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

Quantum state tomography is the conventional method used to characterize density matrices for general quantum states. However, the data acquisition time generally scales linearly with the dimension of the Hilbert space, hindering the possibility of dynamic monitoring of a high-dimensional quantum system. Here, we demonstrate a direct tomography protocol to measure density matrices of photons in the position basis through the use of a polarization-resolving camera, where the dimension of density matrices can be as large as 580$\times$580 in our experiment. The use of the polarization-resolving camera enables parallel measurements in the position and polarization basis and as a result, the data acquisition time of our protocol does not increase with the dimension of the Hilbert space and is solely determined by the camera exposure time (on the order of 10 ms). Our method is potentially useful for the real-time monitoring of the dynamics of quantum states and paves the way for the development of high-dimensional, time-efficient quantum metrology techniques.