The simultaneous amplitude and phase damping of two mode squeezed vacuum state

The Peres-Horodecki criterion of positivity under partial transpose is studied for the simultaneous amplitude and phase damping of a two mode squeezed vacuum state. The criterion turns out to be a necessary and sufficient condition for separability, although the damped state is no longer gaussian. The proof is at the sense of algebraic programming and also perturbation theory. The criterion is stronger than that obtained by the uncertainty principle on the second moments of the partial transposed density matrix. Quantum entanglement or inseparability plays a major role in all branches of quantum information and quantum computation. Peres[1] proposed a criterion for checking the inseparability of a state by introducing the partial transpose operation. This condition is necessary and sufficient for some lower dimensional discrete bipartite systems but is no longer sufficient for higher dimensions[2]. Despite many studies on the discrete states, much attentions have been paid to the continuous variable states [3]. Recently, quantum teleportation of coherent states has been experimentally realized by exploiting a two-mode squeezed vacuum state as an entanglement resource[4]. Due to the decohence of the environment, a pure entanglement state will become mixed. Thus it is important to know if a given bipartite continuous variable state is entangled or not. The decoherence may be caused by coupling to the thermal noise of the environment, amplitude damping, quantum dissipation and phase damping. Besides the phase damping, the other three types of decoherence preserve gaussian property of the state, and a two-mode squeezed vacuum state will evolve to a two mode gaussian mixed state. For the separability of two mode gaussian state, the positivity of the partially transposed state is necessary and sufficient [5][6][7]. However, a gaussian state will evolve to a non-gaussian one by phase damping, and the case of a two mode squeezed vacuum state under the only decoherence of phase damping was perfectly solved[8]. In real experiments, the general situation which should be taken into account is the coexistence of noise, amplitude and phase damping. Theoretically, the separability and entanglement of non-gaussian state are seldom investigated, here we provide an example. Considering the simultaneous damping, the density matrix obeys the following master equation in the interaction picture dρ dt = (L1+L2)ρ. Where L1 is the amplitude damping part L1ρ = ∑ i [ Γ 2 (n+ 1)(2aiρa + i − ai aiρ− ρai ai) + Γ 2 n(2ai ρai − aiai ρ− ρaiai )], (1) with n the average photon number of the thermal environment. And L2 is the phase damping part (e.g. [8]), L2 = ∑ i γ 2 [2ai aiρa + i ai − (ai ai)ρ− ρ(ai ai)]. (2) The state can be equivalently specified by its characteristic function. Every operator A ∈ B(H) is completely determined by its characteristic function χA := tr[AD(μ)] [9], where D(μ) = exp(μa+−μ∗a) is the displacement operator. It follows that A may be written in terms of χA as [10]: A = ∫