  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.
