Defect Holonomy Near Rank-Deficient Mixed States

• URL: http://arxiv.org/abs/2606.02343v2
• Date: Fri, 05 Jun 2026 16:29:53 GMT
• Title: Defect Holonomy Near Rank-Deficient Mixed States
• Abstract summary: We investigate the geometry of mixed quantum states near rank-changing points.<n>We show that rank-deficient states form singular boundary strata where the bundle structure degenerates.<n>By restricting to a punctured state manifold that excludes the singular set, we obtain a well-defined gauge structure.
• Score: 2.679171884680826
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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