| Yes, it is just Malus's law. The key is what angle is relevant. Suppose you did CHSH but instead of pairs of entangled photons with was pairs of non-entangled photons that were polarized in the same direction. They players do the same thing as with entangled photons: use the bit from the referee to pick their measurement angle. A measures at 0 or 45 degrees, where 0 is the axis the photons were polarized on. B measures at -22.5 or 22.5. Let's say the 0 degrees the players are using is the direction of the original polarization axes. When the referee gives a 0 to A then A is measuring on the same axis the photon was polarized on, so will get a 1. When the referee gives a 1 to A then A measures at 45 and Malus gives a 50/50 chance of 1. Player B is always measuring 22.5 from 0, so Malus says B gets a 1 85% of the time. That gives us this: Ref A Ref B A's 1/0 chances B's 1/0 chances
0 0 100/0 85/15
0 1 100/0 85/15
1 0 50/50 85/15
1 1 50/50 85/15
In the last two rows the players win 50% of the time (due to A's 50/50). In the first two rows they win when they get the same bit, which happens 85% of the time. Since all 4 referee results are equally likely, the result is the players win 67.5% of the time.If the player's setup isn't aligned with the initial axis the result will differ. For example let's say their set up is 10 degrees off from the setup described above. Then their angles are 10 and 55 for A and -12.5 and 32.5 for B. If I did the numbers right they will win around 62% of the time. Without entanglement when each player measures a photon the θ for Malus's law is the angle between the axis they measure and the axis the photon was originally polarized with. With entanglement the θ is the angle between the two axis that the players used. |