AI 中文总结
本文用不变量理论严格证明了密度矩阵空间的秩分解是Whitney层化，并将Uhlmann联络解释为分层向量丛上的分层联络，附录还介绍了分层纤维丛与分层联络的新概念。
AI 中文摘要
$n\ imes n$密度矩阵的空间是$n$能级量子系统的完整状态空间。尽管量子信息理论文献中曾断言将该状态空间分解为等秩部分可得到Whitney层化，但严格的证明一直缺失。本文利用不变量理论的方法，证明了矩阵代数的秩分解构成Whitney意义下的(b)-正则层化。此外，我们展示了Uhlmann联络——它描述混合态的绝热和乐，并在纯态上退化为Berry联络——如何在几何上被处理为分层向量丛上的分层联络。最后，附录提供了分层纤维丛及分层联络这一新概念的独立自足阐述，这可能具有独立的研究价值。
英文摘要
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.
