Concrete Representation and Separability Criteria for Symmetric Quantum State

Using the typical generators of the special unitary groups S U(2), the concrete representation of symmetric quantum state is established, then the relations satisfied by those coefficients in the representation are presented. Based on the representation of density matrix, the PPT criterion and CCNR criterion are proved to be equivalent on judging the separability of symmetric quantum states. Moreover, it is showed that the matrix Γ _ρ of symmetric quantum state only has five efficient entries, thus the calculation of ∥Γ _ρ ∥ is simplified. Finally, the quantitative expressions of real symmetric quantum state under the ∥Γ _ρ ∥ separability criterion are obtained.