How does a rectilinear filter used for QKD-BB84 detect both horizontal and vertical polarizations?

Question

When using the BB84 algorithm for QKD, you arbitrarily choose which of two filters (bases) to use when detecting a photon: rectilinear or diagonal. If you choose rectilinear, you can detect polarization with certainty if the photon is rectilinear, i.e., either vertically- or horizontally-polarized.

But how? A polarization filter that allows transmission of vertically-polarized photons will block horizontally-polarized photons and vice-versa, no? Perhaps you use a vertically-polarized filter as the rectilinear filter and then the complete lack of a photon indicates horizontal polarization, but how would you distinguish that from, say, absorption of the photon into the fiber?

Answer 1

I don't know the experimental details, but one option from the theory perspective is that if you know there's supposed to be a photon there, absence of a click conveys the other measurement. Of course, as you say, that's not going to hold up in situations of finite detector efficiency, photon loss during transmission, or even knowing there's been successful photon generation, etc.

Perhaps a better option is to replace the filter with a polarising beamsplitter. So, one polarisation gets transmitted, and the orthogonal one gets reflected, so that you can direct both components into different detectors.