Conditional path probabilities after a quarter-wave plate

A detected lower-path photon with vertical polarization evolves under an angle-dependent Jones matrix. For fast-axis angle $\eta$, its horizontal amplitude is $(1-i)\sin \eta \cos \eta$ and its vertical amplitude is ${\sin }^2\eta +i{\cos }^2\eta$. A later vertical-path probability therefore depends on the plate orientation and on whether the earlier lower-path event has been conditioned on.