obtain a more stringent bound on the mutual information. If we merely apply the SWW bound to the measurement $A_{kj}$, then the bound involves the Holevo quantities of the ensembles that remain when both the values of $k$ and $j$ are known (the final ensembles that result from the efficient measurement). That is $$M(I\! :\! J) \leq \chi - \sum_{kj} P(k,j) \chi_{kj} .$$ One therefore wishes to know whether it is possible to derive a bound which instead involves the Holevo quantities of the ensembles that remain after the inefficient measurement is made, that is, for the receiver who only has access to $j$.

In the first part of this paper we answer this question in the affirmative - for an inefficient measurement where the known outcomes are labeled by $j$, the bound given by Eq.(\[SWW\]) [*remains true*]{}, where now the $\chi_j$ are the Holevo quantities for the ensemble of states $\rho'_{j|i}$ which result from the inefficient measurement.

In the second part of the paper, we consider the average reduction in the von Neumann entropy induced by a measurement: $$\langle \Delta S(\rho) \rangle \equiv S(\rho) - \sum_i P(j) S(\rho'_{j}).$$

Here $\rho'_j$ is the state that results from outcome $j$, given 