Today's post, Decoherence as Projection was originally published on 02 May 2008. A summary (taken from the LW wiki):
Since quantum evolution is linear and unitary, decoherence can be seen as projecting a wavefunction onto orthogonal subspaces. This can be neatly illustrated using polarized photons and the angle of the polarized sheet that will absorb or transmit them.
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Hmm, I think I understand your question now, sorry for the confusion. Photon state at a spacetime point is actually not a single amplitude, but a pair of them, owing to the classical notion of polarization.
No, it is already a pair of complex numbers, one for each polarization. You can write it as (z_left, z_right). What is commonly done is factoring out a pure phase (a complex number of unit magnitude) and ignoring it (it does not affect the probability). For example, the state (i/sqrt(2), i/sqrt(2)) is "the same" as (1/sqrt(2),1/sqrt(2)) and the state (0,i) is "the same" as (0,1). I put "the same" in quotes because once you have more than one photon their relative phase actually matters.
Consider the polarization (0 ; -i). Eliezer describes this polarization as "a complex amplitude for up-down plus a complex amplitude for left-right". I read him as saying that, in the polarization (0 ; -i), the "complex amplitude for up-down" is the single complex number 0, while the "complex amplitude for left-right" is the single complex number -i. Am I misreading him?
ETA: Perhaps this is his intended reading: You can write (0 ; -i) = 0*(1 ; 0) + (-i)*(0 ; 1). Here, the "complex amplitude for up-down" is the ... (read more)