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

  • Howy Jordan
  • Markus J. Pflaum
  • Gerd Rudolph
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Abstract

The space of n× 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.

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Cite

BibTeXRIS
BibTeX
@misc{jordan2026stratification,
  title = {{Stratification of the space of density matrices via invariant theory and Uhlmann's connection}},
  author = {Howy Jordan and Markus J. Pflaum and Gerd Rudolph},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.02773},
  eprint = {2610.02773},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.02773},
}
RIS
TY  - UNPB
TI  - Stratification of the space of density matrices via invariant theory and Uhlmann's connection
AU  - Howy Jordan
AU  - Markus J. Pflaum
AU  - Gerd Rudolph
PY  - 2026
DA  - 2026/10/02
PB  - arXiv
DO  - 10.48550/arxiv.2610.02773
UR  - https://arxiv.org/abs/2610.02773
AB  - The space of n× 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.
ER  -