It is shown that Uhlmann's parallel transport of purifications along a path of mixed states represented by 2 × 2 density matrices is just the path-ordered product of Thomas rotations. These rotations are invariant under hyperbolic translations inside the Bloch sphere that can be regarded as the Poincaré ball model of hyperbolic geometry. A general expression for the mixed state geometric phase for an arbitrary geodesic triangle in terms of the Bures fidelities is derived. The formula gives back the solid angle result well known from studies of the pure state geometric phase. It is also shown that this mixed state anholonomy can be reinterpreted as the pure state non-Abelian anholonomy of entangled states living in a suitable restriction of the quaternionic Hopf bundle. In this picture Uhlmann's parallel transport is just the Pancharatnam transport of quaternionic spinors.
It is shown that Uhlmann's parallel transport of purifications along a path of mixed states represented by 2 × 2 density matrices is just the path-ordered product of Thomas rotations. These rotations are invariant under hyperbolic translations inside the Bloch sphere that can be regarded as the Poincaré ball model of hyperbolic geometry. A general expression for the mixed state geometric phase for an arbitrary geodesic triangle in terms of the Bures fidelities is derived. The formula gives back the solid angle result well known from studies of the pure state geometric phase. It is also shown that this mixed state anholonomy can be reinterpreted as the pure state non-Abelian anholonomy of entangled states living in a suitable restriction of the quaternionic Hopf bundle. In this picture Uhlmann's parallel transport is just the Pancharatnam transport of quaternionic spinors.
The Uhlmann phase extends geometric phases to mixed quantum states via a parallel-transport condition on purification amplitudes, yet its direct implementation under standard Hamiltonian dynamics is obstructed by the non-Hermitian nature of the purification. We establish that for any smooth one-dimensional closed loop of full-rank qubit density matrices, there exists a four-level Hermitian parent Hamiltonian whose doubly degenerate ground-state subspace carries a Wilczek--Zee connection exactly equal to the Uhlmann connection. Consequently, the Uhlmann holonomy is faithfully reproduced by adiabatic evolution in the enlarged system. We further prove that this auxiliary-field construction is obstructed in generic two-dimensional parameter spaces by a Frobenius integrability condition, which we derive explicitly. The one-dimensional Uhlmann phase is thus placed on the same footing as the non-Abelian Berry phase, offering a purely Hermitian, Hamiltonian-based route to simulating mixed-state geometric phases. Numerical integration of the adiabatic dynamics confirms the exact correspondence and validates the convergence to the Uhlmann holonomy in the large-gap limit.
The problem of a geometric phase for an open quantum system is reinvestigated in a unifying approach. Two of the existing methods to define geometric phase, one by Uhlmann's approach and the other by a kinematic approach, which have been considered to be distinct, are shown to be related in this framework. The method is based upon purification of a density matrix by its uniform decomposition and a generalization of the parallel transport condition obtained from this decomposition. It is shown that the generalized parallel transport condition can be satisfied when Uhlmann's condition holds. However, it does not mean that all solutions of the generalized parallel transport condition are compatible with those of Uhlmann's. It is also shown how to recover the earlier known definitions of geometric phase as well as how to generalize them when degeneracy exists and varies in time.
We examine the geodesic between two mixed states of arbitrary dimension by means of their geometric mean operator. We utilize the fiber bundle approach by which the distance between two mixed state density operators ρ1 and ρ2 in the base space M is given by the shortest distance in the (Hilbert Schmidt) bundle space E of their purifications. The latter is well-known to be given by the Bures distance along the horizontal lift in E of the geodesic between the ρ1 and ρ2 in M. The horizontal lift is that unique curve in E that orthogonally traverses the fibers F⊂E above the curve in M, and projects down onto it. We briefly review this formalism and show how it can be used to construct the intermediate mixed quantum states ρ(s) along the base space geodesic parameterized by affine parameter s between the initial ρ1 and final ρ2 states. We emphasize the role played by geometric mean operator M(s)=ρ1−1/2ρ11/2ρ(s)ρ11/2ρ1−1/2, where the Uhlmann root fidelity between ρ1 and ρ(s) is given by F(ρ1,ρ(s))=Tr[M(s)ρ1]=Tr[ρ11/2ρ(s)ρ11/2], and ρ(s)=M(s)ρ1M(s). We give examples for the geodesic between the maximally mixed state and a pure state in arbitrary dimensions, as well as for the geodesic between Werner states ρ(p)=(1−p)I/N+p∣Ψ⟩⟨Ψ∣ with ∣Ψ⟩={∣GHZ⟩,∣W⟩} in dimension N=23. For the latter, we compare expressions in the limit p→1 to the infinite number of possible geodesics between the orthogonal pure states ∣GHZ⟩ and ∣W⟩. Lastly, we compute the analytic form for the density matrices along the geodesic that connects two arbitrary endpoint qubit density matrices within the Bloch ball for dimension N=2.
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.
Using the natural connection equivalent to the SU(2) Yang-Mills instanton on the quaternionic Hopf fibration of S7 over the quaternionic projective space HP1≃S4 with an SU(2)≃S3 fiber the geometry of entanglement for two qubits is investigated. The relationship between base and fiber i.e. the twisting of the bundle corresponds to the entanglement of the qubits. The measure of entanglement can be related to the length of the shortest geodesic with respect to the Mannoury-Fubini-Study metric on HP1 between an arbitrary entangled state, and the separable state nearest to it. Using this result an interpretation of the standard Schmidt decomposition in geometric terms is given. Schmidt states are the nearest and furthest separable ones lying on, or the ones obtained by parallel transport along the geodesic passing through the entangled state. Some examples showing the correspondence between the anolonomy of the connection and entanglement via the geometric phase is shown. Connections with important notions like the Bures-metric, Uhlmann's connection, the hyperbolic structure for density matrices and anholonomic quantum computation are also pointed out.