  Geometric phase has found a broad spectrum of applications in both classical
and quantum physics, such as condensed matter and quantum computation. In this
paper we introduce an operational geometric phase for mixed quantum states,
based on spectral weighted traces of holonomies, and we prove that it
generalizes the standard definition of geometric phase for mixed states, which
is based on quantum interferometry. We also introduce higher order geometric
phases, and prove that under a fairly weak, generically satisfied, requirement,
there is always a well-defined geometric phase of some order. Our approach
applies to general unitary evolutions of both nondegenerate and degenerate
mixed states. Moreover, since we provide an explicit formula for the geometric
phase that can be easily implemented, it is particularly well suited for
computations in quantum physics.
