David Kaiser in Scientific American:
Physicists had long known that the two flavors of polarization—plane or circular—were intimately related. Plane-polarized light could be used to create circularly polarized light, and vice versa. For example, a beam of H-polarized light consisted of equal parts R– and L-polarized light, in a particular combination, just as a beam of R-polarized light could be broken down into equal parts H and V. Likewise for individual photons: a photon in state R, for example, could be represented as a special combination of states H and V. If one prepared a photon in state R but chose to measure plane rather than circular polarization, one would have an equal probability of finding H or V: a single-particle version of Schrödinger’s cat.
In Herbert's imagined set-up, one physicist, Alice (“Detector A” in the illustration), could choose to measure either plane or circular polarization of the photon headed her way . If she chose to measure plane polarization, she would measure H and Voutcomes with equal probability. If she chose to measure circular polarization, she would find R and L outcomes with equal probability.
In addition, Alice knows that because of the nature of the source of photons, each photon she measures has an entangled twin moving toward her partner, Bob. Quantum entanglement means that the two photons behave like two sides of a coin: if one is measured to be in state R, then the other must be in state L; or if one is measured in state H, the other must be in state V. The kicker, according to Bell's theorem, is that Alice's choice of which type of polarization to measure (plane or circular) should instantly affect the other photon, streaming toward Bob . If she chose to measure plane polarization and happened to get the result H, then the entangled photon heading toward Bob would enter the state V instantaneously. If she had chosen instead to measure circular polarization and found the result R, then the entangled photon instantly would have entered the state L.
Next came Herbert's special twist.