  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.
