A complete four-dimensional Bell-state measurement is impossible with linear elements (26). However, one can separate the 16 four-dimensional Bell states into seven classes and select only one state from each class for SDC (27, 28). In our experiment, we only choose five states given in Eq. 1. Then, Bob can determine which state Alice prepared by the measurement setup shown in Fig. 1. When the two photons inject into PBS1, they are separated into two classes by photon’s polarization. For states Ψ11, Ψ12, Ψ13, and Ψ14, one photon is reflected and the other one is transmitted, while for state Ψ23, both photons are reflected or transmitted. After HWP H1 (set at 22.5°) and PBS2, states Ψ11, Ψ12, Ψ13, and Ψ14 are separated into two classes by the relative phase between |00〉 and |11〉, for example, if the relative phase is 0 (Ψ11, Ψ13), then the two photons will both transmit or reflect from PBS2. Here, this separation depends on the two-photon Hong-Ou-Mandel interference (12). Finally, the last part of the measurement setup (consisting of HWPs, BDs, and PBSs) will separate Ψ11 and Ψ1312 and Ψ14) to two classes by the relative phase between |00〉 and |22〉. By carefully checking the coincidental events among those single-photon detectors, Bob can determine which state Alice prepared. For example, coincidence between D1 and D5 means that the state Alice prepared is Ψ11. Thus, Bob can determine which state Alice prepared and decode the information that Alice encoded.

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