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## Main Question or Discussion Point

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(ρ

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, ρ

Please help!

S(ρ

_{AB}||ρ_{A}[itex]\otimes[/itex]ρ_{B}) = Tr(ρ_{AB}logρ_{AB}) - Tr(ρ_{AB}logρ_{A}[itex]\otimes[/itex]ρ_{B}) = Tr_{AB}(ρ_{AB}logρ_{AB}) - Tr(ρ_{A}logρ_{A}) - Tr(ρ_{B}logρ_{B})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, ρ

_{AB}is a density operator acting on the Hilbert space of the bipartite system and the rest should be self-explanatory.Please help!