  Quantum information distribution in a tripartite state plays a fundamental
role in quantum information processes. Here we investigate how a bipartite
unitary transformation $U_{AB}$ redistributes the quantum mutual information
with the third party $C$ in a tripartite pure state $|\psi\rangle_{ABC}$ in a
$d_A\times d_B\times d_C$ dimensional Hilbert space. In particular, we focus on
finding out the optimal unitary transformation $U_{AB}^{\ast}$ that maximizes
the quantum mutual entropy between party $A$ and party $C$,
$I(A:C)=S(\rho_A)-S(\rho_B)+S(\rho_C)$. We show that the mutual entropy
$I(A:C)$ is upper bounded by $2S(\rho_C)$ derived from the Araki-Lieb
inequality. This upper bound can be realized via an optimal unitary
transformation for any pure state with the rank $r_{C}$ of $\rho_C$ satisfying
$r_C\le d_A$. For a generic pure state with $r_C> d_A$, the upper bound can not
be realized by any bipartite unitary transformation. To maximize the mutual
entropy in the latter case, we propose a fast numerical algorithm to produce an
approximate optimal unitary transformation, where our optimization is
transformed into a modified number partition problem. The validness of our
algorithm is confirmed by its comparison with the results from the Adam
algorithm for parameterized unitary transformations. Our approximate algorithm
thus provides a practical protocol to implement redistribution of quantum
mutual information for a tripartite quantum state with high dimensions.
