conditioning on an effect.

And there is another fallacy that pretend there is a theory that can locally pick-up a a sub-ensemble. Because one can also play that game with pairs of socks, and I don't think quantum theory applies to socks.

Again, there is no evidence for this. This is completely interpretation dependent and there is currently no known way to test it even in principle.

This is wrong. In principle you can use entanglement swapping, and realize it by experiment. And if you choose to interpret it as 2&3 does nothing., you are simply wrong.
2&3 pick up a correct sub-ensemble about 1&4, and in a plain an simple non-local way.

So if you are happy with the "shut up and calculate", then compute "non-entangled".

Meanwhile people happy with quantum mechanics and various mirror lazer and polarizer, will build such intricate (and delicate) appliance, to protect 1&4 to be tempered with.
This means something physically happening at 2&3 will add to something physically happening at 1&4 (both space-like separated, mind you)
All this in the lab, not in "interpretation space"

 

[vanhees71]

@DrChinese Let me ask you some direct questions.

1. In which sense do you use the term non-locality? Is it meant as violations of Bell's inequality or something else. If it is only in the sense of violations of Bell's inequalities, then why do you need to talk about entanglement swapping? The violations can be demonstrated without swapping. If it is something else, can you clarify what it is? And do you claim that your references (Zeilinger, Weinberg and so on) use it also that way?

Let's look at Weinberg. In his textbook on quantum mechanics (2nd edition) he defines

As you see it's a very weak form of "locality", but it's in accord with locality as understood in the connection of relativistic QFT. Everything else were very schizophrenic, because Weinberg is the one who used microcausality for decades to establish the foundations of relativistic QFT very clearly.

Unfortunately he doesn't discuss Bell's theorem in connection with relativistic QFT in this book, and I'm not aware of any place, where he does this. If somebody has a reference by him about this, I'd be very interested in it!

2. Consider a pair of two systems A and B in an entangled state. Then there are two standard facts, that have been mentioned and can be found in the literature. The density matrix of B has all the information about the statistics of the possible outcomes of measurements on B. Any measurement on A doesn't change the density matrix of B. The question is: do you dispute any of this? If yes, can you explain why. If not, then why do you say that the statement "Measurement on A does have an effect on B. Or the cause of a result at B is the measurement on A." is interpretation independent? Actually you say it is a proven fact.

That's also not right. It depends of course on what you mean by "measurement on A". Of course, it's clear if you just prepare A and B in an entangled state and just do measurements on B, no matter what's done with A the measurement outcomes (i.e., the usual probabilistic properties of such measurements in the sense of QT) are given by the reduced density matrix, $\hat{\rho}_{B}=\mathrm{Tr}_A \hat{\rho}_{AB}$.

If, however you use experiments on A, e.g., to project out subensembles of (AB), you get a different statistical operator for B. Say you project due to $|\psi \rangle \in \mathcal{H}_A$ you get
$$\hat{\rho}_{B|A \in \psi}=\frac{\mathrm{Tr}_A (|\psi \rangle \langle|\psi| \otimes \hat{1}_{B} \hat{\rho}_{AB})}{\mathrm{Tr}_{AB} (|\psi \rangle \langle|\psi| \otimes \hat{1}_{B} \hat{\rho}_{AB})}.$$
Say $\hat{\rho}_{AB}$ is a Bell state of two photons' polarization like the singlet
$$\hat{\rho}_{AB} = |\Psi \rangle \langle \Psi|=\frac{1}{\sqrt{2}} (|HV \rangle-|VH \rangle)$$
and $|\psi \rangle=|H \rangle$, then
$$\hat{\rho}_B=\frac{1}{2} \hat{1}$$
but with $\psi=|H \rangle$ (i.e., you consider only (AB) pairs, for which you measure A to be H-polarized you get
$$\hat{\rho}_{B|A \in \Psi}=|V\rangle \langle V|.$$
That's one of the amazing features of entangled states: On the one hand if (AB) is prepared in a (pure) Bell state, the properties of B are maximally uncertain, i.e., in our example B is an ideally unpolarized photon, but still there are 100% correlations between certain properties of A and B. In our example, if you select A's with a polarization in a given direction then B's polarization concerning the same direction is the opposite, i.e., if you select A's which are H-polarized wrt. the $x$ direction, then B must be V-polarized wrt. the same $x$ direction.

 

[PeterDonis]

Yes, the total ensemble is in a pure state with the state ket of the form
$$|\Psi_{12} \rangle \otimes |\Psi_{34} \rangle,$$
where $|\Psi_{12} \rangle$ and $|\Psi_{34} \rangle$ are Bell states by preparation.

Now projecting the two photons (23) to another Bell state, you get a subensemble, where (14) are entangled.

Saying it this way is very misleading. The measurement on (23) doesn't just select a subensemble of the original ensemble. It changes the state of the 4-photon system; the 4-photon state is no longer of the form $\ket{\Psi_{12}} \otimes \ket{\Psi_{34}}$. It is now of the form $\ket{\Psi_{14}} \otimes \ket{\Psi_{23}}$.

Of course you are right in saying that this subensemble must indeed be prepared by doing this projective measurement.

By "prepared" I assume you mean the change of the overall state of the 4-photon system due to the (23) measurement, as described above. That is correct, the (23) measurement functions as a preparation of a new state of the 4-photon system. Picking out the subset of runs in which the (23) state coming out of the measurement is one particular Bell state (out of four) then selects a subensemble of the newly prepared 4-photon state. But describing it just as "selecting a subensemble" is misleading because it obfuscates that crucial fact about preparation.

 

[Simple question]

From source A, the rate might be 1 in 100 million; and the same in source B. So roughly, there might be 1 swap out of (100 million)^2 pairs. You might get 10 per seconds, to 1 in 10 minutes; obviously this varies widely based on laser strength, etc.

But we can expect the processes to be improved to reach better rate. Now that I think of it, I would venture that the upper bound would be 50% of the pairs. That is the bigger sub ensemble which also define the complement where all pair have opposite results, and thus preserve the full ensemble 1&2 non-entanglement statistics.

As 50% is also the maximum amount of expected opposite spin of any pair of unrelated photon arriving at 2&3, the universe do not have to conspire a lot to avoid FLT in this case

 

[DrChinese]

@martinbn, I answer your questions below.

I will not respond further to @Nullstein or @vanhees71 or @lodbrok in this thread until they supply specific quotes to support their positions. All 3 have provided references that wasted my time to review, and say nothing remotely similar to their positions. I consider such "references" to be the lazy man's out, and deceptive when it doesn't even support their position anyway (which none did).

My position is standard QM, as written in hundreds of papers and in papers I have referenced here and quoted verbatim. Here it is, as concisely as I can word it.

The Bell test photons which have never interacted (1 & 4) are not entangled and cannot become entangled unless* the Bell State Measurement (BSM) succeeds on the other 2 (regardless of time, place or ordering). That BSM is not considered a classical cause, but it is a "quantum" cause. That's because a classical cause *must* precede any possible effect; while causality in the quantum world does not depend on time ordering (or distance as limited by c).

The BSM is absolutely NOT simply post-selection, although the selection does herald a successful swap event. We know that because without the swap, pairs [1 & 2] and [3 & 4] are maximally entangled and monogamously so. However, with a successful swap, [1 & 4] end up maximally entangled and monogamously so. The "paradox" (which matches the predictions of QM precisely) is in the following variations:

a) Photons 1 and 4 need never co-exist in a common light cone, and yet a distant event (the BSM) causes them to become maximally entangled.
b) The BSM can be performed AFTER photons 1 and 4 are already detected and evidence a violation of a Bell inequality, which is an example of quantum nonlocality (but certainly there are other examples not involving multi-particle entanglement).
c) A BSM can even be performed BEFORE photons 1 and 4 are created, although this requires additional photons and BSMs.
d) The BSM in b) can be made to occur "after" in all references frames.

*We cannot use the word "until" in this context, because it need not precede measurement of [1 & 4].

@DrChinese Let me ask you some direct questions.

1. In which sense do you use the term non-locality? Is it meant as violations of Bell's inequality or something else. If it is only in the sense of violations of Bell's inequalities, then why do you need to talk about entanglement swapping? The violations can be demonstrated without swapping. If it is something else, can you clarify what it is? And do you claim that your references (Zeilinger, Weinberg and so on) use it also that way?2. Consider a pair of two systems A and B in an entangled state. Then there are two standard facts, that have been mentioned and can be found in the literature. The density matrix of B has all the information about the statistics of the possible outcomes of measurements on B. Any measurement on A doesn't change the density matrix of B. The question is: do you dispute any of this? If yes, can you explain why. If not, then why do you say that the statement "Measurement on A does have an effect on B. Or the cause of a result at B is the measurement on A." is interpretation independent? Actually you say it is a proven fact.

1. "Quantum nonlocality" is evidenced by a violation of a Bell inequality. Many authors simply refer to this as "nonlocality", and sometimes I do too. The reason I try to use the additional word "quantum" is because I wish to indicate that there need not be superluminal forces at work, even though there appears to be "something" that occurs superluminally. However, some interpretations have "outs" in which c is respected, so my terminology attempts to account for that. Such interpretations are, of course, what is referred to as "non-realistic" to comply with Bell.

On the other hand, any interpretation in which nonlocal correlations are explained by reference to "updating" of our knowledge while retaining locality should, IMHO, be excluded as being ruled out by swapping experiments. Not all authors yet agree with me on this point, which is part of the reason I enjoy threads like this. Always looking for someone who has a strong counter-argument, but that hasn't happened yet. So far, hand-waving and not a shred of experimental support.

Are violations of Bell inequalities synonymous with nonlocality? Violation of Bell inequalities are not the only types of quantum nonlocality, so to me the answer is "not quite". But they are experimental demonstration that nature is not local realistic, which to me is the same as saying "quantum nonlocal". Anything which is context dependent has the potential to be evidence of quantum nonlocality. Here are some other examples (outside of Bell tests) that come to mind:

a. Nonlocal wave collapse: Experimental Proof of Nonlocal Wavefunction Collapse for a Single Particle Using Homodyne Measurement
b. Hanbury Brown Twiss effect (bunching/anti-bunching)
c. GHZ.

Why do swapping experiments matter to the debate? In a traditional Bell test, the entangled pair share a light cone to the past, and Alice and Bob necessarily operate within the light cone of the particles they measure. They must measure their respective particles after they are entangled. This has the effect of hiding some of the eccentricities of quantum nonlocality. In swapping experiments, you have a lot more flexibility. You can demonstrate that you can entangle particles after the fact, and you can entangle particles outside each other's light cones. That's a dramatic effect!2. Hmmm, does A change as a result of a distant operation on B? First, a caveat: no experiment can precisely determine what order the hypothetical changes would occur in. A to B, or B to A? No one understands the mechanism well enough to convincingly answer that.

My answer is that A changes ("steered") as a result of a distant operation on B (acknowledging that it could be the other way around and you can't discern between the 2 possibilities). Of course, there is some interpretational spin along with this, although I will try to steer clear as best I can.

The simple answer is that by looking at A alone, nothing ever seems to change. If that is your concept of a density matrix, then you won't agree with me. But an entangled photon does not exist in isolation, it is part of a biphoton of [A+B]. That combined matrix certainly changes as a result of a successful swap, and the statistics bear this out. From an earlier reference in this thread:

"In the scenario we present here, measuring the last photon affects the physical description of the first photon in the past, before it has even been measured. Thus, the ”spooky action” is steering the system’s past. Another point of view that one can take is that the measurement of the first photon is immediately steering the future physical description of the last photon. In this case, the action is on the future of a part of the system that has not yet been created."

So apparently these authors agree with me. Virtually any swapping experiment, and many straight Bell tests, say much the same thing. Use of the the phrases "nonlocality", "quantum nonlocality" and/or "action at a distance" run through the Bell literature. The word "nonlocal" appears in the title of about 5000 scientific papers (they aren't proving locality in those papers). So I count it as 5001 for me, and 0 for you. Although 1 good reference might be enough to convince me to change my mind... but where is one that is good enough?

 

[PeterDonis]

@DrChinese, one question about the BSM that is done on photons 2&3 in the entanglement swapping experiments. If we restrict attention to photon 2&3 pairs that meet the narrow time window requirement, the "event ready" signal (i.e., the one that picks out the subset of the runs that will be assessed for entanglement of photons 1&4) is that one particular Bell state (IIRC the singlet state) is observed as the result of the BSM on 2&3. What about the other runs, where that particular Bell state is not observed as the result of the BSM on 2&3? What is observed for photons 2&3, again restricting attention to pairs that meet the narrow time window requirement? Are they in one of the other three Bell states (the experiments as they are currently set up just don't measure which one)?

We may have gone through this in a previous thread but I can't recall for sure.

 

[DrChinese]

@DrChinese, one question about the BSM that is done on photons 2&3 in the entanglement swapping experiments. If we restrict attention to photon 2&3 pairs that meet the narrow time window requirement, the "event ready" signal (i.e., the one that picks out the subset of the runs that will be assessed for entanglement of photons 1&4) is that one particular Bell state (IIRC the singlet state) is observed as the result of the BSM on 2&3. What about the other runs, where that particular Bell state is not observed as the result of the BSM on 2&3? What is observed for photons 2&3, again restricting attention to pairs that meet the narrow time window requirement? Are they in one of the other three Bell states (the experiments as they are currently set up just don't measure which one)?

We may have gone through this in a previous thread but I can't recall for sure.

Good questions. The answers are a bit complicated, and I occasionally get confused between the various permutations, so don't shoot me if a get a + or a - backwards...

1. A good reference (with a few quotes below) is https://arxiv.org/pdf/0809.3991.pdf , see especially the top middle of figure 1. I will use the [2 & 3] pair labeling for the Bell State Measurement (BSM), and the [1 & 4] pair for the Bell test (as in the reference). Everything assumes photons that are linear polarized, all initial pairs [1 & 2] and [3 & 4] entangled. We ignore the many cases where there are clicks at [1] or [4] but do not match a successful BSM on [2 & 3]. BSM successes are rare, since the [1 & 2] pairs come at random intervals, as do the [3 & 4] pairs. For a successful BSM, both need to fire simultaneously*. That might happen perhaps once every 10 seconds.

2. As you know, there are 4 Bell States which may occur during a BSM. They occur with roughly equal likelihood. Also, the initial pairs [1 & 2] and [3 & 4] can themselves be created as perfectly correlated (Type I PDC) or perfectly anti-correlated (Type II PDC). The experimenters themselves know which is which, but by using the same Type on both pairs, there is no adjustment necessary.

• Psi+ for [2 & 3]: Photons [1 & 4] will be perfectly correlated.
• Psi- for [2 & 3]: Photons[1 & 4] will be perfectly anti-correlated.
• Phi+ for [2 & 3]: cannot be distinguished from Phi-.
• Phi- for [2 & 3]: cannot be distinguished from Phi+.

An important factor is that not all of the 4 states can be simultaneously distinguished via BSM. "This is the optimum efficiency possible with linear optics."-Zeilinger et al [J. Calsamiglia and N. Lutkenhaus, Appl. Phys. B (2001)]. As a general rule, the Psi+ and Psi- states can be be distinguished using a beam splitter (BS), 2 polarizing beam splitters (let's label PBS1 and PBS2), and 4 detectors (let's label as PBS1h, PBS1v, PBS2h, PBS2v. Each of the 2 BS outputs are routed to a PBS, and each of the 2 PBS outputs are routed to a detector. Keep in mind, this is all part of the BSM apparatus, used to initiate/herald/cause the swap action.

3. So to answer your question: the statistical split of possible outcomes is approximately as follows:

• 25%: The simultaneous* clicks of PBS1h and PBS1v, or PBS2h and PBS2v, heralds a successful Psi+ swap.
• 25%: The simultaneous* clicks of PBS1h and PBS2v, or PBS1v and PBS2h, heralds a successful Psi- swap.
• 50%: Simultaneous* clicks in any of 4 other combos (such as PBS1h and PBS2h, etc) indicate a Phi+ or Phi-, but you won't know which. There is no way to make sense of the [1 & 4] outcomes as part of the Bell test porting. That makes these useless for consideration. These are not counted, even though they indicate a successful swap.

The approach does vary from one swapping experiment to another. Some experiments look at the Psi- Bell state only, while others look for both Psi states. All approaches ignore at least 2 of the 4 states. It is worthy to remind everyone: in all cases the [2] and [3] photons have the opportunity to overlap and/or interact and/or interfere. They must be indistinguishable in time, so you won't know the source for any of the BSM clicks.

4. What is important to understand is: all qualifying events are included for the CHSH calculation (or whatever statistics are being calculated) as long as the appropriate clicks occur at the BSM apparatus**. They assume that indicates a successful swap occurred, even if it didn't. So the entanglement measure (S) won't be overstated by any swap failures.

"The specific state of photons 1 and 4 after entanglement swapping depends on the result of the BSM, which can either be ψ + or ψ −. The relevant CHSH inequalities for these cases are S=..."

The result of this 2009 experiment: "We achieve a clear violation of the CHSH inequality with Sψ− = 2.37 ± 0.09 and Sψ+ = 2.38 ± 0.09. In all cases the CHSH inequality [S<=2] is violated by more than four standard deviations."

Cheers,

-DrC*Simultaneous meaning: within your narrow time window requirement. That might be on the order of 10 picoseconds.
** Assuming there are matching clicks for the [1 & 4] photons so their match coincidence can be counted as part of the Bell test.

 

[PeterDonis]

Some experiments look at the Psi- Bell state only, while others look for both Psi states. All approaches ignore at least 2 of the 4 states.

Ok, that's what I thought. Thanks for the detailed clarification!