Infinite-dimensional extension of density-matrix stratification and Uhlmann connection

Extend the finite-dimensional stratification theory of density matrices and the associated geometric treatment of Uhlmann’s connection to infinite-dimensional Hilbert spaces, including the geometry of the stratification, the behavior of Uhlmann’s connection on non-generic strata, and parallel lifts of density-matrix evolution equations.

Background

The paper establishes that finite-dimensional density matrices admit a Whitney (b)-regular stratification by rank and that Uhlmann’s connection can be interpreted as a stratified connection over the resulting strata. In the outlook, the authors discuss the corresponding setting for density operators on a separable infinite-dimensional Hilbert space. They specifically note that the extension of the finite-dimensional picture has not yet been achieved and identify the geometry of the stratification, the infinite-dimensional Uhlmann connection, and the evolution equation with its parallel lifts as components of the unresolved program.

References

Up to our knowledge, the extension of the above picture to infinite-dimensional Hilbert spaces was never accomplished.

— Stratification of the space of density matrices via invariant theory and Uhlmann's connection  (2610.02773 - Jordan et al., 2 Oct 2026) in Section 4, “Comments and Outlook”