  A renormalized version of the von Neumann quantum entropy (which is finite
and continuous in general, infinite dimensional case) and which obeys several
of the natural physical demands (as expected for a "good" measure of
entanglement in the case of general quantum states describing bipartite and
infinite-dimensional systems) is proposed. The renormalized quantum entropy is
defined by the explicit use of the Fredholm determinants theory. To prove the
main results on continuity and finiteness of the introduced renormalization the
fundamental Grothendick approach, which is based on the infinite dimensional
Grassmann algebra theory, is applied. Several features of majorization theory
are preserved under then introduced renormalization as it is proved in this
paper. This fact enables us to extend most of the known (mainly, in the context
of two-partite, finite-dimensional quantum systems) results of the LOCC
comparison theory to the case of genuine infinite-dimensional, two-partite
quantum systems.
