Equivalence of continuous-variable stabilizer states under local Clifford operations

In this paper, continuous variable stabilizer state is described by its generator matrix. Then the fact that any stabilizer state could be reduced to the corresponding weighed graph state under local unitary operation in the Clifford group is proved. A matrix equation, which is used to determine whether two stabilizer states are equivalent or not under local unitary operation in the Clifford group, is proposed. Then we apply the method to the five-mode unweighed graph states and demonstrate that our theory could be able to correctly and quickly find the equivalent classes of five-mode unweighed graph states.