Projective quantum measurements on spatial modes of the photon with transmission volume holograms

Transmission volume holograms are evaluated as quantum projectors operating on spatial modes of the photon in mutually unbiased bases (MUBs). With applications to free-space quantum key distribution (QKD) in mind, state spaces based on rectilinear and azimuthal phase modes (i.e. photon linear and orbital angular momenta) are considered. Rectilinear phase modulation is shown to result in both greater mode densities for a receiving aperture in the far field and better efficiency and cross-talk characteristics when volume holograms are used as de-multiplexing elements. Twoand four-dimensional state spaces are defined via rectilinear phase modes and the complex optical fields of the MUBs are calculated and generated with a spatial light modulator in order to record and subsequently illuminate transmission volume holograms. Using holograms prepared in lithium niobate, diffraction efficiencies are measured for the 36 permutations associated with projecting the six MUB states of a two-dimensional state space onto the same six MUB states. Quantum measurements associated with cascaded projection operations in a four-dimensional state space are performed using photo-thermo-refractive glass holograms. Experimental results show approximate agreement with the inner-product relationships that describe quantum projection probabilities.

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