Construction of GHZ-Like States for a Three-Particle （Spin 1／2） System

As the simultaneous eigenstates of a complete set of commuting observables (CSCO) consisting of three-body operators of the form σασ2βσ3γ, the construction of entangled states for a three-particle system is investigated. It is shown that there exist 54 different sets of operators (each containing four commuting three-body operators with the irproduct being -1) and any three members of each set constitute a CSCO. It is found that 54 different sets of maximally entangled states can be constructed, including the usual GHZ (Greenberg-Horne-Zeilinger) state as a special case. Once a preferential representation is adopted, all of them can be expressed as a GHZ-like form. Moreover, there also exist 18 sets of partially entangled states, where only two particles are entangled.

Construction of GHZ-Like States for a Three-Particle (Spin 1/2)System

As the simultaneous eigenstates of a complete set ofcommuting observables (CSCO) consisting of three-body operators ofthe form σ_1ασ_2βσ_3γ,the construction of entangled states for a three-particlesystem is investigated. It is shown that there exist 54 differentsets of operators (each containing four commuting three-body operatorswith their product being -1) and any three members of each setconstitute a CSCO. It is found that 54 different sets of maximallyentangled states can be constructed, including the usual GHZ(Greenberg-Horne-Zeilinger) state as a special case. Once a preferentialrepresentation is adopted, all of them can be expressed as aGHZ-like form. Moreover, there also exist 18 sets of partiallyentangled states, where only two particles are entangled.