  We have studied quantum data compression for finite quantum systems where the
site density matrices are not independent, i.e., the density matrix cannot be
given as direct product of site density matrices and the von Neumann entropy is
not equal to the sum of site entropies. Using the density-matrix
renormalization group (DMRG) method for the 1-d Hubbard model, we have shown
that a simple relationship exists between the entropy of the left or right
block and dimension of the Hilbert space of that block as well as of the
superblock for any fixed accuracy. The information loss during the RG procedure
has been investigated and a more rigorous control of the relative error has
been proposed based on Kholevo's theory. Our results are also supported by the
quantum chemistry version of DMRG applied to various molecules with system
lengths up to 60 lattice sites. A sum rule which relates site entropies and the
total information generated by the renormalization procedure has also been
given which serves as an alternative test of convergence of the DMRG method.
