# Question 11

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 $$\sigma = I_A\otimes \rho_B$$).