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