Geometric phase and spinorial representation of mixed state

• We propose a normalized spinorial representation in connecting the density matrix with mixed state vector.
• We obtain a novel approach to geometric phase of mixed state.
• We find a connecting of the geometric phase for mixed state with Bloch sphere structure.
• We find a like-Herzberg and Longuet–Higgins phase on the origin of the Bloch sphere.

A novel approach to geometric phase of mixed state is proposed by using the normalized spinorial representation in connecting the density matrix with mixed state vector. We find that though the spinor involves N   separate U(1)U(1) phases correspondingly to any N  -level decomposition of the density matrix, both geometric phase and density matrix of mixed state are holonomy ⨂k=1NU(1) gauge invariants. This noncyclic invariant is conceptually useful in analyzing geometric phase and CP violation of open system. Under a quasicyclic case, the geometric phase depends only on the symplectic area spanned in a given closed evolving curve with the classical probabilities relating to the Bloch radius, in which quantifies mixed degree of open system, in the Bloch sphere structure.