  Two representations of mixed states by state-vectors, known as purified state
and thermal vacuum, have been realized on quantum computers. While the two
representations look similar, they differ by a partial transposition in the
ancilla space. While ordinary observables cannot discern the two
representations, we generalize the Berry phase of pure quantum states to mixed
states and construct two geometric phases that can reflect the partial
transposition. By generalizing the adiabatic condition, we construct the
thermal Berry phase, whose values from the two representations can be
different, However, the thermal Berry phase may contain non-geometrical
contributions. Alternatively, we generalize the parallel-transport condition to
include the system and ancilla and show the dynamical phase is excluded under
parallel transport. The geometrical phase accumulated in parallel transport is
the generalized Berry phase, which may or may not differentiate a purified
state from a thermal vacuum depending on the protocol. The generalizations of
the Berry phase to mixed states may be realized and measured on quantum
computers via the two representations to reveal the rich physics of
finite-temperature quantum systems.
