Methods for derivation of density matrix of arbitrary multi-mode gaussian states from its phase space representation

We present a method for derivation of the density matrix of an arbitrary multi-mode continuous variable Gaussian entangled state from its phase space representation. An explicit computer algorithm is given to reconstruct the density matrix from Gaussian covariance matrix and quadrature average values. As an example, we apply our method to the derivation of three-mode symmetric continuous variable entangled state. Our method can be used to analyze the entanglement and correlation in continuous variable quantum network with multi-mode quantum entanglement states.