\rangle$ is always positive[@Ozawa] (for more recent proofs of this result see[@Nielsen; @FJ]). For efficient measurements Eq.(\[genHall\]) is therefore in general stronger than, and gives immediately, Hall’s bound [@Hall; @ib], which states that the mutual information is bounded by the reduction in the von Neumann entropy. The inequality in Eq.(\[genHall\]) is then a generalization of Hall’s bound to inefficient measurements. Since the mutual information is always positive, but for inefficient measurements the reduction in the von Neumann entropy can be negative (that is the entropy of the quantum state can [*increase*]{} as a result of the measurement), the relation $$M(I\! :\! J) \leq \langle \Delta S(\rho)\rangle$$ is not necessarily satisfied for such measurements. However, Eq.(\[genHall\]) tells us that if the entropy of the intial state, $\rho$, does increase, the average increase in the entropy for each of the coding states $\rho_i$ is always [*more*]{} that this by at least the mutual information.

The second result that we obtain from Eq.(\[genHall\]) is that, because the mutual information is nonnegative, we have $$\langle \Delta S(\rho)\rangle \geq \sum_i P(i) \langle \Delta S(\rho_i)\rangle .$$ That is, the reduction in the von Neumann entropy is concave in the initial state. This parallels the fact that 