Tuning the separability in noncommutative space

Abstract

With the help of the generalized Peres–Horodecki separability criterion (Simon’s condition) for a bipartite Gaussian state, we have studied the separability of the noncommutative (NC) space coordinate degrees of freedom. Non-symplectic nature of the transformation between the usual commutative space and NC space restricts the straightforward use of Simon’s condition in NCS. We have transformed the NCS system to an equivalent Hamiltonian in commutative space through the Bopp shift, which enables the utilization of the separability criterion. To make our study fairly general and to analyze the effect of parameters on the separability of bipartite state in NC-space, we have considered a bilinear Hamiltonian with time-dependent (TD) parameters, along with a TD external interaction, which is linear in field modes. The system is transformed (Sp(4,R)) into canonical form keeping the intrinsic symplectic structure intact. The solution of the TD-Schrödinger equation is obtained with the help of the Lewis–Riesenfeld invariant method (LRIM). Expectation values of the observables (thus the covariance matrix) are constructed from the states obtained from LRIM. It turns out that the existence of the NC parameters in the oscillator determines the separability of the states. In particular, for isotropic oscillators, the separability condition for the bipartite Gaussian states depends on specific values of NC parameters. Moreover, particular anisotropic parameter values for the oscillator may cease the separability. In other words, both the deformation parameters (θ, η) and parameter values of the oscillator (mass, frequency) are important characteristics for the separability of bipartite Gaussian states. Thus tuning the parameter values, one can destroy or recreate the separability of states. With the help of a toy model, we have demonstrated how the tuning of a TD-NC space parameter affects the separability.