Question 11

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Question 11.

Dear Eric,

Our definition of the quantum relative entropy (equation (1.6) of Notes 13) is for \(\rho\) a state and \(\sigma\) a positive definite operator, but during the proof of Klein’s inequality in the same notes, we seem to make use that \(\sigma\) has trace 1 although it was not one of the assumptions! (This is when showing that \(r_i\) is a probabililty distribution).

Good point! Klein’s inequality only holds when both arguments of the quantum relative entropy are states. So while we can define \(D(\rho,\sigma)\) only assuming \(\sigma\geq 0\) to use the inequality \(D(\rho,\sigma)\geq 0\) we do need \(\operatorname{tr}\sigma=1\). In other words, it should be one of the assumptions for Klein’s inequality!

By the way, this is why the conditional entropy \[ S(A|B)_\rho = -D(\rho_{AB} \| I_A \otimes \rho_B) \] can be positive or negative (since Klein’s inequality doesn’t hold for \(\sigma = I_A\otimes \rho_B\)).