and $\chi_j$ is the Holevo quantity for the ensemble that the system remains in (from the point of view of the receiver), given outcome $j$. This bound can be at least partially understood by noting that if the system still remains in some ensemble of possible states after the measurement, then future measurements can potentially extract further information about the encoding, and so the information obtained by the first measurement must therefore be less than the maximum possible by at least by this amount. What the SWW bound tells us is that the bound on the information is reduced not only by the amount of information which could be further extracted after outcome $j$ has been obtained, but by the Holevo bound on this information, $\chi_j$.

If the initial state of the system is $\rho_i$, then after outcome $j$ the final state of the system is given by $\tilde{\rho}_{j|i} = A_j \rho_i
A_j^\dagger/\mbox{Tr}[A_j^\dagger A_j \rho_i]$. Thus the states which make up the final ensemble that remains after outcome $j$, are $\{\tilde{\rho}_{j|i}\}$, and the probability of each state in the ensemble is $P(i|j) =
P(j|i)P(i)/P(j)$, with $P(j|i) = \mbox{Tr}[A_j^\dagger A_j \rho_i]$. The Holevo quantity for ensemble $j$ is thus $$\chi_j = 