  The application of geometry to physics has provided us with new insightful
information about many physical theories such as classical mechanics, general
relativity, and quantum geometry (quantum gravity). The geometry also plays an
important role in foundations of quantum mechanics and quantum information. In
this work we discuss a geometric framework for mixed quantum states represented
by density matrices, where the quantum phase space of density matrices is
equipped with a symplectic structure, an almost complex structure, and a
compatible Riemannian metric. This compatible triple allow us to investigate
arbitrary quantum systems. We will also discuss some applications of the
geometric framework.
