Figure 1

Experimental scheme. (a) Whole process of the experimental scheme. First, we prepare an initial system state and auxiliary state ${\psi }_{00}^{\overline{3}}\otimes {\phi }^3$. Then we complete the preparation of nine Bell states through the three-dimensional arbitrary unitary operation of path DOF. Finally, we distinguish the decomposition state $\alpha {ij}^{\overline{3}}$ via a Bell-state analyzer (BSA) and complete the HDBSM of the system state by coincidence. (b) State-preparation scheme. Photons are created via spontaneous parametric down conversion (SPDC) by pumping type-I phase-matched $\beta$-barium borate (BBO) nonlinear crystals with a cw laser. Because SPDC satisfies OAM conservation, thus the state prepared by SPDC comes to be $\left(\right|00\rangle +|11\rangle +|22\rangle )\otimes (|aa\rangle +|bb\rangle +|cc\rangle )/3$. Through the unitary operation on one photon's path DOF, we can change the system state into nine Bell states. (c) Path-OAM BSA. It is the realization of three-dimensional BSM for a single-photon with different DOFs (path and OAM). We convert the OAM DOF into path DOF by splitting the beam according to its OAM, and then the BSM between two DOFs can be completed by appropriate path distribution and unitary operation.