  Calculation of the quantum discord requires to find the minimum of the
quantum conditional entropy $S(\rho^{AB}|\{\Pi^B_{k}\})$ over all measurements
on the subsystem $B$. In this paper, we provide a simple relation for the
conditional entropy as the difference of two Shannon entropies. The relation is
suitable for calculation of the quantum discord in the sense that it can by
used to obtain the quantum discord for some classes of two-qubit states,
without the need for minimization. We also present an analytical procedure of
optimization and obtain conditions under which the quantum conditional entropy
of a general two-qubit state is stationary. The presented relation is also used
to find a tight upper bound on the quantum discord.
