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.

Figure 1

(a) Definition of the discrete spatial modes. (b) Schematic setup for implementing linear quantum operators. A seven-dimensional QFT demonstration is shown as a concrete example. Simulated field evolution after each SLM is presented. Phase-modulation pattern implemented on SLM1 (c) and SLM2 (d) for beam splitting and recombining, respectively.

Figure 2

Simulated fidelity (a) and efficiency (b) for one-to-$N$ beam splitting. (c) The transverse coordinates of the spatial modes used are arranged on a circle.

Figure 3

Experimental setup. The insets show modulation functions on SLM1 and SLM2, together with an enlarged image of a typical phase grating.

Figure 4

Quantum-state tomography via the compressed-sensing method. (a),(b) Projective measurements. (c),(d) Reconstructed density matrices.

Figure 5

Fidelity and trace distance for our reconstructed density matrix versus sampling ratio.

Figure 6

QFT with dimensionality of $15\times 15$ tested with the conjugate Fourier basis. (b) Illustration for generation of high-dimensional Bell states and (c) equivalent order-finding routine with high-dimensional entangled photons, where $f(x)=2^x(\mathrm{mod}N)$ denotes a modular exponential function. A type-I SPDC cone photographed by a near-infrared CMOS camera is shown.

Figure 7

The results for the calibrated DFT matrix under a Fourier basis. For different input Fourier states, the output of the DFT matrix will be single Gaussian spots in different positions.

Figure 8

(a) Matrix-fidelity variation of the calibrated $15\times 15$ DFT demonstration over 1 week. (b)–(d) Output vector of DFT with the input Fourier basis state $|{\omega }_3⟩$.

Figure 9

Histogram for normalized fidelity of different unitary matrices with dimensionality of $15\times 15$. The phase gratings are not optimized in these designs. The histogram is obtained through Monte Carlo simulation with 1000 trials.

Figure 10

Four groups of state tomography with a classical coherent-light source. The three columns correspond to the results for SIC POVMs, experimental density matrices, and theoretical density matrices, respectively. (a) Results for a single-mode eigenstate. (b)–(d) Results for superposed states. The phase information for matrix elements with modulus $|{\rho }_{ij}|<0.02$ is approximately randomly distributed and not displayed to aid clarity.

Figure 11

Two groups of quantum projective measurements and quantum-state tomography. The three columns correspond to the results for SIC POVMs, experimental density matrices, and theoretical density matrices, respectively. The statistical fidelities of SIC POVMs are 0.934 (a) and 0.936 (b). The fidelities of quantum-state tomography are 0.806 and 0.738.