A Note on Geometric Imaginarity

The imaginarity theory emphasizes the significant role of imaginary component in quantum information. We study the geometric imaginarity which characterizes the imaginarity geometrically. We derive the analytical formulae of the geometric imaginarity for mixed states in qubit systems and X states in high dimensional systems. We also study the evolution of the geometric imaginarity under the action of bit flip channel, phase flip channel and amplitude damping channel. The results show that the geometric imaginarity are contractive under these three quantum channels.