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Quantum state tomography is the conventional method used to characterizedensity matrices for general quantum states. However, the data acquisition timegenerally scales linearly with the dimension of the Hilbert space, hinderingthe possibility of dynamic monitoring of a high-dimensional quantum system.Here, we demonstrate a direct tomography protocol to measure density matricesof photons in the position basis through the use of a polarization-resolvingcamera, where the dimension of density matrices can be as large as580$\times$580 in our experiment. The use of the polarization-resolving cameraenables parallel measurements in the position and polarization basis and as aresult, the data acquisition time of our protocol does not increase with thedimension of the Hilbert space and is solely determined by the camera exposuretime (on the order of 10 ms). Our method is potentially useful for thereal-time monitoring of the dynamics of quantum states and paves the way forthe development of high-dimensional, time-efficient quantum metrologytechniques.

Direct tomography of high-dimensional density matrices for general quantum states of photons