A simple and flexible scheme for high-dimensional linear quantum operations on optical transverse spatial modes is demonstrated. The quantum Fourier transformation (QFT) and quantum state tomography (QST) via symmetric informationally complete positive operator-valued measures (SIC POVMs) are implemented with dimensionality of 15. The matrix fidelity of QFT is 0.85, while the statistical fidelity of SIC POVMs and fidelity of QST are ~0.97 and up to 0.853, respectively. We believe that our device has the potential for further exploration of high-dimensional spatial entanglement provided by spontaneous parametric down conversion in nonlinear crystals.
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