Approximation of multipartite quantum states: revised version with new applications

Special approximation technique for analysis of different characteristics ofstates of multipartite infinite-dimensional quantum systems is proposed andapplied to the study of the relative entropy of $\pi$-entanglement and itsregularisation. In particular, by using this technique we obtain simple sufficient conditionsfor local continuity (convergence) of the regularized relative entropy of$\pi$-entanglement. We establish a finite-dimensional approximation property for the relativeentropy of entanglement and its regularization that allows us to generalize tothe infinite-dimensional case the results proved in the finite-dimensionalsettings. We also show that for any multipartite state with finite energy the infimumin the definition of the relative entropy of $\pi$-entanglement can be takenover the set of finitely-decomposable $\pi$-separable states with finiteenergy.