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Quantum entanglement in a non-commutative system

We explore the effect of two-dimensional position-space non-commutativity on the bipartite entanglement of continuous variable systems. We first extend the standard symplectic framework for studying entanglement of Gaussian states of commutative systems to the case of non-commutative systems residing in two dimensions. Using the positive partial transpose criterion for separability of bipartite states we derive a new condition on the separability of a non-commutative system that is dependent on the noncommutative parameter $θ$. We then consider the specific example of a bipartite Gaussian state and show the quantitative reduction of entanglement originating from non-commutative dynamics. We show that such a reduction of entanglement for a non-commutative system arising from the modification of the variances of the phase space variables (uncertainty relations) is clearly manifested between two particles that are separated by small distances.