  The concept of off-diagonal geometric phases for mixed quantal states in
unitary evolution is developed. We show that these phases arise from three
basic ideas: (1) fulfillment of quantum parallel transport of a complete basis,
(2) a concept of mixed state orthogonality adapted to unitary evolution, and
(3) a normalization condition. We provide a method for computing the
off-diagonal mixed state phases to any order for unitarities that divide the
parallel transported basis of Hilbert space into two parts: one part where each
basis vector undergoes cyclic evolution and one part where all basis vectors
are permuted among each other. We also demonstrate a purification based
experimental procedure for the two lowest order mixed state phases and consider
a physical scenario for a full characterization of the qubit mixed state
geometric phases in terms of polarization-entangled photon pairs. An
alternative second order off-diagonal mixed state geometric phase, which can be
tested in single-particle experiments, is proposed.
