Programmable coherent linear quantum operations with high-dimensional optical spatial modes

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