  For any bipartite state, how strongly can one subsystem be quantum correlated
with another? Using the Koashi-Winter relation, we study the upper bound of
purified quantum discord, which is given by the sum of the von Neumann entropy
of the unmeasured subsystem and the entanglement of formation shared between
the unmeasured subsystem with the environment. In particular, we find that the
Luo et al.'s conjecture on the quantum correlations and the Lindblad conjecture
are all ture, when the entanglement of formation vanishes. Let the difference
between the left discord and the right discord be captured by the discord
distance. If the Lindblad conjecture is true, we show that the joint entropy is
a tight upper bound for the discord distance. Further, we obtain a necessary
and sufficient condition for saturating upper bounds of purified quantum
discord and discord distance separately with the equality conditions for the
Araki-Lieb inequality and the Lindblad conjecture. Furthermore, we show that
the subadditive relation holds for any bipartite quantum discord.
