What is the Difference Between Locality and Separability in Quantum Mechanics?

[Demystifier]

Some papers on foundations of quantum mechanics distinguish the notions of locality and separability (or non-locality and non-separability). Can someone explain to me what is the difference between locality and separability? Or can someone point to a reference where this difference is explained?

 

[WaveJumper]

I would guess that the fine line between the two is the scenario where 'infinite speed'(or much greater than the speed of light), at least hypothetically, can exist as a phenomenon, while the universe as a system remains separable(i.e. everything is not one wholeness). I think Non-separability includes Non-locality, while Non-locality, as a term, must not necessarily include Non-separability.

 

[ThomasT]

Some papers on foundations of quantum mechanics distinguish the notions of locality and separability (or non-locality and non-separability). Can someone explain to me what is the difference between locality and separability? Or can someone point to a reference where this difference is explained?

I doubt that I'm able to explain anything to you in any way better than you probably already understand it, while I have learned from your contributions here -- but since you're starting a discussion on the difference between the notions of nonlocality and nonseparability, then here's my two cents.

Nonlocality refers to violations of the principle of locality. Bell tests structure out the possibility of local causal interactions/transmissions between spacelike separated events at A and B during any given coincidence interval.

Nonseparability refers to statistical dependence between events at A and B. The experimental violation of Bell inequalities requires that paired events (A,B) be statistically dependent (detection at one end affects the sample space at the other end).

The most direct way to model locality is via a factorable representation of the joint probability (Bell locality condition of an lhv). The most direct way to model statistical dependence or nonseparability is via a nonfactorable representation of the joint probability (standard qm).

The reason that nonlocality (in nature) cannot be inferred from experimental violations of Bell inequalities is because the Bell locality condition isn't, exclusively, a locality condition.

An article by Peres:

http://arxiv.org/abs/quant-ph/9609016v1

 

[DrChinese]

Some papers on foundations of quantum mechanics distinguish the notions of locality and separability (or non-locality and non-separability). Can someone explain to me what is the difference between locality and separability? Or can someone point to a reference where this difference is explained?

I think you know that this area has been sliced and diced a lot. There are those who point to Bell's paper as the starting point (or the ending point) for the discussion. Unfortunately, as much as I am enamored of the Bell paper for its science: I think some of the terminology and structure of the argument has been something of a breeding ground for some confusion. So while I don't see there as being much debate on the conclusion itself, there is a lot of debate on the explicit and implied elements of the argument.

Fundamentally, I see the separability requirement as a denial of 2 particle state entanglement. So Bell's (2) is a statement of the EPR implied position that entanglement cannot exist for space-like separated particles. That position, in turn, assumes locality, i.e. that there do not exist physical connections (exceeding c) between any 2 particles.

You are asking whether separability and locality are the same. In your non-local view, there is no locality as ALL particles influenced (and are influenced by) all other particles. But that view does not mean that all particles are entangled; clearly you still have that special behavior that is a result of shining a laser into a PDC crystal - which creates entangled photon pairs. If you believe that those pairs are sharing a wave state, you believe in entanglement and deny separability. ON THE OTHER HAND: I don't yet share your non-local view and yet I share your denial of separability. So I would say that locality and separability are NOT the same thing. If you can picture a alternative universe in which there are non-local forces but there is no entanglement of particle states (hardly something that is a direct deduction from non-locality), then they must not be the same thing.

How are they related? Is one a subset of the other? In my mind, entanglement is contained within QM. On the other hand, locality is not. So I see Bell as proving: separability (denial of entanglement) is NOT possible if QM is correct. But at the same time, he ALSO proves the more important result that local realism is not possible if QM is correct. Bell had to assume realism (see his 14 to 15) to prove that separability is incompatible with QM.

Seen another way: I assume that you can at least acknowledge the possibility that we could live in a time-symmetric universe in which locality is otherwise respected. Then we could have locality without separability (or realism) - so again they must not be the same thing. I wouldn't imagine that you could find a definitive discussion of this subject, because everyone has a certain historical twist on the matter. References you might be interested in:

1. More from Peres, Quantum Theory: Concepts and Methods (2002, see page 160 of 464): http://www.fisica.net/quantica/Peres%20-%20Quantum%20Theory%20Concepts%20and%20Methods.pdf

"The title of Bell’s second paper is 'On the Einstein Podolsky Rosen paradox,' but, contrary to the EPR argument, Bell’s is not about quantum mechanics. Rather, it is a general proof, independent of any specific physical theory, that there is an upper limit to the correlation of distant events, if one just assumes the validity of the principle of local causes. This principle (also called Einstein locality, but conjectured well before Einstein) asserts that events occurring in a given spacetime region are independent of external parameters that may be controlled, at the same moment, by agents located in distant spacetime regions."

2. From Marinescu & Marinescu, Quantum Information and Error Correction From Classical to Quantum Concepts(2009, page 139 of 702): http://www.eecs.ucf.edu/~dcm/QCV2.pdf

"Bell's inequality is derived using a very simple model of a physical system that makes only two common sense assumptions: physical properties are independent on observations, the realism principle, and the measurement of different physical properties of different objects carried out by different observers at distinct locations cannot influence each other, the locality principle."

3. Norsen has written on this, of course, and I assume you are already familiar with his work on that - such as: http://arxiv.org/abs/0707.0401 or http://arxiv.org/abs/quant-ph/0601205 (and keeping in mind that I think Norsen is wrong on much of the history and semantics of this issue):

"Bell Locality then entails the following: once we specify a complete description of the pre-measurement state of the particle pair, the probability for Alice to obtain a certain outcome A for a measurement along a certain direction ˆa is independent of the setting (ˆb) and outcome (B) of Bob’s experiment. In particular, the probability in question does not change depending on whether we do or do not specify this information about Bob’s experiment."

So Norsen sees Bell locality as equivalent to a kind of statistical independence, as expressed by the separability formula. I say that should be called Bell separability instead, so as to clarify that violation of separability does not require that there is spooky action at a distance. Norsen sees action at a distance as essentially being a deduction from a Bell Inequality violation, whereas this conclusion is nearly universally rejected elsewhere.

 

[RUTA]

The ontological distinction, as I see it, is that in non-locality you have a superluminal causal mechanism or exchange of information between the entangled particles. In non-separability you don't really have "particles" but just one entity that is responsible for the detector clicks.

 

[Demystifier]

Thank you all!
Now the difference is completely clear to me.

 

[DrChinese]

Thank you all!
Now the difference is completely clear to me.

Ha ha! Now can you explain it to me then...

 

[Demystifier]

Ha ha! Now can you explain it to me then...

To avoid repeating of the nice explanations above, I will try with a short indirect explanation:

Non-separability of QM means that, in the system of many particles, there is no separate wave function for each particle (due to entanglement). In other words, there are statistical correlations among particles. All physicists agree that QM is non-separable.

If we assume that reality exists even without measurements, then, according to the Bell theorem, distant particles must influence each other instantaneously (or faster than light). This is non-locality. Since not all physicists agree that reality exists even without measurements, not all phsicists agree that QM is non-local.

 

[DrChinese]

To avoid repeating of the nice explanations above, I will try with a short indirect explanation:

Non-separability of QM means that, in the system of many particles, there is no separate wave function for each particle (due to entanglement). In other words, there are statistical correlations among particles. All physicists agree that QM is non-separable.

If we assume that reality exists even without measurements, then, according to the Bell theorem, distant particles must influence each other instantaneously (or faster than light). This is non-locality. Since not all physicists agree that reality exists even without measurements, not all phsicists agree that QM is non-local.

Well said! I like it...

 

[ThomasT]

If we assume that reality exists even without measurements, then, according to the Bell theorem, distant particles must influence each other instantaneously (or faster than light). This is non-locality. Since not all physicists agree that reality exists even without measurements, not all phsicists agree that QM is non-local.

Bell's theorem says that factorable lhv entangled state formulations are incompatible with qm and experiments. They're incompatible because the nonseparability of entangled subsystems manifests experimentally via statistical dependency between the separately accumulated data sets via the data matching process.

Violation of Bell inequalities tells us nothing whatsoever about whether Nature is local or nonlocal, or about whether there is a Nature independent of our observations. It tells us only that one of the assumptions (statistical independence) embodied in the formulation of the Bell theorem-inequality has been contradicted.

The defacto standard mainstream scientific assumptions are (1) Nature exists independent of observation, and there are deeper levels of Nature than that revealed via our senses, (2) Nature obeys the principle of locality, and (3) Nature is deterministic.

Whether one wants to call qm local or nonlocal or acausal is irrelevant.

 

[DrChinese]

The defacto standard mainstream scientific assumptions are (1) Nature exists independent of observation, and there are deeper levels of Nature than that revealed via our senses, (2) Nature obeys the principle of locality, and (3) Nature is deterministic.

I would say that there is NOT mainstream agreement on any of these. Reality, locality and determinism are hotly debated across the board. Sorta surprised to see you make these assertions given the discussions you have been a part of here.

 

[zonde]

If we assume that reality exists even without measurements, then, according to the Bell theorem, distant particles must influence each other instantaneously (or faster than light). This is non-locality.

From Bell theorem we can conclude that for real entanglement experiments fair sampling assumption is not applicable.

Idea that distant particles must influence each other instantaneously will fail very quickly if you would examine mind experiment with three entangled photons where conflicting "influences" should take place.

 

[Demystifier]

The defacto standard mainstream scientific assumptions are (1) Nature exists independent of observation, and there are deeper levels of Nature than that revealed via our senses, (2) Nature obeys the principle of locality, and (3) Nature is deterministic.

Experiments (that test Bell inequalities) show that either (1) or (2) is wrong.
If by "science" one means "physics", then (3) is certainly not the mainstream assumption.

 

[Demystifier]

From Bell theorem we can conclude that for real entanglement experiments fair sampling assumption is not applicable.

Fair sampling assumption has more to do with actual experiments (having low detector efficiency) than with the Bell theorem.

Idea that distant particles must influence each other instantaneously will fail very quickly if you would examine mind experiment with three entangled photons where conflicting "influences" should take place.

Who said that the influences are conflicting? They are not.

 

[zonde]

Fair sampling assumption has more to do with actual experiments (having low detector efficiency) than with the Bell theorem.

Fair sampling assumption is required if you want to apply Bell's theorem to actual experiments.

Who said that the influences are conflicting? They are not.

And if you have two photons with polarizers at 45deg relative angle and third photon with polarizer in between first two (22.5 and 22.5 deg)?
Clearly third photon has conflicting "influences" form first two photons.
Or you think otherwise?

 

[Demystifier]

Fair sampling assumption is required if you want to apply Bell's theorem to actual experiments.

With that, I agree.

And if you have two photons with polarizers at 45deg relative angle and third photon with polarizer in between first two (22.5 and 22.5 deg)?
Clearly third photon has conflicting "influences" form first two photons.
Or you think otherwise?

Yes I do. In fact, without both influences, the photon would not even know how to behave, because the information would be incomplete. The behavior of the photon is a single-valued function of both variables (influences).

This is like a 3-body problem in classical Newtonian mechanics. The acceleration of the first body is determined if and only if the positions of both second and third body are known.

 

[zonde]

Yes I do. In fact, without both influences, the photon would not even know how to behave, because the information would be incomplete. The behavior of the photon is a single-valued function of both variables (influences).

I guess you mean that without influence photon will behave in random fashion.

But let's see about two influences in the case I mentioned ... quantitatively.
Two entangled photons from polarizers at 45deg relative angle will have 50% coincidences in idealized case as cos^2(45deg)=0.5 (I assume fair sampling obviously).
Third entangled photon has to have 85% coincidences with the first photon and 85% coincidences with second photon as cos^2(22.5deg)=0.85.
The maximum amount for what all three photons can coincidence is 50% (that's because that is the number for first two photon coincidences). So it means that the rest of 35% from both of 85% coincidences of the third photon should be separate for first photon and second photon. But now for the third photon we have:
50%+35%+35%=120%
Ups, something wrong

Where is the problem?

 

[Demystifier]

I guess you mean that without influence photon will behave in random fashion.

No, I don't. I mean without the influence the photon will not behave at all, because the influence is a part of the physical laws. See the 3-body-problem analogy that I mentioned.

Two entangled photons from polarizers at 45deg relative angle will have 50% coincidences in idealized case as cos^2(45deg)=0.5 (I assume fair sampling obviously).
Third entangled photon has to have 85% coincidences with the first photon and 85% coincidences with second photon as cos^2(22.5deg)=0.85.
The maximum amount for what all three photons can coincidence is 50% (that's because that is the number for first two photon coincidences). So it means that the rest of 35% from both of 85% coincidences of the third photon should be separate for first photon and second photon. But now for the third photon we have:
50%+35%+35%=120%
Ups, something wrong

Where is the problem?

The maximum amount (for what all three photons can coincidence) should not be identified with the actual amount. The actual amount is not 50%, but less.

 

[zonde]

No, I don't. I mean without the influence the photon will not behave at all, because the influence is a part of the physical laws. See the 3-body-problem analogy that I mentioned.

I do not see connection. Singlet photon stream will interact with polarizer anyways. In that case you have Malus law.

The maximum amount (for what all three photons can coincidence) should not be identified with the actual amount. The actual amount is not 50%, but less.

Ok, but if you look at it this way then these 35% is minimum so that those final 120% can only increase but not decrease.
replace 50% with x <= 50% and we have:
x+(85%-x)+(85%-x)=170%-x >= 120%

So this is not a solution to the problem.

 

[Demystifier]

I do not see connection. Singlet photon stream will interact with polarizer anyways. In that case you have Malus law.

You are talking about statistics in ordinary QM. I am talking about hypothetical deterministic influences in a nonlocal hidden-variable formulation of QM.

Ok, but if you look at it this way then these 35% is minimum so that those final 120% can only increase but not decrease.
replace 50% with x <= 50% and we have:
x+(85%-x)+(85%-x)=170%-x >= 120%

So this is not a solution to the problem.

I don't see your point. Are you saying that probabilities in ordinary QM do not sum up to 1?

 

[DrChinese]

Two entangled photons from polarizers at 45deg relative angle will have 50% coincidences in idealized case as cos^2(45deg)=0.5 (I assume fair sampling obviously).
Third entangled photon has to have 85% coincidences with the first photon and 85% coincidences with second photon as cos^2(22.5deg)=0.85.
The maximum amount for what all three photons can coincidence is 50% (that's because that is the number for first two photon coincidences). So it means that the rest of 35% from both of 85% coincidences of the third photon should be separate for first photon and second photon. But now for the third photon we have:
50%+35%+35%=120%
Ups, something wrong

Where is the problem?

First, 3 photon experiments have already been performed - so it is not exactly a thought experiment. See for example:

Multi-Photon Entanglement and Quantum Non-Locality, Jian-Wei Pan and Anton Zeilinger

Second, a 3 particle entangled state does not necessarily operate as you describe above because Malus is not the operative formula (although you would think it would be - at least I did). The 3 particle state is usually the W state if I recall correctly. There is not the kind of "perfect" correlations you would expect as you rotate theta.

In other words, you might expect to get the same result at 60 degrees for all 3 photons but that is not necessarily the case, even if the state was H>H>H> + V>V>V> (which I don't believe it is anyway). It works out in the 2 particle case but not in 3. In the 3 case, you get perfect correlations at some angles (such as 0, 90, etc) but not at others. tez kindly explained this to me a while back.

For more information, try this link which has a few hundred articles on the subject:

W states, three qubits

 

[RUTA]

To avoid repeating of the nice explanations above, I will try with a short indirect explanation:

Non-separability of QM means that, in the system of many particles, there is no separate wave function for each particle (due to entanglement). In other words, there are statistical correlations among particles. All physicists agree that QM is non-separable.

If we assume that reality exists even without measurements, then, according to the Bell theorem, distant particles must influence each other instantaneously (or faster than light). This is non-locality. Since not all physicists agree that reality exists even without measurements, not all phsicists agree that QM is non-local.

Statistical nonseparability ("no measurable property of particle 1 alone, and no measurable property of particle 2 alone, has any definite value." David Albert, Quantum Mechanics and Experience, Harvard Univ Press, 1994, p. 49) does not entail ontological nonseparability. In fact, one can have ontologically separable particles (constitutive locality) in a nonseparable state, explaining outcomes via causal non-locality. [See Healey, R.: Gauging What’s Real: The Conceptual Foundations of Gauge Theories. Oxford University Press, Oxford (2007), p 127, for this distinction.]

 

[zonde]

I don't see your point. Are you saying that probabilities in ordinary QM do not sum up to 1?

I am not directly talking about QM. If you noticed I used only empirical formula cos^2(theta) where theta is relative angle between polarizers.
But my point is that if you assume fair sampling probabilities do not sum up to 1.
So Bell's theorem is not applicable to real experiments and that way non-locality loose it's grounds.

 

[zonde]

First, 3 photon experiments have already been performed - so it is not exactly a thought experiment.

Yes 3 photon entanglement is used in experiments but not the way I described. So the experiment is feasible but not performed as far as I know.

Second, a 3 particle entangled state does not necessarily operate as you describe above because Malus is not the operative formula (although you would think it would be - at least I did). The 3 particle state is usually the W state if I recall correctly. There is not the kind of "perfect" correlations you would expect as you rotate theta.

Correct if I am wrong but you are talking about 3 simultaneous "clicks" in 3 different detectors.
I am talking about 2 simultaneous "clicks" in 3 different detectors. Only outputs of two detectors are correlated ignoring the third in each of three correlation measurements.
That way it should follow the law of two photon entanglement.

But in case you doubt that be aware you can open the case for FTL communication. Namely if what happens to third photon can alter correlation between other two photons then we can design FTL communication device.

 

[ThomasT]

The defacto standard mainstream scientific assumptions are (1) Nature exists independent of observation, and there are deeper levels of Nature than that revealed via our senses, (2) Nature obeys the principle of locality, and (3) Nature is deterministic.

I would say that there is NOT mainstream agreement on any of these. Reality, locality and determinism are hotly debated across the board. Sorta surprised to see you make these assertions given the discussions you have been a part of here.

I don't know why you'd be surprised. I've always advocated the idea of a locally deterministic, observation-independent, deep(er) reality.

If we abstract and generalize from what's known, then (1), (2) and (3) are the most sensible assumptions.
What most scientists might say when asked about these issues is an open question.
In any case, I submit that (1), (2), and (3) are de facto standard assumptions involved in the actual development of the physical sciences.

Experiments (that test Bell inequalities) show that either (1) or (2) is wrong.

I don't think so. Yes, an assumption embodied in Bell's formulation is contradicted. However, the salient feature of Bell's lhv ansatz is the factorability of the
joint (entangled) state. This feature represents both locality and statistical independence. Statistal independence is incompatible with the experimental designs (as well as the qm formulation of entangled states). So, statistical dependence is sufficient to violate Bell inequalities. And since the statistical dependence can be accounted for by local interactions/transmissions, then there is no reason to suppose that Bell inequality violations are due, in any way, to nonlocal interactions or ftl transmissions.

A qualitative understanding of how the correlations are produced is an open question. But there's no reason to abandon locality. And, these tests certainly don't contradict the idea that there is a reality deeper than and outside of our sensory purview. If anything, any and all quantum experimental phenomena only serve to support (1).

If by "science" one means "physics", then (3) is certainly not the mainstream assumption.

It's the de facto standard (albeit mostly tacit) assumption underlying all of our behavior, whether that behavior might be called scientific or not.

The emergence of our universe will remain a mystery, but it's observable evolution certainly seems to support the assumption of determinism, and the search for fundamental dynamical principles.

To revisit the topic: quantum nonseparability doesn't necessarily contradict locality, though it certainly contradicts Bell locality.

 

[RUTA]

I guess you mean that without influence photon will behave in random fashion.

But let's see about two influences in the case I mentioned ... quantitatively.
Two entangled photons from polarizers at 45deg relative angle will have 50% coincidences in idealized case as cos^2(45deg)=0.5 (I assume fair sampling obviously).
Third entangled photon has to have 85% coincidences with the first photon and 85% coincidences with second photon as cos^2(22.5deg)=0.85.
The maximum amount for what all three photons can coincidence is 50% (that's because that is the number for first two photon coincidences). So it means that the rest of 35% from both of 85% coincidences of the third photon should be separate for first photon and second photon. But now for the third photon we have:
50%+35%+35%=120%
Ups, something wrong

Where is the problem?

I explained this to you in another thread. Why are you still asking if you won't pay attention to the answer?

 

[DrChinese]

I am talking about 2 simultaneous "clicks" in 3 different detectors. Only outputs of two detectors are correlated ignoring the third in each of three correlation measurements. That way it should follow the law of two photon entanglement.

So you are talking about measuring Alice and Bob, and hypothsizing the existence (reality) of the third then. As in the usual Bell scenario? Then the results won't add up as you point out and the assumption fails.

 

[DrChinese]

I don't know why you'd be surprised. I've always advocated the idea of a locally deterministic, observation-independent, deep(er) reality.

What most scientists might say when asked about these issues is an open question.
In any case, I submit that (1), (2), and (3) are de facto standard assumptions involved in the actual development of the physical sciences.

I thought you believed in local realism, but I didn''t expect you to say that it is the defacto mainstream view. You should acknowledge that your perspective is NOT the mainstream within the physics community and hasn't been for a long time. Consequently it is also not the defacto view. If you are talking about the view outside of the physics community, well, that isn't too much of an issue in discussions here.

 

[zonde]

I explained this to you in another thread. Why are you still asking if you won't pay attention to the answer?

The post you quoted was not really a question. It was illustration of this statement from other post:
"From Bell theorem we can conclude that for real entanglement experiments fair sampling assumption is not applicable.
Idea that distant particles must influence each other instantaneously will fail very quickly if you would examine mind experiment with three entangled photons where conflicting "influences" should take place."

About your explanation. I guess you are referring to this:

cos^2(theta) where theta is the angle between polarizers doesn't necessarily give you the coincidence rate.

But your statement seems like nit-picking. Theta is related to angles in two different places so of course we need to establish common reference to acquire meaningfull value for theta.

Let me describe realistic situation.
Photons from entangled photon source are transported to three sites where measurements are made via optical fibre. Optical fibre has a lot of turnings so it would be very hard to guess common reference from photon path taken in optical wire.
Another thing is that rotation axis of polarizer can take arbitrary direction at measurement sites say first has axis in direction of source, second - in direction of first site, third - in downward direction (normal to other two rotation axes).

So only practical solution is to do calibration.
So we take arbitrary angle at first site and mark it as common reference.
Second we find for what angle there is coincidence maximum (middle of peak) at other two sites in respect to common reference angle of first site. We mark those maximum angles at second and third site as common reference.

Now we calculate theta as difference in corresponding sites between rotation angles from common reference in clockwise direction when looking from output side of polarizer. If we want to be careful we should check that rotating polarizers in the same direction (clockwise) by the same amount still gives us maximum (to see that our choice of clockwise for all three sites is correct).

 

[zonde]

So you are talking about measuring Alice and Bob, and hypothsizing the existence (reality) of the third then. As in the usual Bell scenario? Then the results won't add up as you point out and the assumption fails.

Well kind of that. I will just add that this assumption that fails is fair sampling assumption.

Just to be on the safe side let me give more detailed description of coincidence measurements.
We have three detectors (A, B and C) at three different sites. We send signals about "clicks" to coincidence counter. Coincidence counter makes records according to such algorithm:
If we have signal from A and B within coincidence window but no signal from C we record A-B coincidence.
If we have signal from B and C but no signal from A we record B-C coincidence.
If we have signal from C and A but no signal from B we record C-A coincidence.
If we have signal from A, B and C we record A-B, B-C and C-A coincidences.

So we do not record A-B-C coincidences you was referring to but nonetheless we find correlations in all three detectors.

And the thing that requires us to hypothesize about existence (reality) of the third photon when we consider correlation in other two detectors is fair sampling assumption.

 

[DrChinese]

Photons from entangled photon source are transported to three sites where measurements are made via optical fibre. ... If we want to be careful we should check that rotating polarizers in the same direction (clockwise) by the same amount still gives us maximum (to see that our choice of clockwise for all three sites is correct).

You are switching between contexts, and that is making the discussion difficult to follow.

The 3 photon case, as I mentioned earlier, is DIFFERENT than the 2 photon case. ALL STANDARD BELL TESTS are actually performed with 2 photons - Alice and Bob. The fair sampling assumption relates to the universe of Alice+Bob pairs. Your scenario - as described - does not match theory or experiment I am familiar with.

Yes, you can entangle 3 photons and send them to 3 detectors. Yes, you can calibrate them for maximum correlation. But they will not follow the cos^2 theta rule you describe, and the reason for that has nothing to do with fair sampling. As I mentioned, the 3 photon case is more complicated than the 2 photon case.

For example: there is the GHZ scenario (groups of 3 photons), which is actually a separate theoretical proof than the Bell inequality (groups of 2 photons).

 

[DrChinese]

If we have signal from A, B and C we record A-B, B-C and C-A coincidences.

So we do not record A-B-C coincidences you was referring to but nonetheless we find correlations in all three detectors.

These seem contradictory. Are you imagining 3 detectors, and sometimes all three fire due to 3 entangled photons? Because this is not what is referred to as the Bell state.

And how is any of this related to fair sampling? As has been mentioned, fair sampling critics assert that detectors are MORE likely to detect photons that will violate Bell's Inequality than those that will not. That being an experimental issue that is rapidly disappearing as technology improves - and guess what? As experimental detection effeciency increases, violation of the Inequalities do NOT decrease as the critics propose. (Of course, those same critics assert that there would be NO violation if all pairs were sampled.)

 

[zonde]

Yes, you can entangle 3 photons and send them to 3 detectors. Yes, you can calibrate them for maximum correlation. But they will not follow the cos^2 theta rule you describe, and the reason for that has nothing to do with fair sampling.

Are you saying that in this case picking arbitrary two detectors and finding out correlation between them it will differ from cos^2(theta) value?

These seem contradictory. Are you imagining 3 detectors, and sometimes all three fire due to 3 entangled photons?

I don't get it. Where do you see contradiction?

As experimental detection effeciency increases, violation of the Inequalities do NOT decrease as the critics propose. (Of course, those same critics assert that there would be NO violation if all pairs were sampled.)

Do you speak about photon polarization experiments or about completely different experiments? Because as far as I know photon polarization experiments are carried out using the same 10% efficiency even if technology allows higher detection rates with reduced noise.