  In this paper, we investigate the quantum entanglement induced by phase-space
noncommutativity. Both the position-position and momentum-momentum
noncommutativity are incorporated to study the entanglement properties of
coordinate and momentum degrees of freedom under the shade of oscillators in
noncommutative space. Exact solutions for the systems are obtained after the
model is re-expressed in terms of canonical variables, by performing a
particular Bopp's shift to the noncommuting degrees of freedom. It is shown
that the bipartite Gaussian state for an isotropic oscillator is always
separable. To extend our study for the time-dependent system, we allow
arbitrary time dependency on parameters. The time-dependent isotropic
oscillator is solved with the Lewis-Riesenfeld invariant method. It turns out
that even for arbitrary time-dependent scenarios, the separability property
does not alter. We extend our study to the anisotropic oscillator, which
provides an entangled state even for time-independent parameters. The Wigner
quasi-probability distribution is constructed for a bipartite Gaussian state.
The noise matrix (covariance matrix) is explicitly studied with the help of
Wigner distribution. Simon's separability criterion (generalized
Peres-Horodecki criterion) has been employed to find the unique function of the
(mass and frequency) parameters, for which the bipartite states are separable.
In particular, we show that the mere inclusion of non-commutativity of
phase-space is not sufficient to generate the entanglement, rather anisotropy
is important at the same footing.
