1-page Bell's theorem

One particle is sent to Alice; the other to Bob; they may be very far apart.

Alice <----------------------------- source --------------------------------> Bob

Each can measure the ‘spin’ of their particle along some direction; each result is + or –. The probability that Alice and Bob get the same (+ or -) result as each other depends on the directions they measure along. For a certain type of source, if they both measure in the same direction, they always get opposite results.

Each of the Observers will ‘choose’ one of three directions: A, B, or C. This ‘choice’ can be made using any procedure or device, however complicated; therefore, it should be considered unpredictable, even though it may be made using deterministic physics.

Distant-Measurement-Independent Result (DMIR): The assumption that THE (single) result of each measurement can’t depend on which direction the other Observer ‘chose’.

Note: If DMIR is false there are 3 possibilities, of which the first two are taken seriously:
1) Nonlocality: An instant (faster-than-light) hidden signal which conveys the information about the measurement angle (which can be ‘chosen’ right before measurement) to the other particle, no matter where it is or how far away.
2) Multiple outcomes of each measurement actually do occur (as in the MWI).
3) “Conspiracy theories” in which the other particle somehow can predict the angle.

Assume DMIR. Then each particle needs to know in advance what result to give for any angle so that they will always give opposite results when both Observers choose the same angle. Hypothetical properties that’d determine the results are called hidden variables.

Notation: Let P(A+ & B-) mean the probability that the hidden variables are such that result + would be found by Alice if she measures along A, and result – if along B.

Since more-general cases are at least as probable as less-general ones:

P(A+ & B-) = P(A+ & B- & C-) + P(A+ & B- & C+) ≤ P(A+ & C-) + P(B- & C+)

It is not possible to measure Alice's particle along more than one direction, but measuring Bob’s particle should reveal the opposite of the result Alice's particle would have given. Let P(A+ // B+) be the probability that result + would be found by Alice in direction A, and result + would be found by Bob in direction B.

P(A+ // B+) ≤ P(A+ // C+) + P(B- // C-)

Quantum mechanically, if A and B are at a 90 degree angle, with the C direction halfway in between them, then P(A+ // B+) = .25,
and P(A+ // C+) = P(B- // C-) = .073

Since .25 > .146, the inequality is violated; DMIR is not consistent with QM.
Violations of such inequalities have been confirmed experimentally; DMIR is false!