The gauge field related to the notion of parallel transport along mixed states is described, in some analogy with the considerations leading to the notion of Berry’s phase for curves of pure states. The notion of parallelity is introduced as follows. First, the “purification” of mixed states is performed by constructing the extended Hilbert space \({\mathcal H}^{ext}={\mathcal H}\otimes {\mathcal H}'\) where \({\mathcal H}\) is the original Hilbert space and dim \({\mathcal H}'\geq \dim {\mathcal H}\), and ascribing to any mixed state \(\rho\) a vector \(\psi\in {\mathcal H}^{ext}\) such that \(\rho (A)=(\psi,A\otimes T'\psi)\) for all observables A. The vector \(\psi\) is not uniquely defined.
Given a path \(s\to \rho_ s\), \(0\leq s\leq 1\), we consider the corresponding path \(s\to \psi_ s\). The purification \(s\to \psi_ s\) is called parallel if for every purification \(s\to \psi '_ s\) of the same path \(s\to \rho_ s\) the inequality (\({\dot \psi}\),\({\dot \psi}\))\(\leq ({\dot \psi}',{\dot \psi}')\) holds for all s.
The standard choice \({\mathcal H}'=dual\) of \({\mathcal H}\) is considered in detail. In this case \({\mathcal W}={\mathcal H}^{ext}\) becomes the space of Hilbert-Schmidt operators over \({\mathcal H}\) with the scalar product \((W_ 1,W_ 2)=tr(W^+_ 1W_ 2)\). The parallelity condition reads (*) \(W^+\dot W-\dot W^+W=0\). Now, \({\mathcal W}\) is viewed as a bundle with the base \(\Omega\) consisting of all density operators and the projection \(\pi\) : \(W\to Tr=WW^+/(W,W)\). If the path in \(\Omega\) is lifted to \({\mathcal W}\) the lift is called parallel if (*) holds. The corresponding connection and curvature forms are constructed. The parallel lifts of the solution \(t\to \rho_ t\) of an evolution equation i\({\dot \rho}=[H,\rho]\) are considered.