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

Fidelity between a bipartite state and another one undergoing local unitary dynamics