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by **gill1109** » Mon Jun 16, 2014 1:03 pm

minkwe wrote:Let lambda be a non-local hidden variable, please write down the integral for the expectation value of the paired-product of outcomes at Alice and Bob.

When lambda is non-local hidden variable, What is wrong with
$A(a, \lambda)$
$B(b, \lambda)$
$E(a,b) = \int A(a, \lambda) B(b, \lambda) \rho(\lambda) d\lambda$

Please show exactly why the above three expressions are as you call it "undefined" or "meaningless" for a non-local hidden variable lambda. I've been asking you for a while, maybe one of your Bell-believer friends can help you explain what exactly in the above expression restricts it to LHV.

There are many kinds of non-local hidden variables models! Perhaps most general would be that Alice's setting is somehow available both at the source and at Bob's measurement station, and vice versa. If both settings are available at the source, then the probability distribution of the hidden variable could depend on the settings. If they are available at both measurement stations, then the measurement functions could depend on both. How about
$E(a,b) = \int A(a, b, \lambda) B(a, b, \lambda) \rho(\lambda | a, b) d\lambda$
for the most general case?

Christian's model (which is Pearle's model interpreted as a non-local model rather than as a detection loophole model) has the distribution of lambda depending on both the settings as one can easily see from his R code where "bad" values of lambda are rejected depending on criteria which depend on a and on b. ie he has
$E(a,b) = \int A(a, \lambda) B(b, \lambda) \rho(\lambda | a, b) d\lambda$
Whether one calls a model like this non-local or conspiratorial is a matter of interpretation, a matter of direction of causality.
Were a and b known in advance and the hidden variable's distribution depending on a and b; the experimenter having no choice at all? (conspiracy or super-determinism)? Or were a and b chosen by the experimenter at the last moment but their values instanteously somehow infecting the hidden variable at the source? (violation of locality)?

Here is Pearle's model:

The hidden variable lambda is the pair (X, Y) where:

X ~ uniform on S^2 … = unit vectors in R^3
Y ~ uniform on (1, 4), independent of X
C := (2 – √Y) / √Y
a and b are Alice, Bob’s settings, in S^2
A := sign(a . X) if |a . X | > C , otherwise 0 (Pearle) or no particle pair (Christian)
B := sign(– b . X) if |b . X | > C , otherwise 0 (Pearle) or no particle pair (Christian)

The hidden variable probability distribution rho used by Christian is the distribution of (X, Y) conditional on |a . X | > C and |b . X | > C where C = (2 – √Y) / √Y

Michel should note that this model is better than his epr-simple in that (a) settings are now direction in space, not just in the plane, (b) the singlet correlation is reproduced exactly, not only roughly. The model violates CHSH but not Larsson modified CHSH, nor CH, nor CHSH with outcomes "0" not discarded.

All these years the experimenters have been testing the wrong inequalities, as Caroline Thompson kept on emphasizing. Till last year, in fact.

by **gill1109** » Mon Jun 16, 2014 12:36 pm

gill1109 wrote:
Xray wrote:
minkwe wrote:
I'm beginning to think that maybe you've been misled by the Wikipedia page about "local hidden variable theories" http://en.wikipedia.org/wiki/Local_hidd ... ble_theory, which wrongly calls equation (1) a probability, likely due to a misunderstanding of Bell's equation (2). Richard is an author of that page, maybe he will fix it, or help you here to explain where "they" got that expression (1) on that page.

gill1109, heinera, Richard Gill, Joy Christian, minkwe, anyone?

On the wikipedia page http://en.wikipedia.org/wiki/Local_hidd ... ble_theory, (that minkwe references directly above) it states

"equation (1) becomes:

(3) P(a,b) = $\int d \lambda \cdot \rho(\lambda) \cdot \cos^2(a - \lambda) \cdot \cos^2(b - \lambda) = \frac{1}{8} + \frac{\cos^2 \phi}{4}$ , where $\phi = b - a.$ "

"The predicted quantum correlation can be derived from this and is shown in fig. 2."

Please provide this Derivation of the predicted quantum correlation from eqn. (3) to help me understand that page

10,000 euros, anyone

Xray

I never read this page of wikipedia before, and certainly never edited it (as far as I can remember). It looks quite a mess. Bad notation, unexplained concepts.

I see that I did add a link to the article two years ago, and an item in the talk page. But my goodness, what a total mess that page is.

by **gill1109** » Mon Jun 16, 2014 8:22 am

Xray wrote:
minkwe wrote:
I'm beginning to think that maybe you've been misled by the Wikipedia page about "local hidden variable theories" http://en.wikipedia.org/wiki/Local_hidd ... ble_theory, which wrongly calls equation (1) a probability, likely due to a misunderstanding of Bell's equation (2). Richard is an author of that page, maybe he will fix it, or help you here to explain where "they" got that expression (1) on that page.

gill1109, heinera, Richard Gill, Joy Christian, minkwe, anyone?

On the wikipedia page http://en.wikipedia.org/wiki/Local_hidd ... ble_theory, (that minkwe references directly above) it states

"equation (1) becomes:

(3) P(a,b) = $\int d \lambda \cdot \rho(\lambda) \cdot \cos^2(a - \lambda) \cdot \cos^2(b - \lambda) = \frac{1}{8} + \frac{\cos^2 \phi}{4}$ , where $\phi = b - a.$ "

"The predicted quantum correlation can be derived from this and is shown in fig. 2."

Please provide this Derivation of the predicted quantum correlation from eqn. (3) to help me understand that page

10,000 euros, anyone

Xray

I never read this page of wikipedia before, and certainly never edited it (as far as I can remember). It looks quite a mess. Bad notation, unexplained concepts.

by **Heinera** » Sun Jun 15, 2014 4:34 am

Xray wrote:
sorry meant to ask:

How does ±1 come out of A(a, -1 <= lambda <= 1) according to Heinera's understanding of the Bell formulation A = ±1

Xray

In that simulation the function is actually A(a,b -1 <= lambda <= 1), and you find the definition in the function "obs" in the R code.

By the way, that simulation is not meant to be taken seriously as a good description of nature. It was written to illustrate how trivially easy one can generate the strong correlations if we drop one of the assumptions of Bell's theorem, in this case locality.

by **Xray** » Sat Jun 14, 2014 4:22 pm

Heinera wrote:
Xray wrote:Hi Heinera

Does the variable lambda have units or does it stand for something else in your model because how does anything other than a come out of A(a, -1 <= lambda <= 1) and we need A(a, …) = ±1 to agree with Bell?

Xray

Why must a function A(a, lambda) of two arguments a and lambda be equal to a?

sorry

meant to ask:

How does ±1 come out of A(a, -1 <= lambda <= 1) according to Heinera's understanding of the Bell formulation A = ±1 :?:

Xray

by **Xray** » Sat Jun 14, 2014 4:15 pm

minkwe wrote:
I'm beginning to think that maybe you've been misled by the Wikipedia page about "local hidden variable theories" http://en.wikipedia.org/wiki/Local_hidd ... ble_theory, which wrongly calls equation (1) a probability, likely due to a misunderstanding of Bell's equation (2). Richard is an author of that page, maybe he will fix it, or help you here to explain where "they" got that expression (1) on that page.

gill1109, heinera, Richard Gill, Joy Christian, minkwe, anyone?

On the wikipedia page http://en.wikipedia.org/wiki/Local_hidd ... ble_theory, (that minkwe references directly above) it states

"equation (1) becomes:

(3) P(a,b) = $\int d \lambda \cdot \rho(\lambda) \cdot \cos^2(a - \lambda) \cdot \cos^2(b - \lambda) = \frac{1}{8} + \frac{\cos^2 \phi}{4}$ , where $\phi = b - a.$ "

"The predicted quantum correlation can be derived from this and is shown in fig. 2."

Please provide this Derivation of the predicted quantum correlation from eqn. (3) to help me understand that page :?:

10,000 euros, anyone :?:

Xray

by **minkwe** » Sat Jun 14, 2014 10:41 am

Heinera wrote:
minkwe wrote:Heinera,
I think you understand what I mean but you also now realize that you were wrong. A local hidden variable lambda between Alice and Bob is a variable shared through a local interaction in the common region of their past light cones (eg from the source). A non-local hidden variable between Alice and Bob is one that is shared by both of them instantaneously, ie could not have been shared through an interaction in the common region of their past light cone. You know exactly what it means.

You use the word "local hidden variable" yet you claim not to know what "non-local hidden variable" means.
The expressions
$A(a, \lambda)$
$B(b, \lambda)$
$E(a,b) = \int A(a, \lambda) B(b, \lambda) \rho(\lambda) d\lambda$
Are exactly the same whether lambda is a non-local or local hidden variable. There is nothing in the above expression that restricts it to LHV.

So let's take your previous example, then, where lambda is actually the detector settings (a,b). How do you propose we should integrate over lambda, when lambda is stuck at one particular value, determined by the two free variables a and b?

I already told you. Read the previous post, and think about what E(a,b) means.

minkwe wrote:There is nothing in Bell's equation 2 that is specific to LHV theories. Nothing whatsoever. If you think there is, you will either explain what exactly in that expression restricts it to LHV theories, or in the alternative, tell us what exactly the expression would be for the expectation value of the paired product of outcomes at Alice and Bob, for a non-local theory. A non-local theory produces outcomes at Alice and Bob doesn't it? Those outcomes can be multiplied with each other can they not? What is the expectation value expression, please?

I'm beginning to think that maybe you've been misled by the Wikipedia page about "local hidden variable theories" http://en.wikipedia.org/wiki/Local_hidd ... ble_theory, which wrongly calls equation (1) a probability, likely due to a misunderstanding of Bell's equation (2). Richard is an author of that page, maybe he will fix it, or help you here to explain where "they" got that expression (1) on that page.

by **Heinera** » Sat Jun 14, 2014 10:34 am

minkwe wrote:Heinera,
I think you understand what I mean but you also now realize that you were wrong. A local hidden variable lambda between Alice and Bob is a variable shared through a local interaction in the common region of their past light cones (eg from the source). A non-local hidden variable between Alice and Bob is one that is shared by both of them instantaneously, ie could not have been shared through an interaction in the common region of their past light cone. You know exactly what it means.

You use the word "local hidden variable" yet you claim not to know what "non-local hidden variable" means.
The expressions
$A(a, \lambda)$
$B(b, \lambda)$
$E(a,b) = \int A(a, \lambda) B(b, \lambda) \rho(\lambda) d\lambda$
Are exactly the same whether lambda is a non-local or local hidden variable. There is nothing in the above expression that restricts it to LHV.

So let's take your previous example, then, where lambda is actually the detector settings (a,b). How do you propose we should integrate over lambda, when lambda is stuck at one particular value, determined by the two free variables a and b?

by **minkwe** » Sat Jun 14, 2014 9:56 am

Heinera,
You say you don't know how to evaluate

$E(a,b) = \int A(a, \lambda(a,b)) B(b, \lambda(a,b)) \rho(\lambda(a,b)) d\lambda(a,b)$
But you evaluate exactly like you evaluate
$E(a,b) = \int A(a, \lambda) B(b, \lambda) \rho(\lambda) d\lambda$
You integrate over all the distinct values of lambda. If there is just one value of lambda for each a,b, with probability 1 the integral simply gives you $A(a, \lambda) B(b, \lambda)$ or if you like $A(a, b)B(a, b)$ . Don't forget what we are calculating --- the expectation value of the result at Alice multiplied by the result at Bob, that is what E(a,b) is. If the result at Alice is A(a,b) and the result at Bob is B(a,b), then $E(a,b) = \int A(a, \lambda) B(b, \lambda) \rho(\lambda) d\lambda = A(a, b)B(a, b)$ There is nothing meaningless here.

In your non-local model, you calculate the average of A*B and equate that to E(a,b), you will do exactly the same thing whether you were dealing with a local model or any model whatsoever. A(a, lambda) is simply a function which takes two arguments and gives a result. It doesn't matter whether those arguments are local or non-local. So long as your model gives a result, your function is well defined contrary to what you are trying to argue here. In your model you have A(a, b, hv), you are arguing speciously now that lambda = hv. But I can easily rewrite your function into A(a, Lambda), where Lambda = (a,b,hv) and A(a,Lambda) will be well defined. Your earlier suggestion that if the two arguments of a function are not independent, then the function is not well defined is just silly. For example, f(a,b) = cos(a-b) is a function of 2 variables. Just because I tell you that b = g(x,a) = (2a - x)^2 does not mean that f(a,b) is not well defined. You are trying to defend the indefensible.

There is nothing in Bell's equation 2 that is specific to LHV theories. Nothing whatsoever. If you think there is, you will either explain what exactly in that expression restricts it to LHV theories, or in the alternative, tell us what exactly the expression would be for the expectation value of the paired product of outcomes at Alice and Bob, for a non-local theory. A non-local theory produces outcomes at Alice and Bob doesn't it? Those outcomes can be multiplied with each other can they not? What is the expectation value expression, please?

Maybe other Bell-believers who agree with your claims will help you to state what exactly the expression would be for the expectation value of the paired product of outcomes for non-local hidden variable theories.

by **minkwe** » Sat Jun 14, 2014 9:49 am

Heinera,
I think you understand what I mean but you also now realize that you were wrong. A local hidden variable lambda between Alice and Bob is a variable shared through a local interaction in the common region of their past light cones (eg from the source). A non-local hidden variable between Alice and Bob is one that is shared by both of them instantaneously, ie could not have been shared through an interaction in the common region of their past light cone. You know exactly what it means.

You use the word "local hidden variable" yet you claim not to know what "non-local hidden variable" means.
The expressions
$A(a, \lambda)$
$B(b, \lambda)$
$E(a,b) = \int A(a, \lambda) B(b, \lambda) \rho(\lambda) d\lambda$
Are exactly the same whether lambda is a non-local or local hidden variable. There is nothing in the above expression that restricts it to LHV.

by **Heinera** » Sat Jun 14, 2014 9:09 am

minkwe wrote:
Heinera wrote:By "non-local hidden variable", I assume you mean that lambda is a function of a and b

No. I mean lambda is a non-local hidden variable. Nothing about a and b. No other information is given about lambda other than the fact that it is non-local.

Sorry, then I have no idea what you mean by a "non-local hidden variable". I have never used that term myself, as far as I can recall. I have a clear opinion of what a "non-local model" means, so I have used the expression "non-local hidden variable model", but in that phrase "non-local" refers to "model". A better way of saying it would probably be "hidden variable non-local model".

by **minkwe** » Sat Jun 14, 2014 7:12 am

Heinera wrote:By "non-local hidden variable", I assume you mean that lambda is a function of a and b

No. I mean lambda is a non-local hidden variable. Nothing about a and b. No other information is given about lambda other than the fact that it is non-local.

So please write down the expectation value of the paired product of outcomes at Alice and Bob. Or explain what is wrong with the three equations.

by **Heinera** » Sat Jun 14, 2014 3:38 am

Xray wrote:Hi Heinera

Does the variable lambda have units or does it stand for something else in your model because how does anything other than a come out of A(a, -1 <= lambda <= 1) and we need A(a, …) = ±1 to agree with Bell?

Xray

Why must a function A(a, lambda) of two arguments a and lambda be equal to a?

by **Heinera** » Sat Jun 14, 2014 3:34 am

minkwe wrote:Let lambda be a non-local hidden variable, please write down the integral for the expectation value of the paired-product of outcomes at Alice and Bob.

When lambda is non-local hidden variable, What is wrong with
$A(a, \lambda)$
$B(b, \lambda)$
$E(a,b) = \int A(a, \lambda) B(b, \lambda) \rho(\lambda) d\lambda$

By "non-local hidden variable", I assume you mean that lambda is a function of a and b, so we can write lambda(a, b). Now lets make the substitution:
$E(a,b) = \int A(a, \lambda(a,b)) B(b, \lambda(a,b)) \rho(\lambda(a,b)) d\lambda(a,b)$
Now, how will you evaluate that? In particular, how should we interpret $d\lambda(a,b)$ ?

by **Joy Christian** » Sat Jun 14, 2014 12:46 am

FrediFizzx wrote:
gill1109 wrote:
FrediFizzx wrote:Is the "setting" represented by a vector? If not, what is its definition?

"Settings" are chosen by experimenters by pressing buttons or turning dials on pieces of apparatus. In mathematical physical theories they could be represented in all kinds of ways. Talking about EPR-B, they are pairs of directions in 3-d space.

So vectors.

lambda does not have to be a vector even when the directions a and b are vectors. For example, I can choose lambda to be the norm function, || . ||. Then

A(a, lambda) = || a ||,

which gives the number 0 < A < infinity; i.e., any number on a dial ranging from 0 to infinity. In other words, a and lambda can be anything in an arbitrary theory.

by **FrediFizzx** » Sat Jun 14, 2014 12:07 am

gill1109 wrote:
FrediFizzx wrote:Is the "setting" represented by a vector? If not, what is its definition?

"Settings" are chosen by experimenters by pressing buttons or turning dials on pieces of apparatus. In mathematical physical theories they could be represented in all kinds of ways. Talking about EPR-B, they are pairs of directions in 3-d space.

So vectors.

by **gill1109** » Fri Jun 13, 2014 11:50 pm

FrediFizzx wrote:Is the "setting" represented by a vector? If not, what is its definition?

"Settings" are chosen by experimenters by pressing buttons or turning dials on pieces of apparatus. In mathematical physical theories they could be represented in all kinds of ways. Talking about EPR-B, they are pairs of directions in 3-d space.

by **FrediFizzx** » Fri Jun 13, 2014 10:36 pm

Is the "setting" represented by a vector? If not, what is its definition?

by **gill1109** » Fri Jun 13, 2014 10:25 pm

FrediFizzx wrote:If a is a vector then lambda must have a vector in it to get a scalar for the final result.

A(a, lambda) is a function of setting and hidden variable. The hidden variable can lie in any space of any kind, large or small, you like.

by **FrediFizzx** » Fri Jun 13, 2014 7:23 pm

If a is a vector then lambda must have a vector in it to get a scalar for the final result.

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