Off-diagonal generalization of the mixed-state geometric phase

Recent investigations in geometric phases in quantum systems have led to cases where the standard definitions break down. On one hand, such situations emerge for orthogonal initial and final pure states connected unitarily, and on the other hand, unitary evolution of a system in a mixed-state may lead to nodal points in parameter space. This paper provides a general treatment of the off-diagonal mixed-state geometric phase comprising unitarities that can be decomposed into a diagonal part leaving the initial basis states unchanged and a permutation part reordering the initial states.