Detecting the inseparability and distillability of continuous variable states in Fock space - art. no. 137903

The partial transposition (PT) operation is an efficient tool in detecting the inseparability of a mixed state. We give an explicit formula for the PT operation for the continuous variable states in Fock space. We then give t he necessary and sufficient condition for the positivity of Gaussian operat ors. Based on this, a number of criteria on the inseparability and distilla bility for the multimode Gaussian states are naturally drawn. We finally gi ve an explicit formula for the state in a subspace of a global Gaussian sta te. This formula, together with the known results for Gaussian states, give s the criteria for the inseparability and distillability in a subspace of t he global Gaussian state.