Stratification of the space of density matrices via invariant theory and Uhlmann's connection

Abstract

The space of $n\times n$ density matrices is the full state space of an $n$-level quantum system. While it has been asserted in the literature on quantum information theory that decomposing this state space into equirank pieces yields a Whitney stratification, rigorous proofs have remained absent. Using methods from invariant theory, this work provides a proof that the rank decomposition of the matrix algebra constitutes a (b)-regular stratification in the sense of Whitney. Furthermore, we demonstrate how the Uhlmann connection - which describes the adiabatic holonomy of mixed states and reduces to the Berry connection on pure states - can be geometrically treated as a stratified connection on a stratified vector bundle. Finally, our appendix provides a self-contained exposition of stratified fiber bundles and the new concept of stratified connections, which may be of independent interest.