A classical challenge to Bell's Theorem?

[Gordon Watson]

This post moved from "Nick Herbert's proof?"

https://www.physicsforums.com/showthread.php?t=589134

at the request of the OP.

When I said randomness I did not refer to unpredictable (experimental) phenomena. When you toss a coin, the result depends deterministically on the initial conditions. That is familiar everyday randomness which is merely practical unpredictability.

QM on the other hand says that nature is intrinsically random. There is no hidden layer "explaining" what actually will happen. The randomness is spontaneous. Inexplicable. Without antecedent. Effects without a cause.

..
Effects without a cause in Herbert's experiment? I presume that you believe that some quantum events have no cause; not classical effects?

So I would welcome any and all comments and calculations on the following scenario, based on a typical Bell-test set-up and the CHSH inequality.

We replace the quantum-entanglement-producing source with a classical source which sends a short pulse of light to Alice and Bob each day (over many years), each pulse correlated by having the same linear-polarization; though each day the common pulse polarization-orientation is different .

Let x denote any variable of your choosing. Then (as in a standard Bell-analysis) Alice's results are represented by (1) A(a, x) = ±1 where a is any analyzer orientation of her choosing; Bob's by (2) B(b, x) = ±1 where b is any analyzer orientation of his choosing; (3) 0 \leq ρ(x); (4) ∫ρ(x) dx = 1.

Please (after Bell, showing all your workings), calculate the expectation: (5) E(AB) = ∫AB ρ(x) dx.

Please provide the maximum value achievable for the CHSH inequality under these conditions.

With thanks in advance,

GW

EDIT added with move: I'd like to understand how physicists and mathematicians deal with the above wholly classical setting in the context set by Bell (1964) when arriving at his theorem. Thanks.

 

[Gordon Watson]

Moved from "Nick Herbert's proof?" https://www.physicsforums.com/showthread.php?t=589134

GW: if you don't tell me the functions A(a,x), B(b,x) and rho(x) I obviously cannot calculate E(A(a)B(b)). However I can tell you that A(a1)B(b1)-A(a1)B(b2)-A(a2)B(b2)-A(a2)B(b1) (a function now only of x) can only take the values +2, 0 and -2. One way to see that is to note that the product of the four terms A(a1)B(b1), A(a1)B(b2), A(a2)B(b2), A(a2)B(b1) is +1, hence an even number of these four terms is equal to +1 and an even number is equal to -1. Therefore if A(a1)B(b1)=+1, then A(a1)B(b2)+A(a2)B(b2)+A(a2)B(b1) = +3 or -1; if A(a1)B(b1)=-1, then A(a1)B(b2)+A(a2)B(b2)+A(a2)B(b1) = +1 or -3. Now just check the 2x2 combinations. Hence the average of A(a1)B(b1)-A(a1)B(b2)-A(a2)B(b2)-A(a2)B(b1) (averaged over x) cannot exceed 2, either.

..
Thanks for this comprehensive reply. It is appreciated.

BUT: Given the wholly classical setting in my example, and the use of Bell's (1964) formulation, I thought there was enough info there for physicists and mathematicians to proceed? Or (at least), explain why they cannot?

 

[Gordon Watson]

Moved from "Nick Herbert's proof?" https://www.physicsforums.com/showthread.php?t=589134

Effects without a cause (not just in Herbert's experiment, but in QM in general): I mean that quantum events in general have no cause. By quantum events I mean results of measuring pure states which are not certain, but for which QM can only tell us the probability.

Thanks. Very puzzling, as worded, so I need to think about it.

To be clear: When the Green light (say) blinks on a detector in a Bell-test, are you saying that this event has, ultimately (when analyzed), no upstream-cause (e.g., in ordinary 3-space)?

Also: To "measure a pure state" is to perturb it, right? A complicating factor?

 

[billschnieder]

if you don't tell me the functions A(a,x), B(b,x) and rho(x) I obviously cannot calculate E(A(a)B(b)). However I can tell you that A(a1)B(b1)-A(a1)B(b2)-A(a2)B(b2)-A(a2)B(b1) (a function now only of x) can only take the values +2, 0 and -2. One way to see that is to note that the product of the four terms A(a1)B(b1), A(a1)B(b2), A(a2)B(b2), A(a2)B(b1) is +1, hence an even number of these four terms is equal to +1 and an even number is equal to -1. Therefore if A(a1)B(b1)=+1, then A(a1)B(b2)+A(a2)B(b2)+A(a2)B(b1) = +3 or -1; if A(a1)B(b1)=-1, then A(a1)B(b2)+A(a2)B(b2)+A(a2)B(b1) = +1 or -3. Now just check the 2x2 combinations. Hence the average of A(a1)B(b1)-A(a1)B(b2)-A(a2)B(b2)-A(a2)B(b1) (averaged over x) cannot exceed 2, either.

Let us change the original scenario a little as follows: We remove the source altogether and replace it with a mystical being who governs a mystical parameter (x) which combines with their chosen angles to produce a +/-1 result. Each day over many years, he instantly decides what parameter (x) is, the instant before Alice and Bob make their measurements, whoever does it first. The only condition being that the same (x) parameter is governing both experiments.

Then (as in a standard Bell-analysis) Alice's results are represented by (1) A(a, x) = ±1 where a is any analyzer orientation of her choosing; Bob's by (2) B(b, x) = ±1 where b is any analyzer orientation of his choosing; (3) 0 ≤ ρ(x); (4) ∫ρ(x) dx = 1.

I wonder what the CHSH inequality will look like. I can bet it will be identical to the one derived by gill1109 above, even though the scenario is manifestly non-local. What gives?

 

[Gordon Watson]

Let us change the original scenario a little as follows: We remove the source altogether and replace it with a mystical being who governs a mystical parameter (x) which combines with their chosen angles to produce a +/-1 result. Each day over many years, he instantly decides what parameter (x) is, the instant before Alice and Bob make their measurements, whoever does it first. The only condition being that the same (x) parameter is governing both experiments.

Then (as in a standard Bell-analysis) Alice's results are represented by (1) A(a, x) = ±1 where a is any analyzer orientation of her choosing; Bob's by (2) B(b, x) = ±1 where b is any analyzer orientation of his choosing; (3) 0 ≤ ρ(x); (4) ∫ρ(x) dx = 1.

I wonder what the CHSH inequality will look like. I can bet it will be identical to the one derived by gill1109 above, even though the scenario is manifestly non-local. What gives?

Bill, you talkin' to me? (In that you cite gill1109.)

1. Not sure about your mystical being? Purpose =? (Is something more needed to clarify the OP?)

2. The CHSH inequality formulation will be the same, imho: since the experimental outcomes are ±1, no matter the settings a, b, etc.

3. To clarify the OP (if that's your issue): Having derived the expectation E(AB) for the classical setting -- from your functions for A and B = ±1 -- what then is the related maximum value that that classical setting might yield for the CHSH inequality? That is: What a, b, c, d settings yield the maximum value in the CHSH formula, and what is that maximum?

4. Is it gill1109's +2?

5. Did you mean to say "the scenario is manifestly LOCAL"?

6. So -- addressing your "What gives" -- just give me your answers to the OP -- or tell me why you can't. Especially as it seems that Bell might think you can; the given situation being wholly classical and involving no more than Bell's proposed (1964, etc.) analytical formulation.

 

[billschnieder]

Bill, you talkin' to me? (In that you cite gill1109.)

1. Not sure about your mystical being? Purpose =? (Is something more needed to clarify the OP?)

2. The CHSH inequality formulation will be the same, imho: since the experimental outcomes are ±1, no matter the settings a, b, etc.

3. To clarify the OP (if that's your issue): Having derived the expectation E(AB) for the classical setting -- from your functions for A and B = ±1 -- what then is the related maximum value that that classical setting might yield for the CHSH inequality? That is: What a, b, c, d settings yield the maximum value in the CHSH formula, and what is that maximum?

4. Is it gill1109's +2?

5. Did you mean to say "the scenario is manifestly LOCAL"?

6. So -- addressing your "What gives" -- just give me your answers to the OP -- or tell me why you can't. Especially as it seems that Bell might think you can; the given situation being wholly classical and involving no more than Bell's proposed (1964, etc.) analytical formulation.

Sorry for hijacking your thread Gordon, I was just responding to the portion by gill1109. To answer your questions, and more on topic.

- Without specifying the method by which the common pulse orientations are chosen, it is not possible to calculate an expectation value.Without ρ(x) we are hopeless to calculate a meaningful E(AB) even if A(a,x) and B(b,x) are clearly specified.

- The maxumum attainable is of course +2 as gill1109 calculated. Aside:

However, note the following extremely important point

A(a1)B(b1)-A(a1)B(b2)-A(a2)B(b2)-A(a2)B(b1)

→ A(a1)[B(b1)-B(b2)] - A(a2)[B(b2)-B(b1)]
→ A(a1)[B(b1)-B(b2)] + A(a2)[B(b1)-B(b2)]
→ [A(a1) + A(a2)]*[B(b1)-B(b2)] ---> **!

if A(a1) = A(a2) = +1, and B(b1) = -B(b2) = +1 Or,
A(a1) = A(a2) = -1 and B(b1) = -B(b2) = -1, we obtain the maxium of 2.If A(a1) = -A(a2) = ±1, OR B(b1) = B(b2) = ±1, we get a value of zero.

And if A(a1) = A(a2) = -1 and B(b1) = -B(b2) = +1 Or
A(a1) = A(a2) = +1 and B(b1) = -B(b2) = -1, we obtain the minimum of 2.

This may seem like a pointless way to arrive at the same result as gill1109 except it is obvious from the emphasized expresion that the original 4 terms (A(a1)B(b1), A(a1)B(b2), A(a2)B(b2), A(a2)B(b1)) of products in the inequality originate from only 4 functions (A(a1), B(b1), A(a2), B(b2)) which must be factorizable. You can not use 4 different runs of an experiment (i, j, k, l) to obtain results from 8 functions (A(a1i), B(b1i), A(a1j), B(b2j), A(a2k), B(b2k), A(a2l), B(b1l)) and expect the inequality to work. It is a simple exercise to verify that for the case where 4 different runs of the experiment are performed, the maximum of the expression will be

A(a1i)B(b1i) - A(a1j)B(b2j) -A(a2k)B(b2k) -A(a2l)B(b1l) <= 4

NOT 2.

Some naively leave out the experiment identifyers (i,j,k,l) and fool themselves into thinking the result can be factorized.

In order for the results from 4 different experiments to be factorizable the following equalities must hold
A(a1i) = A(a1j)
A(a2k) = A(a2l)
B(b1i) = B(b1l)
B(b2j) = B(b2k)

Practically, this means if the experimental results consisted of a list of numbers (+1, -1) for each function and you obtained 8 columns for 4 different experiments, the data MUST be sortable such that 4 of the columns are duplicates, not only in the numbers of +1s and -1s but also in the switching pattern.

Therefore it is not sufficient that A*B for one experiment gives you a certain expection value for the paired product. For the inequality to have a maximum of 2, rather than 4, the value of one pair must constrain the value of a different pair in some manner.

 

[ThomasT]

I'm not saying that there's anything wrong with anything that Gordon Watson and billschnieder have said. And maybe one or both of their approaches will one day explain BI violations in a way that an ignorant layman such as myself might understand. However, currently, I don't think so. I think that a true understanding of why BI violations don't inform wrt the deep reality is more subtle, and yet simpler, than either have yet pinpointed.

Just in the humble, and perhaps quite wrong, opinion of an ignorant layperson.

But, yeah, the factorizability of the entangled state would seem to be the key to it. Because this is a composite of the functions that determine individual detection. And the variable that determines individual detection isn't relevant wrt the rate of coincidental detection.

Just a certain point of view. Maybe it's important, maybe not.

 

[gill1109]

Bill Schnieder: nothing gives. The CHSH inequality is true. Locality in all these discussions concerns only the measurement settings and the measurement outcomes. The hidden variables x or lambda may as well be known throughout the whole universe. You can think of them, if you like, as being "in the measurement apparatus" and "in the particles". But you don't have to think that way. The real assumption in deriving CHSH is the "reality" and time-space location of the outcomes of the unperformed measurements, alongside of those actually performed; and the freedom of the experimenters to choose which measurements to perform.

 

[gill1109]

Bill Schnieder: you wanted outcomes of four experiments to match exactly (your i,j,k,l indices). The way I think about it, in one run of the experiment there are potential outcomes A1, A2, B1, B2. Alice and Bob each toss a coin to choose which outcome to actually observe (A1 or A2, B1 or B2). Then this is repeated many times. We assume that their coin tosses are independent of the physical systems generating A1, A2, B1, B2. Then the average of, say, A1 times B2 over all runs will hardly differ from the average over those runs where Alice chose "1", Bob chose "2". The averages over all runs satisfy CHSH. Hence the observed averages do too, up to statistical variation.

 

[Gordon Watson]

Sorry for hijacking your thread Gordon, I was just responding to the portion by gill1109. To answer your questions, and more on topic.

- Without specifying the method by which the common pulse orientations are chosen, it is not possible to calculate an expectation value.Without ρ(x) we are hopeless to calculate a meaningful E(AB) even if A(a,x) and B(b,x) are clearly specified.

- The maximum attainable is of course +2 as gill1109 calculated.

Bill, I see no hijacking; so, for me, no problem at all. Then, leaving your 'aside' aside for the moment:

The pulse orientations are many (a very large number) and random in orientation. In my terms: 'a uniform distribution' (the same distribution that we'd expect with Bell's λ, though it's a different beast). Is that fair enough?

I think that leaves you needing to specify A and B, consistent with Bell's 'analytical protocol' for the study of local realism: each equal to ± 1. Or telling me why you cannot?

 

[Gordon Watson]

I'm not saying that there's anything wrong with anything that Gordon Watson and billschnieder have said. And maybe one or both of their approaches will one day explain BI violations in a way that an ignorant layman such as myself might understand. However, currently, I don't think so. I think that a true understanding of why BI violations don't inform wrt the deep reality is more subtle, and yet simpler, than either have yet pinpointed.

Just in the humble, and perhaps quite wrong, opinion of an ignorant layperson.

But, yeah, the factorizability of the entangled state would seem to be the key to it. Because this is a composite of the functions that determine individual detection. And the variable that determines individual detection isn't relevant wrt the rate of coincidental detection.

Just a certain point of view. Maybe it's important, maybe not.

Dear ThomasT, the OP is intended to be as 'subtle and as simple' as it gets! (Perhaps it fails?)

But: The OP is wholly classical. And you are not (entirely) an ignorant layman (being familiar with Malus, at least; as well as the point that you make right here, above).

So the challenge remains. That is: Time to do some basic maths! (And cut the words?)

 

[DrChinese]

Gordon, this is a straight classical setup, so of course the CHSH will have its traditional upper limit and no experiment will exceed it (as Richard says, within normal statistical deviation).

Specifically, the value of the function (5) E(AB) is .25 + .5(cos^2(A-B)) which is the classical expectation when there is separability.

To get statistics in which CHSH is violated, you must have entanglement. So really, what point are you trying to make?

 

[Gordon Watson]

Gordon, this is a straight classical setup, so of course the CHSH will have its traditional upper limit and no experiment will exceed it (as Richard says, within normal statistical deviation).

Specifically, the value of the function (5) E(AB) is .25 + .5(cos^2(A-B)) which is the classical expectation when there is separability.

To get statistics in which CHSH is violated, you must have entanglement. So really, what point are you trying to make?

Thanks DrC, nice reply; I much appreciate your having a go. We seem to be in agreement thus far.

However, referring to the OP (and after correcting the typos in your equation above):

1. I see no equations for A and B, with each satisfying ± 1 (the boundary condition required by Bell's formulation)?

2. What maximum did you derive for the CHSH inequality under the subject conditions?

3. What were the related a, b, c and d?

Completion of these tasks should bring us to the point I'm seeking to make from my wholly classical scenario.

Thanks again.

 

[gill1109]

Gordon: the point is that whatever the functions A, B and whatever the angles and whatever the probability distribution of the hidden variables, CHSH will be satisfied.

 

[DrChinese]

Thanks DrC, nice reply; I much appreciate your having a go. We seem to be in agreement thus far.

However, referring to the OP (and after correcting the typos in your equation above):

1. I see no equations for A and B, with each satisfying ± 1 (the boundary condition required by Bell's formulation)?

2. What maximum did you derive for the CHSH inequality under the subject conditions?

3. What were the related a, b, c and d?

Completion of these tasks should bring us to the point I'm seeking to make from my wholly classical scenario.

Thanks again.

1. Does that mean I don't get a 100?

2. Traditional CHSH upper limit is always 2. The candidate local realistic estimate is not a factor, as it is model dependent.

3. I don't know what you are referring to. [...gently prodding us forward as I suspect there is a point just around the corner...]

 

[billschnieder]

1. Does that mean I don't get a 100?

2. Traditional CHSH upper limit is always 2. The candidate local realistic estimate is not a factor, as it is model dependent.

3. I don't know what you are referring to. [...gently prodding us forward as I suspect there is a point just around the corner...]

1. I think Gordon wants to see your equations for A(a,x) and B(b,x) for the scenario he described and show how you arrived at E(AB) = .25 + .5(cos^2(a-b)) from those equations.
3. (a, b, c, d) are the 4 angles for the CHSH experiment I understand.

 

[Gordon Watson]

This is getting to be quite a love-in; for which, Many thanks!

TomT ever-friendly and cautiously seeking. gill1109 helpful, balanced and conventional. DrC getting 100* for nicely trying (when he can be very)! Bill rightly helping DrC to move ahead and get a better score. DrC and I in some sort of general agreement.

If only Bill would read his email and understand why I must run? Hoping to reply to all in about 12 hours; thanks again.

GW */1000 Can do better!

 

[Gordon Watson]

Gordon: the point is that whatever the functions A, B and whatever the angles and whatever the probability distribution of the hidden variables, CHSH will be satisfied.

Thank you. Yes; agreed; for the classical example given in the OP. That is: The OP's classical example remains consistent (under any setting) with the classical (traditional) CHSH.

Indeed, would you agree that no experiment (real or imagined) has ever contradicted a mathematical truism? In fact Feynman's defective analysis of the double-slit experiment ("no one understands") in part arises from his belief in what is NOT (in general) a mathematical truism:

P(x|X) + P(x|Y) = P(x|Z) (!?)

So I would welcome your having a go at the essential challenge in the OP. That is, use Bell's widely accepted local-realistic protocol to analyse what is clearly a local-realistic experiment.

If you can't deliver the requisite A and B, perhaps you could explain why? At least derive the maximum value that the experiment could deliver using the traditional CHSH formula? That way I can check my own calculation, which will be brought into later discussion. Thanks.

 

[Gordon Watson]

1. Does that mean I don't get a 100?

2. Traditional CHSH upper limit is always 2. The candidate local realistic estimate is not a factor, as it is model dependent.

3. I don't know what you are referring to. [...gently prodding us forward as I suspect there is a point just around the corner...]

1. 100/1000? I bet you got those hopeful letters home from school: "Can do better!"

2. I don't understand the second sentence in your #2 at all. "The candidate local realistic estimate is not a factor, as it is model dependent." Please elaborate in case I'm missing something relevant to the OP. Thanks.

3. As Bill explained: a, b, c, d are the traditional measuring-device settings in the traditional CHSH. Alice uses a and c; Bob uses b and d.

Any luck yet with A and B? Or some explanation re your difficulty? I'd value your comments.

 

[Gordon Watson]

1. I think Gordon wants to see your equations for A(a,x) and B(b,x) for the scenario he described and show how you arrived at E(AB) = .25 + .5(cos^2(a-b)) from those equations.
3. (a, b, c, d) are the 4 angles for the CHSH experiment I understand.

Thanks Bill, for correctly clarifying the position. Much appreciated; I'm often away from the Net these days so don't hesitate to help like this and move things along.

Speaking of which, DrC: Any advice/comments re your A and B would be most welcome. And if you'll not be commenting, please let me know.

The challenge is meant to be serious: Use Bell's widely-accepted (and claimed) local-realistic protocol to analyse a genuine local-realistic scenario.

 

[Gordon Watson]

Bill Schnieder: you wanted outcomes of four experiments to match exactly (your i,j,k,l indices). The way I think about it, in one run of the experiment there are potential outcomes A1, A2, B1, B2. Alice and Bob each toss a coin to choose which outcome to actually observe (A1 or A2, B1 or B2). Then this is repeated many times. We assume that their coin tosses are independent of the physical systems generating A1, A2, B1, B2. Then the average of, say, A1 times B2 over all runs will hardly differ from the average over those runs where Alice chose "1", Bob chose "2". The averages over all runs satisfy CHSH. Hence the observed averages do too, up to statistical variation.

Sorry, but this is not clear to me.

You say: "... in one run of the experiment there are potential outcomes A1, A2, B1, B2." I tend to agree.

But you add: "Alice and Bob each toss a coin to choose which outcome to actually observe (A1 or A2, B1 or B2)."

So this confuses me as to what you mean by "potential outcomes"?

In my terms: Alice and Bob have no need toss a coin to choose which device-setting to adopt (respectively) for a and for b. At any such setting the experiment actually DELIVERS the outcomes (A1 or A2; B1 or B2).

Then, in CHSH, they each toss coin (on each run) to determine (respectively) a or c; b or d ... with related outcomes A1 or A2, B1 or B2, C1 or C2, D1 or D2.

PS: I'm fairly sure that we agree about the whole picture, but even your statements about "averaging" confuse me; especially re the bit where Alice chose "1", Bob chose "2"?

?

 

[DrChinese]

1. 100/1000? I bet you got those hopeful letters home from school: "Can do better!"

2. I don't understand the second sentence in your #2 at all. "The candidate local realistic estimate is not a factor, as it is model dependent." Please elaborate in case I'm missing something relevant to the OP. Thanks.

3. As Bill explained: a, b, c, d are the traditional measuring-device settings in the traditional CHSH. Alice uses a and c; Bob uses b and d.

Any luck yet with A and B? Or some explanation re your difficulty? I'd value your comments.

Gordon, I probably don't get where you are going. I am not having any difficulties, so no point in waiting on something which is not going to be forthcoming. I tried to be helpful with the Product State statistics, and will consider showing the math, but honestly that formula is often repeated and there is nothing controversial about it.

a/b/c/d are usually given as 0, 22.5, 45, 67.5 degrees, you can read that anywhere too. I mentioned that the CHSH inequality predicts the maximum value any local realistic model can yield. That would be a best case scenario where the data points are more or less hand picked. As I say, it is independent of the specific model and specific models can give lower values but not higher than 2. QM predicts an "ideal" value of about 2.8 and is always above 2 in Bell tests.

You are the one trying to challenge Bell, so what is the challenge?

 

[Gordon Watson]

Challenge 1: Can Bell's protocol be used to analyse the given classical (and clearly: wholly local and realistic) situation?

If not, why not?

 

[DrChinese]

Challenge 1: Can Bell's protocol be used to analyse the given classical (and clearly: wholly local and realistic) situation?

If not, why not?

The protocol is: We replace the quantum-entanglement-producing source with a classical source which sends a short pulse of light to Alice and Bob each day (over many years), each pulse correlated by having the same linear-polarization; though each day the common pulse polarization-orientation is different .

Sure, we can use something like CHSH or one of the other computational techniques. I like to use coincidence rates at angle settings 0, 120, 240. I know that the P(A-B=120 degrees) is .375 in this case (by substituting in the formula for Product State). The Bell lower limit is .333.

 

[Gordon Watson]

?

Bell's (1964) analytical (and presumed local-realistic) protocol has (equivalently):

A(a, x) = ±1;

B(b, x) = ±1;

E(AB) = ∫AB ρ(x) dx.

I don't see you using any of this anywhere in your analysis of this (clearly) local-realistic experiment?

Why?

In effect: Is there some reason to follow Bell only when it suits you?

 

[ThomasT]

Dear ThomasT, the OP is intended to be as 'subtle and as simple' as it gets! (Perhaps it fails?)

But: The OP is wholly classical. And you are not (entirely) an ignorant layman (being familiar with Malus, at least; as well as the point that you make right here, above).

So the challenge remains. That is: Time to do some basic maths! (And cut the words?)

Hi Gordon. Glad to see you're still thinking about this stuff. It's quite interesting and entertaining to me that each of us has a particular, apparently unique, approach, and that we're having some difficulty in reconciling our approaches. But, while appreciating your kind words, I think it's pretty clear that I'm the ignorant layperson in any of these discussions. So, just consider this post as a fond hello ... and, as might be expected from our past exchanges, I do disagree with your current challenge insofar as I understand it ... which might not be that far. After all, it took me over a year to understand the essence of what DrC was saying, which now makes much sense to me.

Gill is, after reading some of his stuff, imo, a bit of a heavyweight wrt these issues. As are Bill, DrC and yourself ... at least in my view. So, sorry for the intervening posts. I will now fade, once again, into the peanut gallery. Just wanted to say that I love these discussions, and that, as they continue, maybe something will click for me again.

 

[Gordon Watson]

Hi Gordon. Glad to see you're still thinking about this stuff. It's quite interesting and entertaining to me that each of us has a particular, apparently unique, approach, and that we're having some difficulty in reconciling our approaches. But, while appreciating your kind words, I think it's pretty clear that I'm the ignorant layperson in any of these discussions. So, just consider this post as a fond hello ... and, as might be expected from our past exchanges, I do disagree with your current challenge insofar as I understand it ... which might not be that far. After all, it took me over a year to understand the essence of what DrC was saying, which now makes much sense to me.

Gill is, after reading some of his stuff, imo, a bit of a heavyweight wrt these issues. As are Bill, DrC and yourself ... at least in my view. So, sorry for the intervening posts. I will now fade, once again, into the peanut gallery. Just wanted to say that I love these discussions, and that, as they continue, maybe something will click for me again.

Hi Thomas, greetings; with me repeating my old refrain: Let's get into the maths as a way to avoid getting caught up in all the words.

In the case at hand, old Malus is enough to derive the correct results; so you should do that (at least). And CHSH can be "derived" from an IDENTITY, so you should do that. Then ask (like me) how an experiment could contradict a valid "identity" UNLESS there's a fault in the move from the identity to CHSH; i.e., in the derivation? (Which is Bill's approach!)

Then you might ask, just like me: How-come DrC seems not to be using Bell's local-realistic protocol to derive VALID local realistic results?

As for expertise here: Exclude me for sure; and maybe the other old fogeys that are ever with us! Which I think leaves only Gill so far (in this thread). That's why I look forward to, and welcome, Gill's responses.

Now, if you want to avoid the maths: Ask questions (certainly of me), 'cos I'm here (like you) to learn. And, for me, learning what doesn't work is progress!

So, please, no fading! And if you want to go with words, question what you don't understand. For you might identify much that's not understood.

 

[ThomasT]

Hi Thomas, greetings; with me repeating my old refrain: Let's get into the maths as a way to avoid getting caught up in all the words.

The problem is conceptual. What determines the mathematical representation?

If we place certain restrictions on the math based on conceptual/philosophical (eg., locality) requirements, and if those restrictions are contrary to the experimental design, then if we construct inequalities based on those restrictions, then if those inequalities are violated experimentally, then what can we infer from that?

Only that the inequalities are based on a contradiction of an, apparently, incorrect assumption regarding the experimental design, I think.

So, what might this be telling us wrt deep reality. Well, maybe nothing, I think. No way to know.

There's a very simple way to look at Bell-type LR formulations. They, all of them, require that coincidental detection be explicitly expressed/modelled in terms of the hidden variable that determines individual detection. But the variable that determines individual detection is irrelevant wrt coincidental detection. I don't know how to express this strongly enough. λ has nothing to do with coincidental detection. It can be anything, any value of any property. It simply doesn't matter. The rate of coincidental detection is only determined by θ. Nothing else.

Nobody that I've said this to has addressed it. Why not? Isn't this at least interesting? The rate of coincidental detection doesn't vary with λ. And yet standard LR models of entanglement require the rate of coincidental detection to vary with, or at least be expressed in terms of, λ. So, what's wrong with this picture ... what's wrong with this way of modelling quantum entanglement?

This is a rhetorical question in the sense that I think I know what's wrong with it.

Let me further say that given Bell's assumptions and modelling technique, then his conclusions follow. Bell's theorem is mathematically sound. But I think that Bell-type modelling of quantum entanglement is flawed.

It should be clear enough, considering the above, what's wrong with the standard Bell-type modelling of quantum entanglement. We need only ask the question: what is θ measuring? And it seems obvious, to me at least, that θ isn't measuring λ. So, what is θ measuring?

Answer: θ is measuring a relationship between entangled entities that, apparently, isn't varying from pair to pair.

So, how would you model that?

 

[Gordon Watson]

The problem is conceptual. What determines the mathematical representation?

If we place certain restrictions on the math based on conceptual/philosophical (eg., locality) requirements, and if those restrictions are contrary to the experimental design, then if we construct inequalities based on those restrictions, then if those inequalities are violated experimentally, then what can we infer from that?

Only that the inequalities are based on a contradiction of an, apparently, incorrect assumption regarding the experimental design, I think.

So, what might this be telling us wrt deep reality. Well, maybe nothing, I think. No way to know.

There's a very simple way to look at Bell-type LR formulations. They, all of them, require that coincidental detection be explicitly expressed/modelled in terms of the hidden variable that determines individual detection. But the variable that determines individual detection is irrelevant wrt coincidental detection. I don't know how to express this strongly enough. λ has nothing to do with coincidental detection. It can be anything, any value of any property. It simply doesn't matter. The rate of coincidental detection is only determined by θ. Nothing else.

Nobody that I've said this to has addressed it. Why not? Isn't this at least interesting? The rate of coincidental detection doesn't vary with λ. And yet standard LR models of entanglement require the rate of coincidental detection to vary with, or at least be expressed in terms of, λ. So, what's wrong with this picture ... what's wrong with this way of modelling quantum entanglement?

This is a rhetorical question in the sense that I think I know what's wrong with it.

Let me further say that given Bell's assumptions and modelling technique, then his conclusions follow. Bell's theorem is mathematically sound. But I think that Bell-type modelling of quantum entanglement is flawed.

It should be clear enough, considering the above, what's wrong with the standard Bell-type modelling of quantum entanglement. We need only ask the question: what is θ measuring? And it seems obvious, to me at least, that θ isn't measuring λ. So, what is θ measuring?

Answer: θ is measuring a relationship between entangled entities that, apparently, isn't varying from pair to pair.

So, how would you model that?

Let's first agree on this:

θ is measuring a relationship between [STRIKE]entangled entities[/STRIKE] the detector settings a and b that, [STRIKE]apparently[/STRIKE], isn't varying from pair to pair ... when we evaluate a correlation.

 

[ThomasT]

Let's first agree on this:

θ is measuring a relationship between [STRIKE]entangled entities[/STRIKE] the detector settings a and b that, [STRIKE]apparently[/STRIKE], isn't varying from pair to pair ... when we evaluate a correlation.

I'm making some assumptions about what's going on in the deep reality. What you seem to be saying, with the strikeouts, is that θ
is measuring θ. Which doesn't make sense.

So, I'm not sure you understand what I'm saying.

 

[Gordon Watson]

I'm making some assumptions about what's going on in the deep reality. What you seem to be saying, with the strikeouts, is that θ
is measuring θ. Which doesn't make sense.

So, I'm not sure you understand what I'm saying.

It seems to me, Thomas, that I do understand what you're saying; and that you should move to making assumptions about the deeper reality when you rightly understand the full physical significance of θ.

θ is as I say; and more:

When each particle interacts with its detector (here), the respective outputs will be from the set {a+, a-; b+, b-}; + = spin-up; - = spin-down.

Every correlated output {from any of the ab combinations} is also correlated via some function of θ; e.g., with a+b- the bearing of Bob's spin-output b- to Alice's spin-output a+ is θ + ∏.

So where is the confusion for you?

That some function of θ-alone delivers the correlation between all the θ-related outcomes seems to me, like, obvious.

Does this help?

PS: The deeper reality is then related to the tougher question: the relation of the λs to the output-set {a+, a-; b+, b-}.

 

[ThomasT]

It seems to me, Thomas, that I do understand what you're saying; and that you should move to making assumptions about the deeper reality when you rightly understand the full physical significance of θ.

θ is as I say; and more:

When each particle interacts with its detector (here), the respective outputs will be from the set {a+, a-; b+, b-}; + = spin-up; - = spin-down.

The individual outputs will be either that a detection has been registered, or that a detection hasn't been registered. You can denote that however you want, but the conventional notations are +1,-1 or 1,0, corresponding to detection, nondetection, respectively.

I don't know what you mean by the full physical significance of θ. θ just refers to the angular difference between the polarizer settings, afaik.

Every correlated output {from any of the ab combinations} is also correlated via some function of θ; e.g., with a+b- the bearing of Bob's spin-output b- to Alice's spin-output a+ is θ + ∏.

I don't know what this means. The ab combinations are θ. I don't have any idea what the a+b- stuff means or where ∏ comes into it.

So where is the confusion for you?

Well, I don't think I'm confused. P(A,B) is a function that refers to the independent variable θ. And, in the ideal, wrt optical Bell tests, P(A,B) = cos²θ.

That some function of θ-alone delivers the correlation between all the θ-related outcomes seems to me, like, obvious.

Of course it's obvious. Because, in the ideal, this is the QM prediction. Rate of coincidental detection varies as cos² θ.

The deeper reality is then related to the tougher question: the relation of the λs to the output-set {a+, a-; b+, b-}.

The relation of λ to A is denoted as P(A) = cos² |a-λ| .

As I said, I don't think you understand what I'm saying. Namely, that the underlying parameter that determines rate of individual detection is not the underlying parameter that determines rate of coincidental detection.

 

[Gordon Watson]

Moved to https://www.physicsforums.com/showthread.php?p=3840419#post3840419 to avoid confusion with the classical example in the OP.

 

[DrChinese]

?

Bell's (1964) analytical (and presumed local-realistic) protocol has (equivalently):

A(a, x) = ±1;

B(b, x) = ±1;

E(AB) = ∫AB ρ(x) dx.

I don't see you using any of this anywhere in your analysis of this (clearly) local-realistic experiment?

Why?

In effect: Is there some reason to follow Bell only when it suits you?

It's your model, and I supplied the answers to your questions. Bell applies, and the resulting prediction is within the local realistic boundary as we would expect. Is your question how did I arrive at .375?

 

[ThomasT]

This is wrong; a big misunderstanding. This does not hold in entangled experiments. It would hold if λ denoted a polarisation but entangled particles are unpolarised (quoting Bell).

Afaik, wrt optical Bell tests, λ, the hidden variable denotes an underlying polarization that's varying randomly from pair to pair.

I guess I just don't understand your treatment here. As far as I can tell it's not going to get you to a better understanding of why BIs are violated formally and experimentally, and it doesn't disprove Bell's treatment which is based on the encoding of a locality condition which, it seems, isn't, in effect, a locality condition.

And now, since I am a bit confused by your presentation, I think I will just fade back into the peanut gallery. Maybe I'll learn something.

 

[Gordon Watson]

Afaik, wrt optical Bell tests, λ, the hidden variable denotes an underlying polarization that's varying randomly from pair to pair.

I guess I just don't understand your treatment here. As far as I can tell it's not going to get you to a better understanding of why BIs are violated formally and experimentally, and it doesn't disprove Bell's treatment which is based on the encoding of a locality condition which, it seems, isn't, in effect, a locality condition.

And now, since I am a bit confused by your presentation, I think I will just fade back into the peanut gallery. Maybe I'll learn something.

I've responded at https://www.physicsforums.com/showthread.php?t=591572 to avoid confusing our discussion on entanglement with the classical example in the OP.

BUT note: If you carried out your "AFAIK" calculation above (with your "λ and underlying polarisation"), you would reproduce the unentangled classical example given in the OP; which is neither Bell nor Aspect, etc. But, imho, this would help with your understanding of Bell (imho). And maybe bring you back into to this thread?

 

[Gordon Watson]

It's your model, and I supplied the answers to your questions. Bell applies, and the resulting prediction is within the local realistic boundary as we would expect. Is your question how did I arrive at .375?

..
DrC, with respect: I have little idea what you are referring to! So, seeking to understand, I write in the spirit of trying to be helpful! I therefore proceed on the basis that you are sincere and not trying to muddy the waters unnecessarily.

(Please also note a very off-putting habit of yours: You continue to confuse A and B with a and b. These repeated errors are akin to your old one of identifying the particles as Alice and Bob -- which is no problem if we know what your up to: but can be quite confusing when commonly accepted conventions are abused without explanation.)

Now:

1. What is my model? Do you mean the experiment identified in the OP? I take the view that Bell tried to "model" local realism (see below).

2. To be clear about one of my questions: Can you show me, please, the A and B functions that you used in your derivation? [EDIT: DrC, Response here not necessary if the answer to the next question is No!]

3. My next question is: Do they comply with Bell's (1964), A and B = ±1?

4. If not, why not? And so what do you mean by your "Bell applies"? For, further, you could then not have used Bell's integral for E(AB)?

5. One of my questions related to the maximum value of the traditional (your term) CHSHI that you derived from the experiment in the OP. What number did you get?

6. Surely it was not 0.375?

7. I know that (in reference to your analysis) you referred to text-books having all the relevant answers in them. [EDIT: I take it that such answers are relevant to the OP?] I do not know of any such; so, alas, I must always work from first-principles. So, can you list a few such books for me, please?

8. You also mentioned (re#6 above): "The Bell lower limit is .333." What is this, please? And how derived? Is it relevant to any issue here?

9. In the interests of seeking to be very clear and precise, one aspect of this thread, in case you missed it, was addressed to you as follows:

[To DrC, responding to a very confusing post by him.]

?

Bell's (1964) analytical (and presumed local-realistic) protocol has (equivalently):

A(a, x) = ±1;

B(b, x) = ±1;

E(AB) = ∫AB ρ(x) dx.

I don't see you using any of this anywhere in your analysis of this (clearly) local-realistic experiment?

Why?

In effect: Is there some reason to follow Bell only when it suits you?

..
To rephrase my serious interest:

1: Could you explain why (in your view) Bell's protocol is not relevant to what is clearly a local-realistic experiment?

2: Would such a view indicate that Bell may not be as relevant for local-realism as you commonly suppose?

With thanks, as always, this time in anticipation,

Gordon
..

 

[DrChinese]

Gordon,

It is your model, you suggested we consider it per Bell. It's a common one, so ok, so I have. The CHSH upper limit is 2, and I don't know the exact expectation value for CHSH for this model but it is definitely less than 2. Of course the experimental value is above 2.

You asked what P(AB) would be, I indicated .375. If you want to know E(AB), just use lugita15's formula and you get E(AB)=-.25.

Not sure how I am abusing A and B, I didn't even mention them in my last post. Yes, there is a difference between the particle detected by Alice, the angle setting Alice uses, and Alice as the observer. However, I can't help you much on the point because I interchange Alice and Bob, and A and B, etc. as I think it is easier for the analogies I make. I would hope that my choice of capitalization wouldn't confuse too much, I usually try to follow what the other person is doing. In this case, I have been following you preference in this thread to use E(AB) and P(AB) which is what I am writing as well. I do not usually repeat every step of a derivation when you can see that for yourself in the source literature. If you have a specific question, I will usually answer that. Is your question how I arrived at .375? I can show that if it helps. I used the Product State statistics, which apply to the OP example, and that formula has:

P(A,B)=.25+.5(cos^2(A-B))
[Where A and B are the angle settings and there is the unspoken assumption that the example does not have some particular bias that is not mentioned, such that the source is reasonably randomized.]

I mentioned that I commonly use 0 and 120 degrees for my settings. So substituting in the above, I get .25 + .5(.25) = .375 which is the rate of matches (which is what I usually report). Keep in mind that the "proper" correlation rate subtracts mismatches so that is why you could also report .375 - .625 which is the -.25 number mentioned above. I think you are using that to be E(A,B). As long as we know what basis we are reporting on, they are really the same thing.

A bit confused about your comments about Bell not being relevant as I think. I never said the Bell protocol does not work here, and I am not sure why you keep pushing me in that direction. Further, you must keep in mind that a discussion on PhysicsForums is not going to alter what the general physics community thinks of Bell. Bell is highly regarded, in fact an interesting piece of history. By 1970, 5 years after Bell appeared in a now defunct publication, new interpretations were needing to address Bell because its logic was so powerful. See for example something cited by billschnieder:

The Statistical Interpretation of Quantum Mechanics

This devotes considerable attention to EPR, Bell and hidden variables.

 

[bohm2]

There was an article published yesterday on a related topic that some might find interesting and/or add to the confusion:

In this paper I present a stronger Bell argument which even forbids certain non-local theories. The remaining non-local theories, which can violate Bell inequalities, are characterized by the fact that at least one of the outcomes in some sense probabilistically depends both on its distant as well as on its local parameter.

A stronger Bell argument for quantum non-locality
http://philsci-archive.pitt.edu/906...er_Bell_argument_for_quantum_non-locality.pdf

 

[Gordon Watson]

There was an article published yesterday on a related topic that some might find interesting and/or add to the confusion:

A stronger Bell argument for quantum non-locality
http://philsci-archive.pitt.edu/906...er_Bell_argument_for_quantum_non-locality.pdf

..
Dear bohm2,

Welcome to the thread and many thanks for the very interesting article. I've not analysed it in detail (yet) but its sure stored for future reading. It is much appreciated!

To encourage your further involvement here: Note that the classical experiment defined in the OP is designed to minimise many complications with BT (so it is not like that article).

Rather: The OP's classical experiment raises, for me, the question: Given that Bell's protocol/system is used for analysing local-realism in the context of EPR-Bohm (from that other brilliant Bohm): Can Bell's system be used to analyse the (clearly) local-realistic experiment in the OP?

That is: We have outputs A and B that may be equated to ±1. And we have some, here presumably uniform, ρ(x): since x is randomised. These are just those ingredients that Bell (e.g., Bell 1964) focussed upon. Plus his integral: E(AB) = ∫AB ρ(x) dx !

With thanks again,

Gordon

EDIT: So (to be clear), in relation to the (clearly) realistic and Einstein-local experiment in the OP, we seek to follow Bell's protocol (e.g., from Bell 1964) and provide:

(1) A(a, x) = ±1.

(2) B(b, x) = ±1.

(3) 0 ≤ ρ(x); ∫ρ(x) dx = 1.

(4) E(AB) = ∫AB ρ(x) dx = ?

 

[Gordon Watson]

Gordon,

It is your model, you suggested we consider it per Bell. It's a common one, so ok, so I have. The CHSH upper limit is 2, and I don't know the exact expectation value for CHSH for this model but it is definitely less than 2. Of course the experimental value is above 2.

You asked what P(AB) would be, I indicated .375. If you want to know E(AB), just use lugita15's formula and you get E(AB)=-.25.

Not sure how I am abusing A and B, I didn't even mention them in my last post. Yes, there is a difference between the particle detected by Alice, the angle setting Alice uses, and Alice as the observer. However, I can't help you much on the point because I interchange Alice and Bob, and A and B, etc. as I think it is easier for the analogies I make. I would hope that my choice of capitalization wouldn't confuse too much, I usually try to follow what the other person is doing. In this case, I have been following you preference in this thread to use E(AB) and P(AB) which is what I am writing as well. I do not usually repeat every step of a derivation when you can see that for yourself in the source literature. If you have a specific question, I will usually answer that. Is your question how I arrived at .375? I can show that if it helps. I used the Product State statistics, which apply to the OP example, and that formula has:

P(A,B)=.25+.5(cos^2(A-B))
[Where A and B are the angle settings and there is the unspoken assumption that the example does not have some particular bias that is not mentioned, such that the source is reasonably randomized.]

I mentioned that I commonly use 0 and 120 degrees for my settings. So substituting in the above, I get .25 + .5(.25) = .375 which is the rate of matches (which is what I usually report). Keep in mind that the "proper" correlation rate subtracts mismatches so that is why you could also report .375 - .625 which is the -.25 number mentioned above. I think you are using that to be E(A,B). As long as we know what basis we are reporting on, they are really the same thing.

A bit confused about your comments about Bell not being relevant as I think. I never said the Bell protocol does not work here, and I am not sure why you keep pushing me in that direction. Further, you must keep in mind that a discussion on PhysicsForums is not going to alter what the general physics community thinks of Bell. Bell is highly regarded, in fact an interesting piece of history. By 1970, 5 years after Bell appeared in a now defunct publication, new interpretations were needing to address Bell because its logic was so powerful. See for example something cited by billschnieder:

The Statistical Interpretation of Quantum Mechanics

This devotes considerable attention to EPR, Bell and hidden variables.

..
Dear DrC, thanks (as always) for the new information in your response (above); it (as well as your time) is appreciated.

So, in reply: Yes please. Please show me your workings for any of the numbers or results that you have so far produced. That will surely be a step toward sorting out our differences.

But please note: Your confessed use of A & B to denote Alice & Bob, particles 1 & 2, device settings a & b is not only confusing: IT IS wrong when you consider that A & B generally convey specific understandings within the Bell literature, beginning with Bell (1964).

NB: Under any of your uses, or all of them together, this equation of yours makes NO SENSE at all: P(A,B)=.25+.5(cos^2(A-B)).

Please think about it. (Also: P generally defines a probability.)

Also: I'm not clear re lugita15's formula.

PS: I will be away from the Net for several days so I wonder: Would you mind taking your time and answering the specific questions in https://www.physicsforums.com/showpost.php?p=3840882&postcount=37 ?

The advantages that such direct answers offer for me (and maybe some others), are these:

1. All questions may be answered without any reference to QM whatsoever.

2. The experiment under discussion is easily done and is clearly Einstein-local!

3. As far as I can see, this thread, in meaningfully discussing the system that Bell uses to derive his conclusion, requires no knowledge of QM at all! (A conclusion, I'm told, that some have drawn for the whole Bell scene: Bell's theorem is NOT a property of quantum theory. So it's a property of ... ?)

Must run.

GW

 

[Gordon Watson]

...
You asked what P(AB) would be, I indicated .375. If you want to know E(AB), just use lugita15's formula and you get E(AB)=-.25.

..
DrC, I'm hoping that you'll answer my questions directly (to this point, at least; see particularly the recent posts above)? If you will not be answering, please let me know.

Also, if you are withdrawing your offer to expand on your derivations, please also let me know.

Now, re the above quote: I have no clue re "lugita15's formula" -- nor what E(AB) =-.25 relates to?

Why not give your derivation of E(AB) directly (with the explanatory notes that you offered)?

Do you not understand that E(AB) will be some function of a and b (the detector settings)?

Not sure how I am abusing A and B, I didn't even mention them in my last post. Yes, there is a difference between the particle detected by Alice, the angle setting Alice uses, and Alice as the observer. However, I can't help you much on the point because I interchange Alice and Bob, and A and B, etc. as I think it is easier for the analogies I make.

BUT your abuse of A and B continues almost immediately! See *** below!

I would hope that my choice of capitalization wouldn't confuse too much, I usually try to follow what the other person is doing. In this case, I have been following you preference in this thread to use E(AB) and P(AB) which is what I am writing as well.

Please show me where I have used P(AB) in this thread?

And what could it possibly mean?? What does it mean to you, since you clearly differentiate it from E(AB)?

I do not usually repeat every step of a derivation when you can see that for yourself in the source literature. If you have a specific question, I will usually answer that. Is your question how I arrived at .375? I can show that if it helps. I used the Product State statistics, which apply to the OP example, and that formula has:

P(A,B)=.25+.5(cos^2(A-B))
[Where A and B are the angle settings and there is the unspoken assumption that the example does not have some particular bias that is not mentioned, such that the source is reasonably randomized.]

*** So: If A and B are the angle settings, what then is P(AB), please?

I mentioned that I commonly use 0 and 120 degrees for my settings. So substituting in the above, I get .25 + .5(.25) = .375 which is the rate of matches (which is what I usually report). Keep in mind that the "proper" correlation rate subtracts mismatches so that is why you could also report .375 - .625 which is the -.25 number mentioned above. I think you are using that to be E(A,B). As long as we know what basis we are reporting on, they are really the same thing.

...

You commonly use 0 and 120? But how does that help you (in this thread) derive the maximum value attainable for the CHSH inequality for the experiment defined in the OP?

PS: I trust you will persist with your line of reasoning here: So that we might come to agree re the simple facts involved; no opinions being required (at any stage) given the simplicity of the experiment and the related maths.

..

 

[DrChinese]

Please show me where I have used P(AB) in this thread? And what could it possibly mean?? What does it mean to you, since you clearly differentiate it from E(AB)?

*** So: If A and B are the angle settings, what then is P(AB), please?

Arrgh.

Usually, P(AB)=P(A,B)=P(A-B)=P(ab)=P(a,b)=P(a-b) is the probability of matches at angles A/a and B/b. It doesn't matter what you call it, as long as you know what you are referring to. You could also call that an expectation value. Then P(AB)=E(AB), which you used in post #1. This has a range from 0 to 1.

But sometimes, either P(*) or E(*) is referring to the correlation rate, which has a range from -1 to 1. Correlations are matches less mismatches as a percentage. lugita15 discusses this point, I thought clearly, and was trying to follow his lead to make it easier. It doesn't matter to me really, but I prefer to speak in terms of probability. So a correlation of -.25 is the essentially the same thing as a match probability of 37.5%.

Your conclusions are always the same though regardless of basis. The important point is that when a-b=A-B=120 degrees, the expected match percentage is 37.5%. This is above the applicable Bell threshold, which is 33.3% (1/3). So the model respects Bell, and being explicitly classical, you would expect that.

I will be supplying the derivation as I get time, but for now just keep in mind that the formula is .25+(cos^2(a-b)) where a and b are the measurement angle settings and it is averaged over a suitable (random) population of source polarization angles (which I take as being x in your model).

It's your model in post#1, so I assume you know where you are going with this. I don't really want to spend time covering elements of this that you already know and don't have any questions on. So it would be helpful to know what that is.

 

[billschnieder]

Your conclusions are always the same though regardless of basis. The important point is that when a-b=A-B=120 degrees, the expected match percentage is 37.5%. This is above the applicable Bell threshold, which is 33.3% (1/3). So the model respects Bell, and being explicitly classical, you would expect that.

I will be supplying the derivation as I get time, but for now just keep in mind that the formula is .25+(cos^2(a-b))

I suppose then that since it is classical and obeys Bell, you will be able to factorize the formula into 2 functions which give values ±1 (according to Bell). Or better, you should be able to show how starting with such functions of only a, or b with values ±1, you arive at the result .25+(cos^2(a-b)). This is the challenge as far as I understand it.

 

[DrChinese]

I suppose then that since it is classical and obeys Bell, you will be able to factorize the formula into 2 functions which give values ±1 (according to Bell). Or better, you should be able to show how starting with such functions of only a, or b with values ±1, you arive at the result .25+(cos^2(a-b)). This is the challenge as far as I understand it.

Sure, that's it exactly. Of course, it converges on that value in actual trials. And I integrate across a range of x. If you pick only a few specific x values, your results will vary.

You don't have any question about that, do you? I am sure you have done this exercise plenty of times.

PS Somehow the formula got mangled along the way. It should be:

P(ab)=.25+.5(cos^2(a-b)) which has a range from .25 to .75

 

[billschnieder]

Sure, that's it exactly. Of course, it converges on that value in actual trials. And I integrate across a range of x. If you pick only a few specific x values, your results will vary.

You don't have any question about that, do you? I am sure you have done this exercise plenty of times.

PS Somehow the formula got mangled along the way. It should be:

P(ab)=.25+.5(cos^2(a-b)) which has a range from .25 to .75

You are misunderstanding. The challenge is for you to start out with two separate functions each depending only on either a, or b for this classical case which have values ±1, and show that you can arrive at the result P(ab)=.25+.5(cos^2(a-b)) from those functions.

In other words, show that P(ab)=.25+.5(cos^2(a-b)) is factorable into 2 separate functions A(a)=±1, B(b)=±1. If you do not want to do the math yourself, please point to an article or textbook which shows this. Note the cited source MUST be deriving the classical result P(ab)=.25+.5(cos^2(a-b)) from two separable functions of the form A(a)=±1, B(b)=±1.

This is the challenge as I understand it, nothing about trials, nothing about convergence.

 

[DrChinese]

You are misunderstanding. The challenge is for you to start out with two separate functions each depending only on either a, or b for this classical case which have values ±1, and show that you can arrive at the result P(ab)=.25+.5(cos^2(a-b)) from those functions.

In other words, show that P(ab)=.25+.5(cos^2(a-b)) is factorable into 2 separate functions A(a)=±1, B(b)=±1. If you do not want to do the math yourself, please point to an article or textbook which shows this. Note the cited source MUST be deriving the classical result P(ab)=.25+.5(cos^2(a-b)) from two separable functions of the form A(a)=±1, B(b)=±1.

This is the challenge as I understand it, nothing about trials, nothing about convergence.

Umm, OK. Are you telling me YOU don't know this already? That's what I'm asking. (I know Gordon doesn't.)

 

[billschnieder]

Umm, OK. Are you telling me YOU don't know this already? That's what I'm asking. (I know Gordon doesn't.)

Whether I know it or not is not the challenge. The challenge is for you to provide it. So stop focusing on me and provide it already, to get this thread going rather than express frustration at every turn without actually providing what is being asked.

You earlier asked for clarification of what was being asked:

It's your model in post#1, so I assume you know where you are going with this. I don't really want to spend time covering elements of this that you already know and don't have any questions on. So it would be helpful to know what that is.

I have clarified it for you so please calculate away.

 

[DrChinese]

Whether I know it or not is not the challenge. The challenge is for you to provide it. So stop focusing on me and provide it already, ... I have clarified it for you so please calculate away.

You are not the boss of me.

 

[Gordon Watson]

Umm, OK. Are you telling me YOU don't know this already? That's what I'm asking. (I know Gordon doesn't.)

Ignored. Waiting for formal response.