Inseparability inequalities for higher-order moments for bipartite systems

There are several examples of bipartite entangled states of continuous variables for which the existing criteria for entanglement using the inequalities involving the second order moments are insufficient. We derive new inequalities involving higher order correlation, for testing entanglement in non-Gaussian states. In this context we study an example of a non-Gaussian state, which is a bipartite entangled state of the form ψ(xa, xb) ∝ (αxa +βxb)e 2 a +x b . Our results open up an avenue to search for new inequalities to test entanglement in non-Gaussian states. PACS numbers: 03.67.Mn, 42.50.Dv ‡ On leave of absence from Physical Research Laboratory, Navrangpura, Ahmedabad, India Inseparability inequalities for higher-order moments for bipartite systems 2 The detection and characterization of entanglement in the state of a composite is an important issue in quantum information science. Peres has addressed this issue [1] for the first time to show that inseparability of a bipartite composite system can be understood in terms of negative eigenvalues of partial transpose of its density operator. There are several other criteria for inseparability in terms of correlation entropy and linear entropy [2, 3] and in terms of positivity of Glauber-Sudarshan P -function [14]. However all these measures cannot be put to experimental tests. To detect entanglement of any composite system experimentally, one needs to have certain criteria in terms of expectation values of some observables. Using Peres’s criterion of separability, Simon [5] has derived certain separability inequalities, violation of which is sufficient to detect entanglement in bipartite systems. These inequalities involve variances of relative position and total momentum coordinates of the two subsystems and thus can be verified experimentally [6]. Duan et al. [7] have also derived equivalent inequalities independently using the positivity of the quadratic forms. It is further proved that for Gaussian states (states with Gaussian wave functions in coordinate space), violation of these inequalities provides a necessary and sufficient criterion for entanglement. A different form of criterion for entanglement involving second order moments has been derived by Mancini et al. [8]. These inequalities have been tested for entangled states produced by optical parametric oscillators and other systems where the output state can be approximated by Gaussian states [9, 10, 11, 12]. In context of quantum information and communication, non-Gaussian states are equally important as Gaussian states. Several entangled non-Gaussian states have been studied in literature [13, 14, 15]. A way to produce non-Gaussian state is via state reduction method [16, 17, 18]. Thus characterization of entanglement in non-Gaussian state remains an open question. This motivates us to derive new inequalities, when the existing inequalities based on second order correlation fail to test entanglement in these states. Thus these inequalities are expected to involve higher order correlation between position and momentum coordinates. In this paper we consider a bipartite entangled state of bosonic system, which, in turn, is a non-Gaussian state in coordinate space. We consider an entangled state for which the existing inseparability inequalities cannot provide any information about the inseparability of the state. We derive new inseparability inequalities to test its entanglement. We start by deriving the inequalities involving the second order moments. Consider the set of operators U = 1 √ 2 (xa + xb) ; V = 1 √ 2 (pa + pb) [U, V ] = i . (1) Then we would have the uncertainty relation ∆U∆V ≥ 1 2 . (2) We now use Peres-Horodecki criteria of separability in terms of the partial transpose. Under partial transpose, xb → xb, pb → −pb. Hence the condition that the partial Inseparability inequalities for higher-order moments for bipartite systems 3 transpose of a density matrix is also a genuine density matrix, would imply that ∆ (