Fig. 3: Two and three photons in ABS interferometers - platforms A and B1.

a The experiment implements an ABS scheme [6, 2, 2, 3]. The circuit is encoded in a 6-mode universal programmable chip. In particular, we have a six-mode U₀ and then three adaptive transformations U_i. The phase shifters ϕ (rectangles in the figure) and beam-splitter reflectivities θ (circles) are set to angles θ, ϕ = π/4 except for the θ_i of the U_i highlighted in red, orange and yellow that depends on the detection of one photon in the o_j. Pairs of indistinguishable photons generated by parametric down-conversion evolve in such an interferometer and a qubit tomography conditioned on the detection o_i is performed in the green part of the circuit. b Comparison between the numerically simulated kernel and the experimental one, the latter computed via the mutual state fidelity between the states reconstructed at the output of the programmable integrated optical circuit. c Experimental \({\rho }_{i}\) density matrices for the quantum states ρ₁ (top) and ρ₂ (bottom). We retrieved the density matrix by performing the qubit tomography with a tunable beam-splitter and phase-shifter. Uncertainties due to photon-counting statistics are smaller than the image scale. d 3-photon experiment in the 8-mode device. In this scenario we have r = 2 photons detected in 6 adaptive modes. We have a total amount of 15 transformations each of them triggered by the detection of two photons in a pair of the 6 outputs. The optical circuit is divided into an 8-mode unitary U₀, and five transformations U_i, activated and combined according to the configurations of the r photons detected in the 6 output modes. The reflectivity values of the beam-splitters θ_i in red, orange, yellow, teal, and violet, depend on where the ancillary photons are detected, according to the formula displayed in the figure. e Comparison of the 15 × 15 kernels computed according to the theoretical modeling which assumes an imperfect single-photon source and the kernel reconstructed from the states measured at the output of the ABS scheme [8, 3, 2, 15]. Both experiments were carried out with APDs.