A geometric framework for mixed quantum states based on a K\"{a}hler structure

Abstract

In this paper we introduce a geometric framework for mixed quantum states based on a K\"ahler structure. The geometric framework includes a symplectic form, an almost complex structure, and a Riemannian metric that characterize the space of mixed quantum states. We argue that the almost complex structure is integrable. We also in detail discuss a visualizing application of this geometric framework by deriving a geometric uncertainty relation for mixed quantum states. The framework is computationally effective and it provides us with a better understanding of general quantum mechanical systems.