Two scientists (they happen to be twins, named ``Oivil'' and ``Livio'', but never mind\ldots) decide to do the following experiment: They set up a light source, which emits two photons at a time, back-to-back in the laboratory frame. The ensemble is given by:
ρ=21(∣LL⟩⟨LL∣+∣RR⟩⟨RR∣),(4.128)
where ``L'' refers to left-handed polarization, and ``R'' refers to right-handed polarization. Thus, ∣LR⟩ would refer to a state in which photon number 1 (defined as the photon which is aimed at scientist Oivil, say) is left-handed, and photon number 2 (the photon aimed at scientist Livio) is right-handed.
These scientists (one of whom is of a diabolical bent) decide to play a game with Nature: Oivil (of course) stays in the lab, while Livio treks to a point a light-year away. The light source is turned on and emits two photons, one directed toward each scientist. Oivil soon measures the polarization of his photon; it is left-handed. He quickly makes a note that his brother is going to see a left-handed photon, sometime after next Christmas.
Christmas has come and gone, and finally Livio sees his photon, and measures its polarization. He sends a message back to his brother Oivil, who learns in yet another year what he knew all along: Livio's photon was left-handed.
Oivil then has a sneaky idea. He secretly changes the apparatus, without telling his forlorn brother. Now the ensemble is:
ρ=21(∣LL⟩+∣RR⟩)(⟨LL∣+⟨RR∣).(4.129)
He causes another pair of photons to be emitted with this new apparatus, and repeats the experiment. The result is identical to the first experiment.
(a) Was Oivil just lucky, or will he get the right answer every time, for each apparatus? Demonstrate your answer explicitly, in the density matrix formalism.
(b) What is the probability that Livio will observe a left-handed photon, or a right-handed photon, for each apparatus? Is there a problem with causality here? How can Oivil know what Livio is going to see, long before he sees it? Discuss! Feel free to modify the experiment to illustrate any points you wish to make.