Berry phase with density matrix approach Berry phase, coming from Schrodinger equation, has well known form in terms of closed integral
$$\gamma = \int_C A(\xi) d\xi $$ 
with Berry connection $$A(\xi) = i < \psi(\xi) | \partial_{\xi} | \psi(\xi) > $$ and $\psi(\xi)$ is eigenstate of Schrodinger equation $$H |\psi> = E |\psi>.$$ 
Suppose instead you have system described by mixed state.
Is there generalization of Berry phase for density matrix $\rho$?
 A: I do not think the notion of a berry phase makes sense for a general mixed state. This is because in general geometric phases in general are got by varying parameters in the Hamiltonian. This makes them inherently a geometric feature of $\textit{coherent}$ evolution. 
Another way of seeing, is to consider the following particular situation: suppose you have one qubit in state $\psi(t=0) $ with Hamiltonian $H$. Then you change parameters in the $H$ so that the evolution of qubit is supposed to return to the original state $ \lambda \psi(t=0)$ where $\lambda$ is some $U(1)$ phase factor.Throughout the evolution we are staying on the surface of the Bloch sphere. Now if you want to bring in the possibility of mixed states then one way is to introduce noise in the evolution but as you change the parameters in the Hamiltonian the Bloch sphere is being deformed and in general your state may end up inside the Bloch sphere. In other words the possibility of returning to your state modulo global phases is in not possible for arbitrary mixed states. 
