The Schr\"odinger equation is an important class of mathematical physics equations, and it has significant applications in engineering. The conservative difference scheme for the coupled nonlinear Schr\"odinger equation is studied based on the high-order finite difference method, the Crank-Nicolson, and the Leap-frog method. Moreover, the proposed numerical formulation is decoupled, linear, and it satisfies the discrete mass and energy conservation laws. The existence, uniqueness, stability and convergence of the numerical formulation are discussed, and it is shown that the numerical formulation is of the accuracy $O(\tau^2+h^4)$. The numerical experiments are given, and their results verify the efficiency of the scheme.