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

TL;DR

This paper investigates the maximal and minimal fidelity between bipartite quantum states under local unitary transformations, providing formulas, bounds, and insights relevant to entanglement and state accessibility.

Contribution

It formulates the fidelity optimization problems as semi-definite programs and derives explicit solutions for specific state classes, advancing understanding of local unitary effects.

Findings

Closed-form fidelity formulas for pure and Werner states

Bounds on fidelity for general mixed states

Insights into entanglement and state commutativity

Abstract

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.