I'm having problems understanding the trace of tensor products when the density matrix is expressed in its reduced density operators. The proof of subadditivity is quite simple. S(ρAB||ρA[itex]\otimes[/itex]ρB) = Tr(ρABlogρAB) - Tr(ρABlogρA[itex]\otimes[/itex]ρB) = TrAB(ρABlogρAB) - Tr(ρAlogρA) - Tr(ρBlogρB) This carries on to finalize the proof. But this last step is the step where I'm at a loss. How is the trace over AB (the second term in the first step) expanded into the respective partial traces over A and B (the second and third term in the last step)? Here, ρAB is a density operator acting on the Hilbert space of the bipartite system and the rest should be self-explanatory. Please help!