Local realism ruled out? (was: Photon entanglement and )

[Eye_in_the_Sky]

... I who harbours a misconception

Previously, I began a discussion with akhmeteli saying:

Hello, akhmeteli. It appears to me there may be some misconception in the way you are thinking about Bell's theorem.

At the conclusion of our discussion, I said:

Thank you, akhmeteli, for answering my questions. Originally, it appeared to me that there may have been some misconception in the way you were thinking about Bell's Theorem. But from the answers you have given, I do not detect any such misconception.

Indeed ...

local determinism → D

and

QM → ~D ,

where D is a certain condition.

Now I see it has been I who harbours a misconception. The first proposition in the above is INCORRECT. The correct statement is:

local determinism Λ PC → D ;

this is the weak version of deriving a Bell inequality.

The strong version reads like this:

locality Λ CF Λ PC → D .

The first is only a corollary of the second, because

local determinism → locality Λ CF ,

but not conversely.
____________________

In case anyone is wondering:

PC ≡ perfect (anti-) correlation ,

CF ≡ counterfactuality ,

D ≡ a Bell inequality .
------------------------------------------------------
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Oh Mom … there it is again!

... So, there are two 'theorems', a weak one and a strong one:

Weak Theorem: local determinism → D ;

Strong Theorem: locality Λ PC Λ CF → D .

... whoops!

 

[Eye_in_the_Sky]

Bell expresses locality as the factorability of the joint probability.

You keep saying that, but I went back to the original Bell paper again recently to check something else, and I really don't think your statement is correct.

The passage from Bell's paper addressing locality is from section II ...

SpectraCat, ThomasT's claim does not apply to the part of Bell's paper that you quoted. The part you quoted is the beginning of "stage 2" in Bell's two-stage argument. At that spot, at the beginning of "stage 2", all outcomes are assumed to be predetermined (yet unknown). ThomasT's claim applies to "stage 1", not "stage 2".

So where then in Bell's paper is "stage 1" to be found? It is to be found in the first paragraph of section II as follows:

Consider a pair of spin one-half particles formed somehow in the singlet spin state and moving freely in opposite directions. Measurements can be made, say by Stern-Gerlach magnets, on selected components of the spins σ₁ and σ₂. If measurement of the component σ₁∙a, where a is some unit vector, yields the value +1 then, according to quantum mechanics, measurement of σ₂∙a must yield the value -1 and vice versa. Now we make the hypothesis [2], and it seems one at least worth considering, that if the two measurements are made at places remote from one another the orientation of one magnet does not influence the result obtained with the other. Since we can predict in advance the result of measuring any chosen component of σ₂, by previously measuring the same component of σ₁, it follows that the result of any such measurement must actually be predetermined.
-------------------------------------------------------------------------
[2] "But on one supposition we should, in my opinion, absolutely hold fast: the real factual situation of the system S₂ is independent of what is done with the system S₁, which is spatially separated from the former."

Note, however, that ThomasT's claim can only be applied to the above argument after that argument has been reformulated in terms of the joint-probability-function of the particle pair as calculated at the level of a physical theory. At this level, Einstein's locality statement [2] is transferred over to a mathematical condition which the joint-probability-function must then satisfy. That mathematical condition has come to be called "Bell Locality".

ThomasT's claim, then, boils down to the following:

"Bell Locality" is not a faithful representation of a principle of "Local Causality".
___________________________

Way back in post #239, I posted a diagram and two quotes on Bell's "Local Causality Criterion", thinking it might stimulate some discussion. When I saw that it did not do so, I decided I had better follow up on it with some more information on the matter. Thereafter, I decided I ought to attempt to present a 'clean presentation' of the entire matter. So far, this has proved to be exceedingly difficult for me.

Unfortunately, my time is running out, and by next week I will definitely have to stop posting here in the forum for quite some time.

So maybe I will have to compromise in some way.

 

[Eye_in_the_Sky]

I would use "entangled state" instead of "twin-state" because that way you imply certain things that might not be very appropriate.
But otherwise I think we are on the same line here with addition that instead of the same result (maximally similar) there might be maximally opposite result as well depending from setup.

Yes we are 'on the same line' here. "Twin-state" is idiomatic for

(|x>|x> + |y>|y>) / √2 .

I would say that I do not quite understand how do you incorporate "angular momentum" in this context. If you associate "angular momentum" with polarization of individual photon then ...

For the moment, let us restrict our considerations to the singlet spin-½ pair with Stern-Gerlach magnets. In this context, would you say that the following statement is true?

"PC" is an essential ingredient of any theory which incorporates in it the notion of "angular momentum" as construed in conventional terms.

If the statement is true in the spin-½ context, then it might be possible (... at this stage I do not quite see how) to construct an argument for its truth in the optical context. On the other hand, if even in the spin-½ context the statement is false, then surely it is also false in the optical context.

 

[zonde]

"Experimental violation of a Bell's inequality with efficient detection" (2001) Rowe et al.

http://www.nature.com/nature/journal/v409/n6822/full/409791a0.html

Not actually 100% detection but high enough. This uses Be ions rather than photons.

"Our measured value of the appropriate Bell's 'signal' is 2.25+/- 0.03, whereas a value of 2 is the maximum allowed by local realistic theories of nature. In contrast to previous measurements with massive particles, this violation of Bell's inequality was obtained by use of a complete set of measurements. Moreover, the high detection efficiency of our apparatus eliminates the so-called 'detection' loophole."

This experiment makes joined detection for photons scattered from both ions. It can be picked out here:
"The state of an ion, |down> or |up>, is determined by probing the ion with circularly polarized light from a 'detection' laser beam. During this detection pulse, ions in the |down> or bright state scatter many photons, and on average about 64 of these are detected with a photomultiplier tube, while ions in the |up> or dark state scatter very few photons. For two ions, three cases can occur: zero ions bright, one ion bright, or two ions bright. In the one-ion-bright case it is not necessary to know which ion is bright because the Bell's measurement requires only knowledge of whether or not the ions' states are different. Figure 2 shows histograms, each with 20,000 detection measurements. The three cases are distinguished from each other with simple discriminator levels in the number of photons collected with the phototube."

So the photon interference happens and result of measurement is not discrete sum of two photon ensembles but result of interference of two photon ensembles.
For easier visualization I can suggest to compare this experiment with double slit experiment where two ions play the role of slits. Difference is that each ion separately produces sharp bands but presence of other ion shifts the bands to one side.

So my point is that you don't even need any specific LR theory to account for results of this experiment in local realistic fashion.

 

[Frame Dragger]

So my point is that you don't even need any specific LR theory to account for results of this experiment in local realistic fashion.

As long as you realize that you're in a terribly small minority. Frankly, I'd make your point from the outset and seek to justify it, not the other way around.

Bell is a test for LR matching QM's predictions, and there is a REASON why dBB is the only HV theory to be left after Bell (that is meaningful in any way, which is debatable)

You're seeming to advocate the notion of ensembles of particles creating apparent interference patterns, but that it is not a property of a single photon (or particle). If you're formulating this through LHVs, just say it so we can all go on our way.

 

[SpectraCat]

SpectraCat, ThomasT's claim does not apply to the part of Bell's paper that you quoted. The part you quoted is the beginning of "stage 2" in Bell's two-stage argument. At that spot, at the beginning of "stage 2", all outcomes are assumed to be predetermined (yet unknown). ThomasT's claim applies to "stage 1", not "stage 2".

So where then in Bell's paper is "stage 1" to be found? It is to be found in the first paragraph of section II as follows:


Note, however, that ThomasT's claim can only be applied to the above argument after that argument has been reformulated in terms of the joint-probability-function of the particle pair as calculated at the level of a physical theory. At this level, Einstein's locality statement [2] is transferred over to a mathematical condition which the joint-probability-function must then satisfy. That mathematical condition has come to be called "Bell Locality".

You have lost me here ... what is equation 2 if not a reformulation "in terms of the joint-probability-function of the particle pair as calculated at the level of a physical theory", which also allows for the possibility of hidden variables?

I agree that, if there are no hidden variables, then that expression reduces to P(A,B)=P(A)P(B), as ThomasT says .. is that what you mean? If so, what is wrong with that as a definition of locality? I have checked back through his posts (although not exhaustively), and it seems ThomasT says cryptic things like "if you don't see something wrong with this, then you should", instead of explaining what he actually means. Perhaps I am just dense, but I don't see this ...

 

[Eye_in_the_Sky]

(|x>|x> + |y>|y>) / √2

Something is bothering me here.

Should the sign be "+" as I have written it? Or should the sign be "-"?

 

[Eye_in_the_Sky]

I see it now.

(|x>|x> + |y>|y>) / √2

Okay, I see it now.

When the polarization basis vectors of both particles are referenced to the same set of axes, the "+" sign applies.

Alternatively, one may prefer to write the joint state with respect to two different sets of axes such that each particle propagates in the "+z" direction of its own set. Say, for example, the two sets are related by a half turn about the x-axis. In that case, the "-" sign applies. But then, one is required to put tags (e.g. subscripts) on the basis vectors because the two pairs of linear polarization basis vectors are no longer the same; i.e.

|x>₁ corresponds to |x>₂ ,

but

|y>₁ corresponds to -|y>₂ ,

and the state is written

(|x>₁|x>₂ - |y>₁|y>₂) / √2 .

 

[ThomasT]

I agree that, if there are no hidden variables, then that expression reduces to P(A,B)=P(A)P(B) ... what is wrong with that as a definition of locality?

Maybe it's an ok definition of locality. Maybe the locality condition has nothing to do with why Bell inequalities are violated. Maybe common cause hidden variable assumption has nothing to do with why Bell inequalities are violated. Maybe there's another assumption underlying the construction of inequalities that is the real reason for their violation.

Bell's theorem seems to be based on the notion that the correlation between P(A,B) and Θ must be a linear one, as, for example, the archetypal Bell inequality, (1-P(Θ)) + (1-P(Θ)) <= 1-P(2Θ), where P(Θ) is the normalized rate of detection wrt some angular difference, Θ, of the polarizers.

I'm not sure where this notion (the assumption of linear correlation between P(A,B) and Θ) comes from, but it would account for violation of inequalities based on it.

Does it come from the notion that LR formalization of entanglement entails assignment of definite value(s) to λ? If so, that is a problem since λ can't be tracked (ie., it has no definite value at any given time).

Well, maybe LR formalization requires this, and maybe not. However, I think that a LR understanding of entanglement doesn't require it.

Consider a source producing pairs of counter-propagating photons entangled in polarization. The polarization, λ, is varying randomly from pair to pair with photon_A and photon_B of each pair polarized identically due to, presumably, emission by the same atom.

The normalized rates of detection are:

1. With no polarizers,

P(A) = P(B) = P(A,B) = 1


2. With 1 polarizer, b, at B,

P(A) = 1
P(B) = cos²(|b - λ|_avg) = cos²(45⁰) = .5
P(A,B) = .5


3. With 2 polarizers, b₁ and b₂, at B,

P(A) = 1
P(B) = cos²|b₁-b₂|
P(A,B) = cos²|b₁-b₂|


4. With 1 polarizer, a, at A, and 1 polarizer, b, at B,


P(A) = cos²(|a - λ|_avg) = cos2(45⁰) = .5
P(B) = cos²(|b - λ|_avg) = cos2(45⁰) = .5
P(A,B) = cos²|a-b|


The applicability of Malus Law above seems quite (LRly) understandable to me. If it's applicable in 3, then why not in 4?

If the foregoing makes any sense then consider also that the QM calculation wrt 4 incorporates Malus Law for the same reason that Malus Law applies to 3 -- crossed polarizers analyzing randomly polarized light. Nothing nonlocal about 3, is there? So, should QM be deemed a local theory?

Anyway, requiring a definite value for λ isn't a realistic requirement

So what is Bell's theorem supposed to be telling us that we couldn't have surmised without it?

 

[Frame Dragger]

Maybe it's an ok definition of locality. Maybe the locality condition has nothing to do with why Bell inequalities are violated. Maybe common cause hidden variable assumption has nothing to do with why Bell inequalities are violated. Maybe there's another assumption underlying the construction of inequalities that is the real reason for their violation.

Bell's theorem seems to be based on the notion that the correlation between P(A,B) and Θ must be a linear one, as, for example, the archetypal Bell inequality, (1-P(Θ)) + (1-P(Θ)) <= 1-P(2Θ), where P(Θ) is the normalized rate of detection wrt some angular difference, Θ, of the polarizers.

I'm not sure where this notion (the assumption of linear correlation between P(A,B) and Θ) comes from, but it would account for violation of inequalities based on it.

Does it come from the notion that LR formalization of entanglement entails assignment of definite value(s) to λ? If so, that is a problem since λ can't be tracked (ie., it has no definite value at any given time).

Well, maybe LR formalization requires this, and maybe not. However, I think that a LR understanding of entanglement doesn't require it.

Consider a source producing pairs of counter-propagating photons entangled in polarization. The polarization, λ, is varying randomly from pair to pair with photon_A and photon_B of each pair polarized identically due to, presumably, emission by the same atom.

The normalized rates of detection are:

1. With no polarizers,

P(A) = P(B) = P(A,B) = 1


2. With 1 polarizer, b, at B,

P(A) = 1
P(B) = cos²(|b - λ|_avg) = cos²(45⁰) = .5
P(A,B) = .5


3. With 2 polarizers, b₁ and b₂, at B,

P(A) = 1
P(B) = cos²|b₁-b₂|
P(A,B) = cos²|b₁-b₂|


4. With 1 polarizer, a, at A, and 1 polarizer, b, at B,


P(A) = cos²(|a - λ|_avg) = cos2(45⁰) = .5
P(B) = cos²(|b - λ|_avg) = cos2(45⁰) = .5
P(A,B) = cos²|a-b|


The applicability of Malus Law above seems quite (LRly) understandable to me. If it's applicable in 3, then why not in 4?

If the foregoing makes any sense then consider also that the QM calculation wrt 4 incorporates Malus Law for the same reason that Malus Law applies to 3 -- crossed polarizers analyzing randomly polarized light. Nothing nonlocal about 3, is there? So, should QM be deemed a local theory?

Anyway, requiring a definite value for λ isn't a realistic requirement

So what is Bell's theorem supposed to be telling us that we couldn't have surmised without it?

Science doesn't look kindly on surmise when a test and a theorem can be constructed instead...

I think you should just accept that you believe there is an underlying flaw, or ensemble of local hidden variables, and that Bell is hogwash. You'd be in the minority, but you're entitled to your opinion after all.

 

[ThomasT]

Science doesn't look kindly on surmise when a test and a theorem can be constructed instead...

I think you should just accept that you believe there is an underlying flaw, or ensemble of local hidden variables, and that Bell is hogwash. You'd be in the minority, but you're entitled to your opinion after all.

There's disagreement wrt the meaning of Bell's theorem and violation of Bell inequalities. Is it locality, or hidden variables, or some other assumption that we should be focusing on?

Since a local (if not quite realistic) understanding of Bell test results (via appropriate application of Malus Law) seems possible, I proposed that maybe the problem is the assumption that correlation between P(A,B) and Θ must be linear when, on its face, this assumption contradicts classical and quantum optical application of Malus Law.

This assumption follows from the requirement that LR model specify definite value of λ. But, since λ has no definite value at any given time, then this is an unwarranted requirement. Only the assumption of locally caused relationship or common property wrt entangled disturbances is necessary for local understanding of Bell test results and correct application of Malus Law to Bell test setups.

So, I propose that the reason why Bell inequalities are violated, and why this doesn't tell us anything about Nature, is due to their being based on the unwarranted assumption that, wrt a LR understanding, P(A,B) and Θ must be linearly correlated.

 

[Frame Dragger]

There's disagreement wrt the meaning of Bell's theorem and violation of Bell inequalities. Is it locality, or hidden variables, or some other assumption that we should be focusing on?

Since a local (if not quite realistic) understanding of Bell test results (via appropriate application of Malus Law) seems possible, I proposed that maybe the problem is the assumption that correlation between P(A,B) and Θ must be linear when, on its face, this assumption contradicts classical and quantum optical application of Malus Law.

This assumption follows from the requirement that LR model specify definite value of λ. But, since λ has no definite value at any given time, then this is an unwarranted requirement. Only the assumption of locally caused relationship or common property wrt entangled disturbances is necessary for local understanding of Bell test results and correct application of Malus Law to Bell test setups.

So, I propose that the reason why Bell inequalities are violated, and why this doesn't tell us anything about Nature, is due to their being based on the unwarranted assumption that, wrt a LR understanding, P(A,B) and Θ must be linearly correlated.

There is a disagreement on this forum; out in the world, there is very little as to what Bell means. Whether you accept or reject it is another matter, but it's hardly controversial. Most people, myself included, believe that BI's DO tell us something about nature, but most importantly they tell us what theories can match QM and in what fashion. The fact that it's all counterintuitive and weird doesn't change matters, at least, not for most. Some theoreticians do need to worry about alternatives, but to be blunt, it's looking bleak for them right now.

 

[DrChinese]

So, I propose that the reason why Bell inequalities are violated, and why this doesn't tell us anything about Nature, is due to their being based on the unwarranted assumption that, wrt a LR understanding, P(A,B) and Θ must be linearly correlated.

I still don't know what this means. Bell does not assume anything about LR other than LR itself and general equivalence to the predictions of QM (which of course leads to contradictions). So you still have not made much of a case for your perspective. And as Frame Dragger says, this is looking pretty bleak.

 

[ThomasT]

There is a disagreement on this forum; out in the world, there is very little as to what Bell means.

That might be true. Things like this are explored on PF in order to get a better understanding of them. Wasn't von Neumann's no HV theorem noncontroversial, sort of taken for granted, until knowledge of its flawed assumption became mainstream?

Whether you accept or reject it is another matter, but it's hardly controversial. Most people, myself included, believe that BI's DO tell us something about nature ...

Now is your chance to put in your own words what you think violations of BI's tell us about Nature, and why you think they tell us that.

... but most importantly they tell us what theories can match QM and in what fashion.

Isn't discovering the existence of underlying FTL propagations at least as important?

The fact that it's all counterintuitive and weird doesn't change matters, at least, not for most.

The application of Malus Law to Bell test preparations isn't counterintuitive, and no weirder than the results with a standard polariscope setup.

What is weird and counterintuitive is the assumption that the correlation between |a-b| and P(A,B) should be a linear one if the crossed polarizers, a and b, are jointly analyzing identically polarized members of randomly polarized pairs.

 

[DrChinese]

What is weird and counterintuitive is the assumption that the correlation between |a-b| and P(A,B) should be a linear one if the crossed polarizers, a and b, are jointly analyzing identically polarized members of randomly polarized pairs.

The only linear relationship I can think of in this context is a common Local Realistic boundary condition. I.e. what values a local realistic theory could predict and NOT run afoul of a Bell Inequality. Is that what you are referring to?

If so, I have some comments on that surrounding experiment. If not, can you explain what linear correlation you are referring to?

 

[Eye_in_the_Sky]

You have lost me here ...

... Hopefully the following approach will make what I am trying to say clearer.
___________________

If I were asked to write down a theorem associated with stage 2 of Bell's argument, I would write down something like this:

Theorem 2: Suppose T is a fundamentally deterministic theory which has the PC-feature. Then, Bell's inequality holds in T, if T is local.

[NOTE: I have merely exchanged the term "realistic" (in the expression "local realistic") with the words "fundamentally deterministic".]
___________________

If I were asked to write down a theorem associated with stage 1 of Bell's argument in the case where that argument is formulated along the lines of the original language of EPR, I would write down something like this:

Theorem 1 (old version): If Quantum Mechanics is local, and counterfactual definiteness is a valid principle, then Quantum Mechanics is incomplete.

On the other hand, if I were asked to write down a theorem associated with stage 1 of Bell's argument in the case where that argument is formulated in terms of the joint-probability-function of the particle pair as calculated at the level of a physical theory, I would write down something like this:

Theorem 1: Suppose T is a complete stochastic theory which has the PC-feature. Then, if T satisfies the "Bell Locality" condition, T is fundamentally deterministic. In that case T is local in the sense of Theorem 2.
___________________

As I tried to point out in post #452, the "Bell Locality" condition pertains to "stage 1", not "stage 2". That is to say, "Bell Locality" was designed specifically for "stage 1". It was designed to elevate "Theorem 1 (old version)" to the rank of Theorem 1, the new version. What does this accomplish? It allows us to link Theorem 1 to Theorem 2, thereby yielding:

Theorem 3: Suppose T is a complete stochastic theory which has the PC-feature. Then, Bell's inequality holds in T, if T satisfies the "Bell Locality" condition.

Compare this to:

Theorem 2: Suppose T is a fundamentally deterministic theory which has the PC-feature. Then, Bell's inequality holds in T, if T is local.

As you can see, Theorem 3 is a generalization of Theorem 2. This is because the category of "complete stochastic" includes the category of "fundamentally deterministic" as a particular case – i.e. it is the case of a stochastic theory for which all of the irreducible probabilities are always either 0 or 1. In that case, "Bell Locality" becomes "locality" in the sense of Theorem 2.

But Theorem 3 can still be refined. This is because "locality" (in the sense of special relativity) implies "Bell Locality". Or equivalently, a violation of "Bell Locality" implies "nonlocality".

So, I would rewrite Theorem 3 as:

Bell's Inequality Theorem: Suppose T is a complete stochastic theory which has the PC-feature. Then, if Bell's inequality is violated in T, T is nonlocal.

[NOTE: In such a theory T, there is no assumption of "hidden variables" of any kind.]
___________________

Of course, I have not stated definitions of "complete stochastic" and "Bell Locality". Nor have I established the truth of Theorem 1. Nor have I shown that "locality" (in the sense of special relativity) implies "Bell Locality".

My purpose in the above was just to identify the conceptual context in which the "Bell Locality" condition applies and to specify its point of application within that context. That point is in "stage 1, Theorem 1".

 

[SpectraCat]

... Hopefully the following approach will make what I am trying to say clearer.
___________________

If I were asked to write down a theorem associated with stage 2 of Bell's argument, I would write down something like this:

Theorem 2: Suppose T is a fundamentally deterministic theory which has the PC-feature. Then, Bell's inequality holds in T, if T is local.

[NOTE: I have merely exchanged the term "realistic" (in the expression "local realistic") with the words "fundamentally deterministic".]
___________________

If I were asked to write down a theorem associated with stage 1 of Bell's argument in the case where that argument is formulated along the lines of the original language of EPR, I would write down something like this:

Theorem 1 (old version): If Quantum Mechanics is local, and counterfactual definiteness is a valid principle, then Quantum Mechanics is incomplete.

On the other hand, if I were asked to write down a theorem associated with stage 1 of Bell's argument in the case where that argument is formulated in terms of the joint-probability-function of the particle pair as calculated at the level of a physical theory, I would write down something like this:

Theorem 1: Suppose T is a complete stochastic theory which has the PC-feature. Then, if T satisfies the "Bell Locality" condition, T is fundamentally deterministic. In that case T is local in the sense of Theorem 2.
___________________

As I tried to point out in post #452, the "Bell Locality" condition pertains to "stage 1", not "stage 2". That is to say, "Bell Locality" was designed specifically for "stage 1". It was designed to elevate "Theorem 1 (old version)" to the rank of Theorem 1, the new version. What does this accomplish? It allows us to link Theorem 1 to Theorem 2, thereby yielding:

Theorem 3: Suppose T is a complete stochastic theory which has the PC-feature. Then, Bell's inequality holds in T, if T satisfies the "Bell Locality" condition.

Compare this to:

Theorem 2: Suppose T is a fundamentally deterministic theory which has the PC-feature. Then, Bell's inequality holds in T, if T is local.

As you can see, Theorem 3 is a generalization of Theorem 2. This is because the category of "complete stochastic" includes the category of "fundamentally deterministic" as a particular case – i.e. it is the case of a stochastic theory for which all of the irreducible probabilities are always either 0 or 1. In that case, "Bell Locality" becomes "locality" in the sense of Theorem 2.

But Theorem 3 can still be refined. This is because "locality" (in the sense of special relativity) implies "Bell Locality". Or equivalently, a violation of "Bell Locality" implies "nonlocality".

So, I would rewrite Theorem 3 as:

Bell's Inequality Theorem: Suppose T is a complete stochastic theory which has the PC-feature. Then, if Bell's inequality is violated in T, T is nonlocal.

[NOTE: In such a theory T, there is no assumption of "hidden variables" of any kind.]
___________________

Of course, I have not stated definitions of "complete stochastic" and "Bell Locality". Nor have I established the truth of Theorem 1. Nor have I shown that "locality" (in the sense of special relativity) implies "Bell Locality".

My purpose in the above was just to identify the conceptual context in which the "Bell Locality" condition applies and to specify its point of application within that context. That point is in "stage 1, Theorem 1".

Wait ... you thought that would make it clearer??

In all seriousness, that may be clearer in the sense that you have laid it all out, but it's going to take a while for me to wade through it all. Some definitions would help .. I can probably look them up in past posts, but it would be easier if you could reiterate the following:

What is the PC-feature?

What do you mean by a complete stochastic theory in the context of Theorems 1 and 3 (new version)?

 

[Eye_in_the_Sky]

Wait ... you thought that would make it clearer??

In all seriousness, that may be clearer in the sense that you have laid it all out ...

Sorry about that. ... Yes, "clearer" ONLY in the sense that it has all been laid out.

Some definitions would help ..

I had hoped to avoid the labour of having to define the terms, thinking that it would suffice to present things in a way that it could all be followed at the linguistic level of merely matching words.

As I said:

"My purpose ... was just to identify the conceptual context in which the 'Bell Locality' condition applies and to specify its point of application within that context."

I can probably look them up in past posts, but it would be easier if you could reiterate the following:

What is the PC-feature?

Okay, the "PC-feature" is easy enough. In words it goes like this:

When Alice and Bob's settings are the same, their outcomes are opposite with probability equal to 1.

Thus, "PC" is short for "perfect anti-correlation at equal settings".
___________________________________

What do you mean by a complete stochastic theory in the context of Theorems 1 and 3 (new version)?

As for "complete stochastic", that is something rather more involved. As yet the term has not been defined in this thread.

But since you are asking, I will put down the words.

To say that a stochastic theory is complete means:

1) With respect to a given spacelike hypersurface S, the theory correctly identifies all possible "states" of the system, each of which constitutes "a complete description" in terms of local beables along S. Let Λ_S denote the set of all such states.

2) For any given state λЄΛ_S, the value which the theory assigns to P(X|Y,λ) – i.e. "the probability of X, given Y when the state is λ" – takes into account all of the relevant information contained the condition Y and the complete state λ.

I have selected the term "complete stochastic" and assembled its definition on the basis of what is written in the following reference:

Travis Norsen, "Bell Locality and the Nonlocal Character of Nature"

(In fact, in that reference you will find a proof of what I have referred to as Theorem 1.)

Next ... it is essential to recognize the following consequence of the above definition:

From the definition above, it follows that in a complete stochastic theory all probabilities of the form P(X|Y,λ) assigned by the theory are irreducible. That is to say, these elements of randomness ascribed by the theory belong to the "real physical situation" as an intrinsic property. These probabilities do not in any way arise on account of a lack of information concerning the relevant facts upon which physical predictions are to be made.

And for further emphasis, here is how Maudlin puts it:

"... any theory which takes stochastic laws seriously at the ontological level must take ascriptions of probability equally seriously. If we believe that a photon approaching a polarizer has a 50 percent chance of passing and a 50 percent chance of being absorbed, and that these probabilities are reflections not of our ignorance but of a basic indeterminism in nature, then we must take an event’s having a particular probability as a basic physical fact. In this case a change from 50 percent probability of passage to 99 percent probability is a physical change."

The above quote, I have taken from this reference:

http://arxiv.org/PS_cache/arxiv/pdf/0707/0707.0401v1.pdf

 

[DrChinese]

I have selected the term "complete stochastic" and assembled its definition on the basis of what is written in the following reference:

Travis Norsen, "Bell Locality and the Nonlocal Character of Nature"

(In fact, in that reference you will find a proof of what I have referred to as Theorem 1.)

Travis is brilliant, and I reference some of his work from my own web page. But I would not consider his as a good reference for definitions like this. If you start from the perspective that locality/separability is the fundamental premise of Bell, all you end up seeing is a proof of non-locality - which he does.

If you start from a premise of realism/counterfactual definiteness (CD) - as I tend to - then you see a proof of contextuality (non-realistic interpretations). I believe it is correct to see the conjunction of assumptions in Bell: locality + realism. Look to other theorems for more than this.

 

[ThomasT]

The only linear relationship I can think of in this context is a common Local Realistic boundary condition. I.e. what values a local realistic theory could predict and NOT run afoul of a Bell Inequality. Is that what you are referring to?

Yes, that's it. It's the specific reason why BI's are violated, right? OK, so the discussion regarding the meaning of Bell's theorem and violation of BI's has to do with what, exactly, this boundary condition comes from.

If so, I have some comments on that surrounding experiment.

OK.

 

[DrChinese]

Yes, that's it. It's the specific reason why BI's are violated, right? OK, so the discussion regarding the meaning of Bell's theorem and violation of BI's has to do with what, exactly, this boundary condition comes from.

Well... Let's look at this as a boundary. Bell assumed that the LRT would need to match the predictions of QM. So if you follow that approach, there is no boundary. LR is simply ruled out, and that is the end of it. Of course, you must prove that the predictions of QM are supported to get this result.

Next, there is the boundary you describe. This comes from a relaxed assumption. The relaxed assumption is that the perfect correlations of EPR are in effect, but the Local Realistic Theory does not match QM. This yields the Bell Inequality, which is essentially the "closest" any LR can come to the predictions of QM. Of course it still wouldn't match the predictions of QM. But it would not violate Bell's Theorem/Inequality.

So the boundary condition is a result of Bell pointing out that QM and LR are incompatible as to their predictions. It really has nothing to say about assumptions within those theories OTHER than the LR being realistic/separable. Amd there is certainly no requirement that the LR have the linear relationship you mention. The function could be anything, since it doesn't match QM (or experiment).

 

[yoda jedi]

What do YOU mean by the word "covariant"?
Anyway, with the usual definition of that word, there is a way to make the wave function covariant:
http://xxx.lanl.gov/abs/1002.3226

context independent.

 

[Eye_in_the_Sky]

Back in post #468, I wrote down a definition of the term "complete" in connection with the notion of a complete stochastic theory.

... There is a 'bug' in that definition. I will show you where:

1) With respect to a given spacelike hypersurface S, the theory correctly identifies all possible "states" of the system, each of which constitutes "a complete description" in terms of local beables along S. Let Λ_S denote the set of all such states.

The words in bold face do not belong in the definition. Those words should be deleted. The definition should read as follows.

To say that a stochastic theory is complete means:

1) With respect to a given spacelike hypersurface S, the theory correctly identifies all possible "states" of the system, each of which constitutes "a complete description" along S. Let Λ_S denote the set of all such states.

2) For any given state λЄΛ_S, the value which the theory assigns to P(X|Y,λ) – i.e. "the probability of X, given Y when the state is λ" – takes into account all of the relevant information contained the condition Y and the complete state λ.

... Soon, I will go back over everything and try to determine whether or not those words which 'snuck' into the definition were entirely superfluous. If they were not, I will try to determine their proper place.

 

[Eye_in_the_Sky]

...

... but it's going to take a while for me to wade through it all.

... Please, ONLY do so if it pleases you to do so.

As I said:

"As I said:

'My purpose ... was just to identify the conceptual context in which the "Bell Locality" condition applies and to specify its point of application within that context.' "

 

[Eye_in_the_Sky]

If you start from a premise of realism/counterfactual definiteness (CD) - as I tend to - then you see a proof of contextuality (non-realistic interpretations).

Dr. Chinese, I do not understand what you mean by this. Can you explain it?
_______________________

I believe it is correct to see the conjunction of assumptions in Bell: locality + realism.

In connection with "stage 2" of Bell's argument, I agree with you. But in connection with "stage 1" I do not see it.

Now that I have fixed-up the definition of a "complete" stochastic theory, Quantum Mechanics can be admitted as a candidate. The λ's all have the form

λ = [ψ₁(x,t_o) + ψ₂(x,t_o)] ⊗ |singlet> ,

where the spacelike hypersurface S is given by t=t_o in the mutual rest frame of Alice and Bob.

Over the next month or so, I will put some time into trying to make a determination of whether or not Theorem 1 (as I have written it) is in fact valid. I will also check to see that I have properly understood the true meaning of "Bell Locality".

 

[DrChinese]

Dr. Chinese, I do not understand what you mean by this. Can you explain it?

I had said: If you start from a premise of realism/counterfactual definiteness (CD) - as I tend to - then you see a proof of contextuality (non-realistic interpretations).

Understand that my argument is not rigorous. I am simply saying that when you start from one side, that is what you tend to see and ignore much other material. That is certainly what Travis does, as he denies that realism is a part of the Bell argument despite my pointing out to him the exact spot it is introduced many times.

So a good example of my argument is Mermin's example of the "instruction sets". That is the CD assumption. Don't need to assume separability for that, just the usual realistic requirement. In my mind, this argument applies without regard to locality. As I say that is just a perspective, and should not be taken too literally. However, there are a number of authors - certainly as respected as Norsen - who make this argument more strongly. I'll see if I can dig up a reference. But keep in mind that neither of the "Bell only requires locality assumption" or "Bell only requires realism assumption" schools is considered generally accepted. The general conclusion is that both assuptions are present in Bell.

 

[Eye_in_the_Sky]

I had said: If you start from a premise of realism/counterfactual definiteness (CD) - as I tend to - then you see a proof of contextuality (non-realistic interpretations).

Understand that my argument is not rigorous.

A non-rigorous argument can have merit.

I am simply saying that when you start from one side, that is what you tend to see and ignore much other material.

If you "start from one side" and derive Bell's inequality from it (and the derivation is correct), then you have found sufficient conditions for Bell's inequality to hold. Relative to those conditions, all other conditions are not necessary.

I am simply saying that when you start from one side, that is what you tend to see and ignore much other material. That is certainly what Travis does, as he denies that realism is a part of the Bell argument despite my pointing out to him the exact spot it is introduced many times.

That would be because you are pointing out the spot in "stage 2" of Bell's argument. But (according to Travis) already at the very beginning of that stage, 'realism' has been established as a consequence of three other premises: 'completeness', 'PC', and 'Bell Locality'. Travis's proof is "stage 1" of Bell's argument as Bell ultimately intended it to be: 'realism' follows as a consequence of the "stage 1" argument.

A "consequence" ... do you understand that? ... a "consequence".

So in order to debunk Travis's claim, one needs to directly address the argument of "stage 1" and show that 'realism' cannot be derived from the conjunction of 'completeness', 'PC', and 'Bell Locality' (as Travis claims it can) – i.e. either there is some flaw in the argument, or 'realism' has been smuggled into it.

So a good example of my argument is Mermin's example of the "instruction sets". That is the CD assumption.

"Counterfactual definiteness" is a weaker premise than "instruction sets".

"Counterfactual definiteness" is the assumption that there would have been definite outcomes in the counterfactual cases (without necessarily assigning specific values to those outcomes).

"Instruction sets" is the assumption in which the definite outcomes in (at least some of) the counterfactual cases are assigned specific values.

I am not familiar with Mermin's example. Is this it?

David Mermin’s EPR gedanken experiment

Yes, I think it must be.

Don't need to assume separability for that, just the usual realistic requirement. In my mind, this argument applies without regard to locality.

What you are saying is wrong.

"Instruction sets" always require "separability". This is because each particle is assigned its own separate set of instructions. The joint state is separable.

Moreover, Mermin's example is "local". This because each particle is assigned its instructions at the source and there is no communication between wings.

So, Mermin's example is a particular instance of the general principle that "local classical instruction sets" cannot account for all of the quantum correlation predictions.

As I say that is just a perspective, and should not be taken too literally. However, there are a number of authors - certainly as respected as Norsen - who make this argument more strongly. I'll see if I can dig up a reference.

Do you mean an argument in support of the following claim?

Some form of 'realism' must necessarily be assumed in order to arrive at Bell's inequality.

If so, then by all means find some references.

But keep in mind that neither of the "Bell only requires locality assumption" or "Bell only requires realism assumption" schools is considered generally accepted.

There is no such thing as a "Bell only requires realism assumption" school. Belief in the existence of such a school is DELUSIONAL.

 

[DrChinese]

1. A non-rigorous argument can have merit.

2. A "consequence" ... do you understand that? ... a "consequence".

So in order to debunk Travis's claim, one needs to directly address the argument of "stage 1" and show that 'realism' cannot be derived from the conjunction of 'completeness', 'PC', and 'Bell Locality' (as Travis claims it can) – i.e. either there is some flaw in the argument, or 'realism' has been smuggled into it.

3. "Counterfactual definiteness" is a weaker premise than "instruction sets".

"Counterfactual definiteness" is the assumption that there would have been definite outcomes in the counterfactual cases (without necessarily assigning specific values to those outcomes).

"Instruction sets" is the assumption in which the definite outcomes in (at least some of) the counterfactual cases are assigned specific values.

4. I am not familiar with Mermin's example. Is this it?

David Mermin’s EPR gedanken experiment

Yes, I think it must be.


5. "Instruction sets" always require "separability". This is because each particle is assigned its own separate set of instructions. The joint state is separable.

Moreover, Mermin's example is "local". This because each particle is assigned its instructions at the source and there is no communication between wings.

So, Mermin's example is a particular instance of the general principle that "local classical instruction sets" cannot account for all of the quantum correlation predictions.


6. Do you mean an argument in support of the following claim?

Some form of 'realism' must necessarily be assumed in order to arrive at Bell's inequality.

If so, then by all means find some references.


7. There is no such thing as a "Bell only requires realism assumption" school. Belief in the existence of such a school is DELUSIONAL.

1. Sure, and I make them all the time.


2. Well actually I don't have to do anything to debunk Norsen. He has some followers and I respect that. There are plenty of others who disagree, and some have in fact already debunked his general line of thinking.

The point is that IF you assume completeness - which Bell doesn't - then perhaps you can get X conclusion. EPR did exactly that.


3. Sorry, to me CD = realism and yes I know that it doesn't to some people. If you can give me a specific example of a relevant difference, that would be wonderful. Meanwhile, most attempts to explain the difference end up being a semantic exercise that puts me to sleep. If it has a value but it is unknown (and perhaps unknowable), that is one thing. If it has no definite value, that is another.

I guess there are shades of gray in between, but they actually don't matter. Because Bell assumes realism, that there is a specific outcome possible for an observation that is not performed. Which is the definition of Bell realism. Same essential definition as for EPR's element of reality, by the way.

QM itself is not CD (or realistic) in the formalism (HUP's non-commuting operators).


4. Yes, or my own version:

http://drchinese.com/David/Bell_Theorem_Easy_Math.htm

Naturally, it is more eloquent.


5. Whoa, I wouldn't agree that the instruction set implies separability! That's sort of the point, actually. I am saying - non-rigorously - that where you start colors what you conclude. You see separability, while I see realism.

Maybe the system is non-local realistic (and not necessarily separable because they share a single instruction set) ! But if it were, you couldn't replicate the QM predictions. Now I know you are going to object about BM, but that is not what I am talking about; as BM is not only non-local realistic but it is ALSO contextual. So clearly, somehow, there is a group of people who see the need to support contextuality along with realism. (I non-rigorously accept contextuality and reject realism. But of course, maybe I am wrong.) But either way, Bell stands.


6. Sure, how about this member of your "non-existent" school:

A Bell Theorem with no locality assumption (2006), C. Tresser.
http://arxiv.org/abs/quant-ph/0608008

A pint of beer says you debate the merit of the paper BEFORE you acknowledge the existence of the school... and that you are flat out incorrect on this point.


7. Well, I think we found your hot spot. See 6.

 

[yoda jedi]

There is no such thing as a "Bell only requires realism assumption" school. Belief in the existence of such a school is DELUSIONAL.

...If one uses a broader and more common definition of locality... (C. Tresser)

that`s the problem, people confusing ontology with semantics, distorting, stretching or whatever...

imagine:
"or using a shorter and less common definition of realism" or
"realism according groblacher" or " a very bizzarre notion of locality"

accommodative opinions.

Some form of 'realism' must necessarily be assumed.

indeed, with "NOTHING" nothing can be conceived.

 

[Frame Dragger]

indeed, with "NOTHING" nothing can be conceived.

Well, you could be into that whole "let there be light" bit, but really I like your explanation much better.