Optical qudit-type entanglement creation at long distances by means of small cross-Kerr nonlinearities

Entanglement represents an important resource for quantum information processing, but its generation itself requires physical resources that are limited. We propose a scheme for generating a wide class of entangled qudit-type states of optical field modes at sites separated by noisy medium when only weak optical nonlinearities are available at both sites. The protocol is also based on exploiting a weak probe field, transmitted between the sites and used for generation of quantum correlations between two spatially separated field modes. The idea of probabilistic entanglement enhancement by measurement is discussed, and corresponding scheme for measuring the probe field state with linear optics and photodetectors not resolving photon numbers is proposed. It is shown that the protocol is applicable in the case when decoherence, limited efficiency and dark counts of photodetectors, and uncertainty of nonlinear coupling constants are present.