Scaling of success probabilities for linear optics gates

By using the abstract linear-optical network derived by Scheel and Lütkenhaus (2004 New J. Phys. 6 51) we show that for the lowest possible ancilla photon numbers the probability of success of realizing a (single-shot) generalized nonlinear sign-shift gate on an (N + 1)-dimensional signal state scales as 1/N2. We limit ourselves to single-shot gates without conditional feed-forward. We derive our results by using determinants of Vandermonde-type over a polynomial basis which is closely related to the well-known Jacobi polynomials.