  Special approximation technique for analysis of different characteristics of
states of multipartite infinite-dimensional quantum systems is proposed and
applied to study of the relative entropy of entanglement and its
regularisation. We prove several results about analytical properties of the
multipartite relative entropy of entanglement and its regularization (the lower
semicontinuity on wide class of states, the uniform continuity under the energy
constraints, etc.). We establish a finite-dimensional approximation property
for the relative entropy of entanglement and its regularization that allows to
generalize to the infinite-dimensional case the results proved in the
finite-dimensional settings.
