Figure 6

The ratio (15) plotted as a function of the number of paths n and the OAM dimension d. Orange dots represent exact values of \(r_{\mathrm{Reck}}\) for general unitaries implemented with Reck et al. scheme [17]. Violet dots correspond to the ratio \(r_{\mathrm{perm.}}\) for permutations, which form a special subset of unitaries. The orange surface is given by formula (30) that expresses the approximate scaling of ratio \(r_{\mathrm{Reck}}\). The violet surface is likewise given by formula (17) derived for permutations. The bottom right corner of the plot corresponds to \(n = d = 2048\), for which \(r_{\mathrm{Reck}} \sim 10^{-4}\) for general unitaries and \(r_{\mathrm{perm.}} \sim 10^{-3}\) for permutations. The gray plane divides the cases with \(n > d\) and \(n \leq d\)