S(\tilde{\rho}_j) - \sum_i P(i|j) S(\tilde{\rho}_{j|i}),$$ where $\tilde{\rho}_j = A_j \rho A_j^\dagger/\mbox{Tr}[A_j^\dagger A_j \rho]$. If at least one of the measurement operators $A_j$ are higher than rank 1, then the measurement is incomplete. If the measurement is complete, then for each $j$ all the final states $\tilde{\rho}_{j|i}$ are identical, $\chi_j$ is zero and the SWW bound reduces to the Holevo bound.

The most general kind of measurement can also be inefficient. A measurement is described as inefficient if the observer does not have full information regarding which of the outcomes actually occurred. The name [*inefficient*]{} comes from that fact that the need to consider such measurements first arose in the study of inefficient photo-detectors. [@inefficient] An inefficient measurement may be described by labeling the measurement operators with two indices, so that we have $A_{kj}$. The receiver has complete information about one of the indices, $j$, but no information about the other, $k$. [@note2] As a result the final state for each $j$ (given the value of $i$) is now $$\rho'_{j|i} = \sum_k P(k|j) \frac{A_{kj} \rho_i A_{kj}^\dagger}
                                    {\mbox{Tr}[A_{kj}^\dagger A_{kj} \rho_i]} .$$ Since inefficiency represents a loss of information, we wish to ask whether it is possible to take this into account and 