Generalizations of Berry phase and differentiation of purified state and thermal vacuum of mixed states

Two state-vector representations of mixed states, known as purified state and thermal vacuum, have been realized on quantum computers. Although the two look similar, they differ by a partial transposition in ancilla space. To differentiate representations, we generalize Berry phase pure states to construct geometric phases that can reflect transposition. By generalizing adiabatic condition, Berry, showing different values from representations. However, it may contain non-geometrical contributions. Alternatively, parallel-transport condition include system ancilla. The geometrical accumulated during parallel transport is generalized phase, which also vacuum depending protocol. generalizations be measured computers via reveal rich physics finite-temperature systems.