Fig. 3: Tomography fidelity and matrix distances for DEMESST and OLI sampling methods.

Sampling was performed on approximate entangled W states of 2--4 modes. The top (a)–(c) show fidelities to an ideal (exactly equal population coefficients) W state for 2–4 modes, with dashed horizontal lines indicating the final converged fidelity obtained from the OLI method. These final fidelities are, for 2–4 modes, 0.966 ± 0.005, 0.949 ± 0.004, and 0.912 ± 0.007 for OLI and 0.96 ± 0.01, 0.954 ± 0.004, and 0.911 ± 0.007 for DEMESST and are in good agreement. The bottom (d)–(f) show Frobenius norm matrix distances between the state at a given measurement number versus the final measured state. Error bars indicate the standard error. The rates of convergence are close to 1/\(\sqrt{x}\) or a power of −0.5, as expected. As the mode number increases, the DEMESST method performs increasingly more efficiently by requiring fewer measurements to reach a given level of convergence or error threshold.