Quantum coherence measurement of qudit systems using three-dimensional photonic circuits

Knowing that quantum coherence plays a central role in quantum phenomena, we propose, in this work, three-dimensional photonic circuits with compact designs that allow measurement of the quantum coherence of N-dimensional quantum systems from the probabilities of detecting the photons that emerge from their outputs. Here, we present two proposals: one based on the positive operator-valued measure (POVM) proposed by Carmeli et al. [New J. Phys. 20, 063038 (2018)] and the other based on the sum of the input-state partial visibilities, obtained by interfering pairs of base states. Based on the form of the POVM that we define for this task, the proposed interferometers are given by a sequence of different unitary operations. These operations act independently in each vertical sector, or layer, of the circuit, which simplifies its determination. We have thus obtained circuits whose number of beam splitters obeys a quadratic polynomial function and has a linear optical depth, which confirms its compactness and its reduced complexity.