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.(adsbygoogle = window.adsbygoogle || []).push({});

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!

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# Proof of subadditivity of quantum entropy

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