Fig. 1: Experimental set-up.

The heralded single photons are created via type-I SPDC and are injected into the optical network. The first polarizing beam splitter (PBS1), half-wave plates (HWP1–3), and beam displacer (BD1) are used to generate the pure qudit states. To prepare mixed states, quartz crystals (QCs) are inserted to destroy spatial coherence of the photons. The measurements are realized by wave plates and BDs. The photons are detected by APDs. We use the sequential measurement {A, a, α} as an example. The measurements of A, C, b, c, α, β, and γ can be realized by the set-up involving a PBS, BDs, and several wave plates, whereas the set-ups for realizing a and B can be simplified and involve only a PBS and HWPs. The set-up only corresponds to one of the eight possible paths and the input–output statistics have been obtained by repeating the experiment, with the appropriate choices of optical elements, for eight times and recording the number of photons in the detection time 2 s.