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arXiv:2301.01210 · Physical review. B./Physical review. B

Geometric phases of mixed quantum states: A comparative study of interferometric and Uhlmann phases

Xu-Yang Hou, Xin Wang, Zheng Zhou, Hao Guo, Chih-Chun Chien

Published Apr 19, 2023• 11 citations
arXivHTMLPDF

Abstract

Two geometric phases of mixed quantum states, known as the interferometric phase and Uhlmann phase, are generalizations of the Berry phase of pure states. After reviewing the two geometric phases and examining their parallel-transport conditions, we specify a class of cyclic processes that are compatible with both conditions and therefore accumulate both phases through their definitions, respectively. Those processes then facilitate a fair comparison between the two phases. We present exact solutions of two-level and three-level systems to contrast the two phases. While the interferometric phase exhibits finite-temperature transitions only in the three-level system but not the two-level system, the Uhlmann phase shows finite-temperature transitions in both cases. Thus, using the two geometric phases as finite-temperature topological indicators demonstrates the rich physics of topology of mixed states.

DOI
10.1103/physrevb.107.165415

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quant-pharXiv:2608.22919•Aug 25, 2026•0 citations
Two geometric phases of mixed quantum states, known as the interferometric phase and Uhlmann phase, are generalizations of the Berry phase of pure states. After reviewing the two geometric phases and examining their parallel-transport conditions, we specify a class of cyclic processes that are compatible with both conditions and therefore accumulate both phases through their definitions, respectively. Those processes then facilitate a fair comparison between the two phases. We present exact solutions of two-level and three-level systems to contrast the two phases. While the interferometric phase exhibits finite-temperature transitions only in the three-level system but not the two-level system, the Uhlmann phase shows finite-temperature transitions in both cases. Thus, using the two geometric phases as finite-temperature topological indicators demonstrates the rich physics of topology of mixed states.
Two mixed-state geometric phases, known as the Uhlmann phase and interferometric geometric phase (IGP), of spin coherent states (CSSs) and spin squeezed states (SSSs) are analyzed. Exact solutions and numerical results of selected examples are presented. For the j=3/2j=3/2j=3/2 CSS, the Uhlmann phase exhibits finite-temperature topological phase transitions characterized by abrupt jumps. The IGP for the same state similarly shows discontinuous jumps as the temperature varies. In the case of the j=1j=1j=1 one-axis SSS, both the Uhlmann phase and IGP display discrete finite-temperature jumps. By contrast, the j=1j=1j=1 two-axis SSS shows no such transitions because the Uhlmann phase and IGP both vary smoothly with temperature. We also briefly discuss potential realizations and simulations related to these phenomena in spin systems.
Geometric phases play a fundamental role in understanding the geometric structure of quantum states, yet extending the Uhlmann phase to non-Hermitian systems poses significant challenges due to parameter-dependent inner product structures. In this work, we develop a comprehensive theory of the Uhlmann phase for quasi-Hermitian systems, where the physical Hilbert space metric varies with external parameters. By constructing a generalized purification that respects the quasi-Hermitian inner product, we derive the corresponding parallel transport condition and Uhlmann connection. Our analysis reveals that the parameter-dependent metric modifies the Uhlmann connection and leads to a finite-temperature distribution of Uhlmann-phase regions that differs from the standard Hermitian case. Applying this formalism to solvable two-level models, we uncover tunable finite-temperature Uhlmann-phase diagrams, where the parameter-dependent metric shifts and deforms the zeros of the Uhlmann amplitude, thereby reshaping the temperature intervals in which nontrivial Uhlmann phases occur. Furthermore, by extending established interferometric protocols originally developed for Hermitian systems, the geometric amplitude can be recast as a measurable Loschmidt amplitude between purified states, providing a practical and experimentally accessible pathway to investigate quasi-Hermitian mixed-state geometric phases and their finite-temperature transitions. This work establishes a unified framework for understanding mixed-state geometric phases in quasi-Hermitian quantum systems and, through its natural relation to the mixed-state quantum geometric tensor, opens new avenues for exploring local geometry in quasi-Hermitian thermal states.
Geometric phases have been extensively investigated in a wide range of quantum systems, often revealing deep connections to the underlying topology of many-body states. In this work, we examine two geometric phases defined for mixed quantum states−-−the interferometric geometric phase and the Uhlmann phase−-−extracted from two-site reduced density matrices of the transverse-field Ising model with nearest-neighbor interacting spins. By applying coordinated local unitary rotations to the spins, we compute the geometric phases associated with the two-site states across the critical point. We find that the interferometric phase is a more reliable indicator of the quantum phase transition in this model than the Uhlmann phase.
Note that equations and expressions has been omitted here and is instead presented in the work. This thesis focuses on the geometric phase in pure and mixed quantum states. For the case of a pure quantum state, Berry's adiabatic approach (4.1.10) and Aharonov & Anandan's non-adiabatic generalization of Berry's approach (4.2.8) are included in this work. Mixed quantum state involves Uhlmanns approach (5.1.42), which is used extensively in Section 7 and Sjöqvist's et al. approach (5.2.22), used extensively in Section 6. Sjöqvist's approach states that the Uhlmann phase is an observable and provides the experimental groundworkusing an interferometer.􀀀This was later proven, by Du et al.[45] to reproduce experimental data (Figure 19) on page 56. The Uhlmann phase can be used to observe the behaviour of topological kinks. This was tested on 3 models, the Creutz-ladder, the Majorana chain andthe SSU-model. It is found that the Uhlmann phase is split into two regimes with the dividing parameter being the temperature. This temperature is called the critical temperature, Tc. If the temperatureis is below the critical temperature, the Uhlmann phase yields Π and if thetemperature is above the critical temperature, the Uhlmann phase yields zero.
Dynamic quantum phase transitions (DQPT) following quantum quenches exhibit singular behavior of the overlap between the initial and evolved states. Here we present a formalism to incorporate a geometric phase into quench dynamics of mixed quantum states, a process named the Uhlmann quench, based on the Uhlmann parallel transport. To overcome the incompatibility between the Uhlmann parallel-transport condition and Hamiltonian dynamics, we formulate the evolution of purification of the density matrix in a form which not only respects the dynamics according to the density matrix but also follows the Uhlmann parallel-transport condition to generate a geometric phase after a quantum quench. For cyclic processes exemplified by a spin-1/2 system, geometric DQPTs (GDQPTs) can emerge with both singular behavior in the dynamic analog of the free energy and jumps of the geometric phase. Moreover, the Uhlmann phase reflecting the holonomy is generated at the end of each cycle. The Uhlmann quench thus paves the way for investigating the interplay between quantum dynamics and geometric processes in mixed states.
The Berry phase arises naturally from the adiabatic dynamics of a pure quantum state, whereas the Uhlmann phase is usually formulated kinematically for a prescribed path of mixed states. For a qubit subject to a uniformly gapped conical drive, we obtain the Uhlmann connection and holonomy in closed form for the equilibrium Gibbs cycle. We then solve the corresponding Lindblad equation exactly in the rotating frame and show that its steady state forms a nonequilibrium limit cycle that lags the instantaneous Gibbs cycle. As the driving becomes slow, the Uhlmann phase approaches the equilibrium Uhlmann phase, which cooling then reduces to the Berry phase: in the joint adiabatic and low-temperature limit, the dynamical Uhlmann phase therefore converges to the Berry phase. This grounds the Uhlmann--Berry correspondence in the physical dynamics of an open system. We also characterize the finite-driving corrections, together with the discontinuous πππ jump that the Uhlmann phase undergoes on the equatorial cycle as the temperature is varied. Finally, we examine an isolated transversal gap closing, where this zero-temperature correspondence can break down: the pure-state path becomes open, its closure is ambiguous, and gap-opening regularizations close it along a geodesic set by the direction of the bias field, splitting the Berry phase into a one-parameter family of values. The Gibbs path instead closes smoothly through the maximally mixed state, so the Uhlmann holonomy requires no regularization and remains unique and continuous at every finite temperature. In the low-temperature limit the Uhlmann phase selects a single member of the Berry family---the geodesic closure in the osculating plane, fixed by the velocity and acceleration of the driving field. The Uhlmann--Berry correspondence of the gapped regime thus survives the gap closing as a geodesic selection rule.