Abstract:

The characterization of quantum entanglement is an unsolved problem. It is interested to measure entanglement based on the geometric properties of vectors because a quantum state is represented as a unit vector in a Hilbert space. Some scholars have defined an entanglement measure on the two-body pure state system$C^{2}\otimes C^{2}$based on the modulus length of the wedge product of two vectors, whose modulus length corresponds geometrically to the area of an oriented parallelogram on a plane. In the work, we give the entanglement measures on the two-body pure state system$C^{3}\otimes C^{3}$and$C^{d}\otimes C^{d}$by using the modular length of the wedge product of vectors. They geometrically correspond to the volume of an oriented parallelepiped and$d\times(d-1)\times\cdots\times4$oriented parallelepipeds. In addition, We propose a geometric criterion for determining separable states. The results show that the entanglement measure$E$defined based on the geometric background of mathematics is a simple and intuitive measurement method.

Key words: Entanglement measure, Wedge product, Geometric criterion