Evolution and Entanglement of Gaussian States in the Parametric Amplifier

Abstract

We obtain the linear time-dependent constants of motion of the parametric amplifier and use them to determine the evolution of a general two-mode Gaussian state in the tomographic-probability representation. By means of the discretization of the continuous variable density matrix, we calculate the von Neumann and linear entropies to measure the entanglement properties between the modes of the amplifier. We compare the obtained results for the nonlocal correlations with those associated to a linear map of discretized symplectic Gaussian-state tomogram onto a qubit tomogram. We use this qubit portrait procedure to establish Bell-type inequalities, which provide a necessary condition to determine the separability of quantum states, which can be evaluated through homodyne detection. We define the other no-signaling nonlocal correlations through the portrait procedure for noncomposite systems.