This paper develops a rigorous, self-contained differential-geometric framework for tracking mixed quantum states in open physical systems, providing an analytical structural extension to the pure-state Kâhler geometry of Paper 1. Because open quantum dynamics induced by environment-driven dissipation and decoherence break the global manifold homogeneity, we model the total space of density operators S(H) as a stratified space. We prove that the individual rank-k subsets constitute embedded smooth real manifolds of dimension 2k(N + 1) − k2 − 1, satisfying the algebraic conditions of Whitney Regularity Condition B to guarantee continuous tangent space alignment across interfaces. To assign coordinate-independent connection and metric tensors over this stratified landscape, we formalize the global principal U (n) fiber bundle structure using Uhlmann’s purification theorem. By lifting mixed density operators to pure-state amplitudes W ∈ GL(n, C) on a bipartite workspace, we show that minimizing the ambient flat Hilbert–Schmidt metric along the vertical fibers induces the non-degenerate Bures metric on the base manifold. We derive the exact matrix elements of the corresponding Symmetric Logarithmic Derivative (SLD) fields in an arbitrary non-degenerate eigenbasis, establishing the mixed-state information scaling boundary gQF IM = 4gB. Finally, we characterize the non-Abelian Yang–Mills curvature FU = dAU + AU ∧ AU of the Uhlmann connection, demonstrating how non-commutativity emerges naturally under environmental coupling. By isolating the measure-zero boundary configurations where eigenvalues or matrix rank drop, we generalize our control immersion criterion to identify both rank deficient and eigenvalue-vanishing caustic singularities. This metric-determinant verification pipeline allows researchers to systematically diagnose, map, and structurally bypass trajectory bottlenecks during Lindbladian control path optimization on real-world noisy quantum backends
For full-rank density matrices the state space is contractible, the Uhlmann bundle is topologically trivial, and the holonomy admits no quantized invariants; although the Uhlmann phase is genuinely geometric, this topological poverty has kept it from serving as a robust physical diagnostic of mixed-state matter. Mixed states of fixed rank below the Hilbert-space dimension, however, are ubiquitous: reduced states of constrained subsystems, states confined to invariant sectors, and states supported on decoherence-free subspaces can have a support smaller than the Hilbert space, a support that can vary with parameters. Their geometry is far richer: the support carries genuine curvature, non-Abelian holonomy, and Chern topology. We formulate Uhlmann's theory directly on the manifold of rank-
k density matrices: minimal purifications make the fixed-rank stratum the base of a principal
U(k)-bundle whose Uhlmann connection is uniquely determined by a Sylvester equation, and in an eigenframe this connection takes a closed form reducing to the Berry, Wilczek--Zee, and faithful Uhlmann connections at
k=1, at equal weights, and at
k=N. The fixed-rank bundle inherits the topology of the Grassmannian, and non-factorizable higher Chern topology requires failure of the global eigenline splitting, which in the present spectral setting requires degeneracy, minimally realized by a rank-2 Yang monopole whose quantized second Chern number links the geometry to the four-dimensional quantum Hall response. Solvable models, from a genuinely non-Abelian holonomy to a dissipatively driven orbit whose Uhlmann holonomy is steered by the reservoir through an elliptic-integral spectral dressing, illustrate the content: the supporting subspace carries the topology, the spectral weights shape the transport, and Uhlmann holonomy provides a direct geometric probe of the resulting mixed-state structures.
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.
A unifying framework for identifying distance and holonomy for decompositions of density operators is introduced. Parallelity between quantum ensembles is defined by minimizing this distance over allowed decompositions. The minimum is a property of a pair of states and coincides with the Bures distance. The parallelity condition imposes a connection (rule for parallel transport) that results in the Uhlmann holonomy for sequences of density operators. A distance and holonomy for spectral decompositions of density operators is identified as a sub-group restriction of the full decomposition freedom. These spectral concepts are gauge invariant (decomposition independent) properties of mixed quantum ensembles, as long as the corresponding density operators are non-degenerate. A gauge invariant spectral geometric phase for discrete sequences of mixed quantum states is obtained as the phase of the trace of the spectral holonomy. This geometric phase differs from the interferometric mixed state geometric phase in the continuous limit.
We examine the geometric interpretation of the quantum geometric tensor proposed in [Phys. Rev. B {\bf 110}, 035404 (2024)] for mixed quantum states, focusing on its imaginary part, which is proportional to a weighted sum of the Berry curvatures of the eigenstates of the density operator. While the real part naturally decomposes into Fisher--Rao and weighted Fubini--Study contributions, we show that the imaginary part does not, in general, coincide with the curvature of a connection on a globally defined U(1) line bundle whose holonomy yields a mixed-state geometric phase. Using a two-level system as an explicit example, we demonstrate that its surface integral depends on the choice of surface bounded by the same closed path, with ambiguities that are not integer multiples of
2π.
We investigate the geometry of mixed quantum states near rank-changing points, showing that these singularities function as effective geometric defects. The Uhlmann connection is well-defined on the full-rank sector of the density-matrix manifold, while rank-deficient states form singular boundary strata where the bundle structure degenerates. By restricting to a punctured state manifold that excludes the singular set, we obtain a well-defined gauge structure and identify an asymptotically robust invariant: the Uhlmann holonomy around noncontractible loops encircling the defect on a restricted two-dimensional punctured submanifold. In an exactly solvable qutrit model, a restricted submanifold emerges on which the connection is locally flat yet carries nontrivial monodromy, analogous to flat connections with Aharonov--Bohm-type transport. The holonomy depends only on the ratios of the vanishing eigenvalues under frozen radial dependence of the eigenbasis geometry and a fixed angular loop. In contrast, the Uhlmann curvature may diverge path-dependently when eigenvalues shrink with distinct powers, with a leading spectral-prefactor scaling law, establishing that the holonomy survives as a universal asymptotic invariant while the curvature remains non-universal. Within the effective SU(2) defect sector, the conjugacy class of the holonomy, equivalently the Wilson loop variable, provides a continuous, non-quantized classification of the asymptotic monodromy surrounding the rank-deficient defect. This non-quantization does not imply a lack of robustness: the asymptotic holonomy is an invariant of the restricted punctured submanifold and is insensitive to smooth deformations of the loop or the radial profile within the fixed spectral-ratio sector.