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.