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

TL;DR

This paper extends the concept of Berry phase to mixed quantum states, introducing two geometric phases that can distinguish between purified states and thermal vacuums on quantum computers, revealing new finite-temperature quantum physics insights.

Contribution

It generalizes Berry phase to mixed states and constructs two geometric phases sensitive to partial transposition, enabling differentiation of state representations on quantum computers.

Findings

Thermal Berry phase can differ between representations.

The generalized Berry phase can include non-geometrical contributions.

Parallel transport excludes dynamical phase, isolating geometric effects.

Abstract

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.