On a connection governing parallel transport along \(2 \times{}2\) density matrices (Q1802931)

The authors investigate the Uhlmann connection in the bundle \(Gl(n,\mathbb{C})\to Gl(n,\mathbb{C})/U(n)\), for the case \(n=2\). They study the underlying bundle structure including the boundary of pure states. A relation to the canonical connection in the quaternionic Hopf bundle is established. Finally it is proved that the Uhlmann connection fulfils the source-free Yang-Mills equation with respect to the Riemannian metric on the manifold \({\mathcal D}_ 2\) of density matrices induced by the Bures metric.