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, to exhibit the entanglement between the noncommutative co-ordinates, the parameters (mass and frequency) have to satisfy a unique constraint equation. We have considered the most general form of an anisotropic oscillator in NCS, with both spatial and momentum noncommutativity. The system is transformed to the usual commutative space (with usual Heisenberg algebra) by a well-known Bopp shift. The system is transformed into an equivalent simple system by a unitary transformation, keeping the intrinsic symplectic structure (Sp(4,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Sp(4,{\mathbb {R}})$$\end{document}) intact. Wigner quasiprobability distribution is constructed for the bipartite Gaussian state with the help of a Fourier transformation of the characteristic function. It is shown that the identification of the entangled degrees of freedom is possible by studying the Wigner quasiprobability distribution in phase space. We have shown that the co-ordinates are entangled only with the conjugate momentum corresponding to other co-ordinates.