- #511
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Let me try to clarify what, exactly, is impossible in a local hidden variables theory.
First, what is the correlation function, which I'm calling C(\alpha, \beta)?
If you have a list, or run, of 4-tuples of numbers: \alpha_n, A_n, \beta_n, B_n where for each n, \alpha_n and \beta_n are angles, and A_n and B_n are each either +1 or -1, then you can compute a correlation C(\alpha, \beta) as follows:
C(\alpha, \beta) = \frac{1}{N_{\alpha \beta}} \sum A_n B_n
where the sum is over those runs n such that \alpha_n = \alpha and \beta_n = \beta, and where N_{\alpha \beta} is the total number of such runs.
Suppose that we generate such a list as follows: We create a sequence of spin-1/2 twin pairs. On run n, one particle is detected by Alice using a spin-measurement device aligned in the x-y plane at angle \alpha_n away from the x-axis, and the other is detected by Bob at an angle \beta_n. If Alice measures spin-up, then A_n = +1. If Alice measures spin-down, then A_n = -1. If Bob measures spin-up, then B_n = +1. If Bob measures spin-down, then B_n = -1.
The prediction of quantum mechanics is that in the limit as the number of trials at each angle goes to infinity, is that
C(\alpha, \beta) = - cos(\beta - \alpha)
At this point, let's specialize to specific values for \alpha and \beta. Assume that \alpha and \beta are always chosen to be from the set
{ 0°, 120°, 240°}. A way to explain the correlations using deterministic local hidden variables is to assume that corresponding to run number n there is a hidden variable \lambda_n, which is (or determines) a triple of values \lambda_n = \langle A_{0\ n}, A_{120\ n}, A_{240\ n}\rangle. Then A_n and B_n are deterministic functions of the angles \alpha and \beta and the "hidden variable" \lambda_n:
A_n = A_{\alpha_n\ n}
B_n = B_{\beta_n\ n} \equiv - A_{\beta_n\ n}
Now, here's where the impossibility claim arises: If we have a sequence of triples \langle A_{0\ n}, A_{120\ n}, A_{240\ n}\rangle, then we can compute correlation functions as follows:
C'(\alpha \beta) = \frac{1}{N} \sum A_{\alpha\ n} B_{\beta\ n}
where N is the total number of runs.
I'm using a prime to distinguish this correlation function from the previous. The difference between the two is that C(\alpha \beta) is computed using those runs in which Alice happens to choose detector angle \alpha, and Bob happens to choose detector angle \beta. In contrast, C'(\alpha \beta) is computed using all runs, since by assumption, A_{\alpha\ n}, B_{\beta\ n} determines what Alice and Bob would have gotten on run n had they chosen settings \alpha and \beta.
The impossibility claim is that there is no sequence of triples \langle A_{0\ n}, A_{120\ n}, A_{240\ n}\rangle such that the corresponding C'(\alpha, \beta) agrees with the quantum mechanical prediction for the correlation.
This impossibility claim is NOT contradicted by actual experiments, because an actual experiment cannot measure the triple \langle A_{0\ n}, A_{120\ n}, A_{240\ n}\rangle; it can only measure two of the three values.
First, what is the correlation function, which I'm calling C(\alpha, \beta)?
If you have a list, or run, of 4-tuples of numbers: \alpha_n, A_n, \beta_n, B_n where for each n, \alpha_n and \beta_n are angles, and A_n and B_n are each either +1 or -1, then you can compute a correlation C(\alpha, \beta) as follows:
C(\alpha, \beta) = \frac{1}{N_{\alpha \beta}} \sum A_n B_n
where the sum is over those runs n such that \alpha_n = \alpha and \beta_n = \beta, and where N_{\alpha \beta} is the total number of such runs.
Suppose that we generate such a list as follows: We create a sequence of spin-1/2 twin pairs. On run n, one particle is detected by Alice using a spin-measurement device aligned in the x-y plane at angle \alpha_n away from the x-axis, and the other is detected by Bob at an angle \beta_n. If Alice measures spin-up, then A_n = +1. If Alice measures spin-down, then A_n = -1. If Bob measures spin-up, then B_n = +1. If Bob measures spin-down, then B_n = -1.
The prediction of quantum mechanics is that in the limit as the number of trials at each angle goes to infinity, is that
C(\alpha, \beta) = - cos(\beta - \alpha)
At this point, let's specialize to specific values for \alpha and \beta. Assume that \alpha and \beta are always chosen to be from the set
{ 0°, 120°, 240°}. A way to explain the correlations using deterministic local hidden variables is to assume that corresponding to run number n there is a hidden variable \lambda_n, which is (or determines) a triple of values \lambda_n = \langle A_{0\ n}, A_{120\ n}, A_{240\ n}\rangle. Then A_n and B_n are deterministic functions of the angles \alpha and \beta and the "hidden variable" \lambda_n:
A_n = A_{\alpha_n\ n}
B_n = B_{\beta_n\ n} \equiv - A_{\beta_n\ n}
Now, here's where the impossibility claim arises: If we have a sequence of triples \langle A_{0\ n}, A_{120\ n}, A_{240\ n}\rangle, then we can compute correlation functions as follows:
C'(\alpha \beta) = \frac{1}{N} \sum A_{\alpha\ n} B_{\beta\ n}
where N is the total number of runs.
I'm using a prime to distinguish this correlation function from the previous. The difference between the two is that C(\alpha \beta) is computed using those runs in which Alice happens to choose detector angle \alpha, and Bob happens to choose detector angle \beta. In contrast, C'(\alpha \beta) is computed using all runs, since by assumption, A_{\alpha\ n}, B_{\beta\ n} determines what Alice and Bob would have gotten on run n had they chosen settings \alpha and \beta.
The impossibility claim is that there is no sequence of triples \langle A_{0\ n}, A_{120\ n}, A_{240\ n}\rangle such that the corresponding C'(\alpha, \beta) agrees with the quantum mechanical prediction for the correlation.
This impossibility claim is NOT contradicted by actual experiments, because an actual experiment cannot measure the triple \langle A_{0\ n}, A_{120\ n}, A_{240\ n}\rangle; it can only measure two of the three values.
)
...okay, a completely new definition of scientific refutability... well... let’s see, if you in this brand new epic light want to refute The Flat Earth Society... eh... you must (did I really get that right?? ) first prove that Earth is flat??