Entangled photons – Theoretical calculation of the correlation function

Without exchanging their measuring data, Alice and Bob cannot state whether their results are correlated with one another or not. The correlation function can, however, be calculated from quantum theory based on the assumption that Alice and Bob observe the entangled state Omega.

We have to determine the four combined probabilities: (□□, □■, ■□, ■■). The top row of representations relates to Alice’s white measurement result, the bottom row to Alice’s black measurement result.

On the right-hand side at the top, we see the probability distribution P(White, White) – a white oval, and implicitly also P(White, Black) – black crescents – of Bob.

On the right-hand side at the bottom the probability distribution P(Black, White) can be seen – a white oval, and implicitly also P(Black -Black) – black crescents – of Bob is shown.
Moreover, the probability distribution for Bob’s photon depends on the difference in the angle (β-α) of his and Alice’s measuring axis.
Should, for example, the difference in the angle be zero, then the probability of P (White, White) is 100% for Alice’s “White” measurement.

We now see two curves for Bob’s probability distribution, depending on the difference in the angle both for matching measurement results, and for for different measurement results.

If we work out the difference between the two probability distributions, we obtain the theoretically calculated correlation function C.