This is probably my misunderstanding of the notation... The definition of a density matrix is in the attached file. (Sorry, the latex editor is not rendering properly when I preview my post). This definition is a sum over only one index 'j', which will invariably lead to a diagonal matrix. But the density matrix is often not diagonal, for instance a basic superposition (pure) state. What am I missing in this definition? Shouldn't it be over indices i and j, for instance, in a two dimensional system?