I would like to know if the density matrix spectrum is always discrete or if it is possible it has a continuum spectrum. It is clear that a pure density matrix has a discrete spectrum but it is not obvious in general. I have heard that all compact operator has discrete eigenvalues and if it has an accumulation point it must be zero. It seems to me to be the case for reduced density matrix but as I am not a good mathematician I cannot see if a reduced matrix is or not a compact operator Where could I find some discussion about that topic? Thanks.