sum up these results by saying that the statement that information gain increases with initial uncertainty can fail to hold only if the measurement is asymmetric in its sensitivity.

An information bound for general quantum operations
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We now show that the bound proved by SWW can be generalized to obtain a more stringent bound for channels in which the receivers measurement is inefficient. To show this it turns out that we can use the same method employed by SWW, but with the addition of an extra quantum system which allows us to include the inefficiency of the measurement.

For a quantum channel in which the encoding ensemble is $\varepsilon =
\{P(i),\rho_i\}$, and the measurement performed by the receiver is described by operators $A_{kj}$ ($\sum_{kj} A_{kj}^\dagger A_{kj} = 1$), where the measurement is in general inefficient so that the receiver knows $j$ but not $k$, then the mutual information, $M(I\! :\! J)$, is bounded such that $$M(I\! :\! J) \leq  \chi  - \sum_j P(j) \chi_j ,$$ where $P(j)$ is the overall probability for outcome $j$, $\chi = S(\rho) - \sum_i P(i) S(\rho_i)$ is the Holevo quantity for the initial ensemble and $$\chi_j =  S(\sigma_{j})  - \sum P(i|j) S(\sigma_{j|i}) ,$$ is 