This paper extends Uhlmann's parallel-transport geometry for mixed quantum states beyond the conventional full-rank setting to the Global Hybrid Operator Density of a GENERIC-consistent Gay–Balmaz–Tronci hybrid quantum–classical closure, building on the well-posedness and thermodynamic-consistency results established in two companion papers. The central object is a Hybrid Uhlmann connection whose holonomy around a closed recovery cycle yields three main results: the Global Hybrid Winding Theorem, showing that the holonomy spectrum changes if and only if a classical-sector memory kernel is genuinely non-Markovian (sharpened to an exact, non-perturbative statement for a fixed physical direction); an Abelian Blindness theorem, proving the holonomy's determinant channel is exactly and unconditionally insensitive to occupation loss (leakage), even as the underlying state approaches or crosses the rank-deficient boundary; and a Directional Singularity theorem, showing that the full non-Abelian spectrum instead develops a genuine, non-removable singularity there, with the limiting value depending on the direction of approach to the boundary. Together these results show that a single geometric quantity — the holonomy spectrum — can separate non-Markovian memory from leakage, two error mechanisms that existing diagnostics require distinct, purpose-built tools to detect. The theorems are verified on a structure-preserving numerical integration scheme and accompanied by a concrete, falsifiable experimental protocol for superconducting qutrit hardware, extending a recently demonstrated scalar Uhlmann-phase measurement to full non-Abelian holonomy reconstruction.
This paper extends Uhlmann's parallel-transport geometry for mixed quantum states beyond the conventional full-rank setting to the Global Hybrid Operator Density of a GENERIC-consistent Gay–Balmaz–Tronci hybrid quantum–classical closure, building on the well-posedness and thermodynamic-consistency results established in two companion papers. The central object is a Hybrid Uhlmann connection whose holonomy around a closed recovery cycle yields three main results: the Global Hybrid Winding Theorem, showing that the holonomy spectrum changes if and only if a classical-sector memory kernel is genuinely non-Markovian (sharpened to an exact, non-perturbative statement for a fixed physical direction); an Abelian Blindness theorem, proving the holonomy's determinant channel is exactly and unconditionally insensitive to occupation loss (leakage), even as the underlying state approaches or crosses the rank-deficient boundary; and a Directional Singularity theorem, showing that the full non-Abelian spectrum instead develops a genuine, non-removable singularity there, with the limiting value depending on the direction of approach to the boundary. Together these results show that a single geometric quantity — the holonomy spectrum — can separate non-Markovian memory from leakage, two error mechanisms that existing diagnostics require distinct, purpose-built tools to detect. The theorems are verified on a structure-preserving numerical integration scheme and accompanied by a concrete, falsifiable experimental protocol for superconducting qutrit hardware, extending a recently demonstrated scalar Uhlmann-phase measurement to full non-Abelian holonomy reconstruction.
This paper develops a mixed-state geometric framework for hybrid quantum–classical dynamics by constructing a Hybrid Uhlmann connection on the Global Hybrid Operator Density of the GENERIC-consistent Gay-Balmaz–Tronci closure. Building on the Stiefel-bundle parallel transport formulation of Avitabile and Bridges together with an exactly unitary Cayley integrator, it proves the Global Hybrid Winding Theorem, establishing that the Uhlmann holonomy spectrum is invariant under memoryless evolution and changes if and only if the recovery cycle carries non-Markovian memory. The paper further characterizes the rank-deficient leakage boundary, proving that the determinant (abelian) channel is exactly blind to leakage while the full non-abelian holonomy develops an intrinsic directional singularity as occupations vanish. Numerical simulations illustrate these geometric signatures, and the paper proposes a general experimental readout protocol for future implementation across quantum computing architectures.
This paper establishes the first rigorous well-posedness theory for the exact-factorization hybrid quantum–classical closure introduced by Gay-Balmaz and Tronci. By constructing an admissible operator-valued state space together with a vacuum control condition that resolves the structural singularity at vanishing classical density, the work proves local existence, uniqueness, and a Beale–Kato–Majda-type continuation criterion, together with sufficient conditions for global-in-time existence under physically relevant hybrid Hamiltonians. Numerical verification across fifteen representative quantum-computing platforms confirms preservation of the hybrid closure, operator positivity, and bounded continuation behavior. These analytical results provide the mathematical foundation for subsequent thermodynamic, geometric, and topological extensions of hybrid quantum–classical dynamics, including the irreversible GENERIC framework developed in the companion paper
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.
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.
This manuscript establishes the first rigorous well-posedness theory for the exact-factorization hybrid quantum–classical closure of Gay-Balmaz and Tronci. The central result, the Covariance-Symplectic Obstruction Theorem, shows the uniformly hyperbolic admissible sector is generically empty, with local well-posedness instead established on a weaker "uniformly symmetrizable" sector via an explicit microlocal symmetrizer and a Friedrichs–Kato existence argument. A companion Rank-Deficiency Theorem classifies six standard device Hamiltonians (transmon-resonator, tunable coupler, Rydberg array, trapped-ion, optomechanical, and Kerr-cat architectures) as excluded from the hyperbolic sector, connecting the abstract theory to concrete qubit platforms.
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.