The information of high-dimensional time-bin encoded photons

We determine the shared information that can be extracted from time-bin entangled photons using frame encoding. We consider photons generated by a general down-conversion source and also model losses, dark counts and the effects of multiple photons within each frame. Furthermore, we describe a procedure for including other imperfections such as after-pulsing, detector dead-times and jitter. The results are illustrated by deriving analytic expressions for the maximum information that can be extracted from high-dimensional time-bin entangled photons generated by a spontaneous parametric down conversion. A key finding is that under realistic conditions and using standard SPAD detectors one can still choose frame size so as to extract over 10 bits per photon. These results are thus useful for experiments on high-dimensional quantum-key distribution system.