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3D compact photonic circuits for realizing quantum state tomography of qudits in any finite dimension

In this work, we propose three-dimensional photonic circuit designs that guarantee a considerable reduction in the complexity of circuits for the purpose of performing quantum state tomography of N-dimensional path qudits. The POVM (Positive Operator-Valued Measure) chosen in this work ensures that, for odd dimensions, such process is minimal. Our proposal consists of organizing the waveguides that form the circuit as a square array formed by N vertical sectors composed of N waveguides each, arranged in the vertical direction. Based on the symmetry of the chosen POVM, the interferometer acting on the initial quantum system can be divided into a sequence of three different unitary operations. These operations act independently on each vertical sector, or layer, of the circuit, which simplifies their determination. We have thus obtained circuits such that the number of beam splitters obeys a polynomial function of degree 3 with the quantum system dimension, whereas in current proposals this quantity grows with a polynomial function of degree 4. Besides that, the optical depth is reduced from a quadratic to a linear function of the quantum system dimension in our scheme. These results confirm the remarkable reduction of the complexity of the photonic circuits in our proposal.