Entanglement in phase-space distribution for an anisotropic harmonic oscillator in noncommutative space

The bipartite Gaussian state, corresponding to an anisotropic harmonic oscillator in a noncommutative space (NCS), is investigated with the help of Simon’s separability condition (generalized Peres-Horodecki criterion). It turns out that, exhibit entanglement between co-ordinates, parameters (mass and frequency) have satisfy unique constraint equation. We considered most general form NCS, both spatial momentum noncommutativity. system transformed usual commutative (with Heisenberg algebra) by well-known Bopp shift. into equivalent simple unitary transformation, keeping intrinsic symplectic structure ( $$Sp(4,{\mathbb {R}})$$ ) intact. Wigner quasiprobability distribution constructed for state Fourier transformation characteristic function. shown that identification entangled degrees freedom possible studying phase space. co-ordinates are only conjugate other co-ordinates.