As Berry first stated, when a quantum system is prepared in an energy eigenstate and adiabatically driven in a cycle, it acquires, in addition to the dynamical phase, a phase that depends solely on the path traced in the ray space. Being independent of the specific dynamics giving rise to the path, this phase is of geometrical nature. Following Berry’s work, the notion of geometric phases has been extended far beyond the original context, encompassing definitions applicable to arbitrary unitary evolutions. They possess significance not only in the fundamental understanding of quantum mechanics and its mathematical framework but also in explaining various physical phenomena, and hold promise for practical applications.

However, in practice, a pure state of a quantum system is an idealized concept, and every experimental or real-world implementation must account for the presence of an environment that interacts with the observed system. This interaction necessitates a description in terms of mixed states and non-unitary evolutions. The definition of a geometric phase applicable in such scenarios remains an open problem, giving rise to multiple proposed solutions. Characterizing these geometric phases becomes a multifaceted task, encompassing motivations that span from fundamental aspects of quantum mechanics to technological applications.