  The fidelity and local unitary transformation are two widely useful notions
in quantum physics. We study two constrained optimization problems in terms of
the maximal and minimal fidelity between two bipartite quantum states
undergoing local unitary dynamics. The problems are related to the geometric
measure of entanglement and the distillability problem. We show that the
problems can be reduced to semi-definite programming optimization problems. We
give close-form formulae of the fidelity when the two states are both pure
states, or a pure product state and the Werner state. We explain from the point
of view of local unitary actions that why the entanglement in Werner states is
hard to accessible. For general mixed states, we give upper and lower bounds of
the fidelity using tools such as affine fidelity, channels and relative entropy
from information theory. We also investigate the power of local unitaries and
quantification for the commutativity of quantum states, and the equivalence of
the two optimization problems.
