The dynamic of quantum entanglement of two dimensional harmonic oscillator in non-commutative space

In the present paper, we study the influence of non-commutativity on entanglement in a system of two oscillators-modes in interaction with its environment. The considered system is a two-dimensional harmonic oscillator in non-commuting spatial coordinates coupled to its environment. The dynamics of the covariance matrix, the separability criteria for two Gaussian states in non-commutative space coordinates, and the logarithmic negativity are used to evaluate the quantum entanglement in the system, which is compared to the commutative space coordinates case. The result is applied for two initially entangled states, namely the squeezed vacuum and squeezed thermal states. It can be observed that the phenomenon of entanglement sudden death appears more early in the system for the case of squeezed vacuum state than in the case of squeezed thermal state. Thereafter, it is also observed that non-commutativity effects lead to an increasing of entanglement of initially entangled quantum states, and reduce the separability in the open quantum system. It turns out that a separable state in the usual commutative quantum mechanics might be entangled in non-commutative extension.

Automated summary

This paper examines the influence of non-commutativity on entanglement in a system of two oscillators-modes, using various methods to evaluate quantum entanglement and compare the cases of commutative and non-commutative space coordinates. The results show that non-commutativity can lead to earlier entanglement sudden death, increase entanglement of initially entangled quantum states, and reduce separability in the open quantum system. It is also found that separable states in usual commutative quantum mechanics may become entangled in non-commutative extension.