Bell's Inequality Explanation for Intelligent Non-Scientist

[johana]

Then that has no relationship to Bell's A and B which by definition can only have values $\pm 1$.

I did say it's a probability function and that it does indeed return only \pm 1 values, as is evident from the equations. It's just like a coin has its probability function and it only returns heads or tails.

You are missing the point my example is illustrating, which is that hidden variables is a function, and not just any function, but probability function. Think about it. What else can it be? There is no other way to obtain meaningful result, is there?

Your notation is recipe for confusion. $A(a,\lambda) = cos^2((a-b)/2)$ You have b in the a function. Please read Bell's paper.

The important thing is that it actually calculates comparable result. Is this better:

A(a,\lambda_A) = cos^2(a)
B(b,\lambda_B) = cos^2(b)

A(60, \lambda_A) = P(+) = cos^2(60) = 0.25
B(30, \lambda_B) = P(+) = cos^2(30) = 0.75

P(++ | --) = (0.25 * 0.75) + (0.75 * 0.25) = 0.375
P(+- | -+) = (0.25 * 0.25) + (0.75 * 0.75) = 0.625

E(60,30) = P_{++} + P_{--} - P_{+-} - P_{-+} = 0.375 - 0.625 = -0.25

 

[Avodyne]

No, it is not better. $A(a,\lambda_A)$ can have the values $+1$ or $-1$ only. Therefore it cannot equal $\cos^2(a)$.

 

[johana]

No, it is not better. $A(a,\lambda_A)$ can have the values $+1$ or $-1$ only. Therefore it cannot equal $\cos^2(a)$.

A(a,\lambda_A) = cos^2(a), S={+1,-1}

It's a probability function with sample space +1 and -1. What's the problem? Is there anything else I need to do with the notation to make that more clear?

 

[Avodyne]

The problem is that $A(a,\lambda_A)$ is most definitely not a probability function. It is the actual result that one would get if the polarizer was set to $a$ and the incoming photon carried hidden information $\lambda_A$. This actual result can have only the values $+1$ or $-1$.

The whole point of hidden-variable theory is that there is always an actual result that could be specified for every possible measurement, whether that measurement was made or not, and that we could predict those results if only we knew the values of the hidden variables.

The values of the hidden variables are assigned a probability distribution $\rho(\lambda_A,\lambda_B)$, but the results of measurements $A(a,\lambda_A)=\pm 1$ and $B(b,\lambda_B)=\pm 1$ are always definite.

 

[DrChinese]

A(a,\lambda_A) = cos^2(a), S={+1,-1}

It's a probability function with sample space +1 and -1. What's the problem? Is there anything else I need to do with the notation to make that more clear?

1. Since the above doesn't match up with anything we can make sense of, yes.

The function you have above doesn't return +1 or -1. It is not rotationally invariant either, since it depends only on a.2. Also regarding the formula you presented:

E(60,30)=P+++P−−−P+−−P−+ = 0.375 - 0.625 = -0.25

I can see this is supposed to be some kind of correlation related to settings of 60 and 30 degree measurement settings. But:

a) If this is Type II entangled pairs, it is not correct as the correlation of those will be sin^2(theta=30)-cos^2(theta=30)= -.5.

b) If the source stream of pairs is polarized at 0 degrees (and therefore not entangled), then the correlation formula is correct as you have it.

c) And in fact if the source stream of pairs is such that every pair is polarized identically at some random angle, the resulting expectation value would be:

(cos^2(theta=30)-sin^2(theta=30))/2= -.25.

Note that b) and c) are different from the correlated expectation value for entangled pairs a). Here is an example of a local realistic theory that fails (ie one in which entangled pairs are supposed to be randomly polarized at some unknown angle). I think this is what you were trying to say, but it is not clear (and maybe you were saying something else entirely).

 

[billschnieder]

A(a,\lambda_A) = cos^2(a), S={+1,-1}

It's a probability function with sample space +1 and -1. What's the problem? Is there anything else I need to do with the notation to make that more clear?

Johana, maybe this will help: can you tell us exactly what $cos^2(a)$ is supposed to represent? In other words, describe the experiment which produces such a result.

 

[johana]

Johana, maybe this will help: can you tell us exactly what $cos^2(a)$ is supposed to represent? In other words, describe the experiment which produces such a result.

It's your friendly neighborhood Malus' law. Send 10,000 0° polarized photons (as DrChinese already pointed out) through 60° polarizer A, 25% will go "+" way and 75% "-" way. Then send 10,000 0° polarized photons through 30° polarizer B, 75% will go "+" way and 25% "-" way. Ok? The point is, how else could you possibly get random +/- output as such unless hidden variable was actually a probability function just like that?

 

[Avodyne]

But this is not what $A(a,\lambda_A)$ means in Bell's formula. It means the result for a single photon with hidden variable $\lambda_A$ that passes through a polarizer set at angle $a$. This single result must be $+1$ or $-1$.

$A(a,\lambda_A)$ does not mean the average value over 10,000 photons.

 

[Nugatory]

The point is, how else could you possibly get random +/- output as such unless hidden variable was actually a probability function just like that?

We produce random outputs from a completely deterministic system in which none of the hidden variables are probability functions all the time; throwing a six-sided die or tossing a coin are obvious examples. If we knew the exact initial conditions and had enough time to work the problem, we'd be able to explain the outcome using classical mechanics.

The point of Bell's theorem is that if such an explanation exists for the apparent randomness of quantum mechanical results, that explanation is necessarily non-local.

Here is a question for you (and it is not a rhetorical question):

Have you read and understood the EPR paper and Bell's paper? If you haven't read them, you're wasting your time and ours. If you have read them, and there are parts of the arguments that you don't follow, ask and we can have a more focused and productive discussion.

 

[johana]

But this is not what $A(a,\lambda_A)$ means in Bell's formula. It means the result for a single photon with hidden variable $\lambda_A$ that passes through a polarizer set at angle $a$. This single result must be $+1$ or $-1$.

$A(a,\lambda_A)$ does not mean the average value over 10,000 photons.

Malus' law is a probability function with sample space S={+,-}. It means every single photon either goes "+" way or "-" way. Number of photons is arbitrary, average probability applies equally to each individually or all of them together.

 

[johana]

We produce random outputs from a completely deterministic system in which none of the hidden variables are probability functions all the time; throwing a six-sided die or tossing a coin are obvious examples. If we knew the exact initial conditions and had enough time to work the problem, we'd be able to explain the outcome using classical mechanics.

That's exactly what local hidden variable is supposed to achieve, to predict outcome based on initial conditions and classical mechanics, including classical probability, such as used for predicting six-sided dice roll or coin toss outcome.

We toss a coin 10,000 times and we get this kind of random sequence:

H H T H T T H T H T T H T H H H T T H T

...we send 10,000 unpolarized photons through a polarizer and we get the same kind of random sequence:

+ - - + - + + - + + - + - - + + + - + - -

Outcome of coin toss sequence can be defined with probability function. Is there any other type of function in classical physics, beside probability, which is able to define or represent such RANDOM outcome? I don't think so, and so I conclude: if local hidden variable exists it would have to be a probability function. That's all.

 

[Nugatory]

Closed - this discussion is no longer adding any value.