A Pure State, Differential Geometry,

The set D~1(H) of pure states is the foundation of studying the space D(H) of density states. This paper studys some problems about pure states D~1(H) in density states D(H) from the geometric point of view. We give the Remannian metric g of the pure states D~1(H) as a submanifold of u~*(H), define the Hermitian structure of complex manifold D~1(H) by g, and find that h and the Fubini-study metric on the complex projective spaces differ by a constant multiple. It is one of very important problems to effectively decide whether a given composite state is entangled or not. In this paper, the length of a pure state on composite quantum systems is defined, and it can be used as a measurement of the entanglement. We also give a necessary and sufficient condition to decide whether a given composite pure state is entangled or not.