Tuning the separability in noncommutative space

Abstract

We study the Separability of the noncommutative (NC) space coordinate degrees of freedom with the generalized Peres-Horodecki separability criterion (Simon's condition) for a bipartite Gaussian state. Non-symplectic nature of the transformation between the usual commutative space and NC space restricts the use of Simon's condition in NCS. We transform the NCS system to an equivalent Hamiltonian in commutative space through Bopp shift, which enables the utilization of the separability criterion in NC space. For afairly general study, we consider a bilinear Hamiltonian with time-dependent (TD) parameters, along with a TD external interaction, which is linear in field modes. The system is transformed into canonical form keeping the intrinsic symplectic structure ($Sp(4,\mathbb{R})$) intact. The solution of the TD-Schr\"{o}dinger equation is obtained with the help of 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 NC parameters. Moreover, anisotropic parameter values for the oscillator affects the separability. In other words, both the deformation parameters ($\theta,\;\eta$) and parameter values of the oscillator are important for the separability of bipartite states. Thus tuning the parameter values, one can destroy or recreate the separability of states. With the help of toy models, we have demonstrated TD-NC space parameters effect on separability.