Upper Bounds on Success Probabilities in Linear Optics

Abstract

We shortly review the basic ideas of generating nonlinear operators with single‐photon sources, linear optical elements, and appropriate measurements of auxiliary modes. This theory allows for an algorithmic construction of useful single‐mode and multi‐mode quantum gates necessary for all‐optical quantum information processing. From this amplitude‐operator point of view all known schemes that realize the nonlinear sign shift gate can be generated, and generalizations to some multi‐mode gates are easily obtained.We will use these ideas of measurement‐induced nonlinearities to develop an abstract way of defining linear‐optics networks. That allows us to treat the problem of maximizing the success probability of generating these conditional quantum gates in a semi‐analytical way. In particular, we show that for a wide class of auxiliary states the success probability of generating a nonlinear sign shift gate does not exceed 1/4 which is the strongest bound to date. Particularly interesting is the observation that the Hilbert space dimension of the ancilla state can be kept to a minimum. We generalize the theory to show that the nonlinear sign shift on single‐mode states with up to N photons can be performed with a success probability that scales as 1/N2.