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Special approximation technique for analysis of different characteristics ofstates of multipartite infinite-dimensional quantum systems is proposed andapplied to study of the relative entropy of entanglement and itsregularisation. We prove several results about analytical properties of themultipartite relative entropy of entanglement and its regularization (the lowersemicontinuity on wide class of states, the uniform continuity under the energyconstraints, etc.). We establish a finite-dimensional approximation propertyfor the relative entropy of entanglement and its regularization that allows togeneralize to the infinite-dimensional case the results proved in thefinite-dimensional settings.

Approximation of multipartite quantum states and the relative entropy of entanglement