The evolution of the optical NOON state is investigated. The environments of
the evolution are the thermal noise and amplitude damping due to fiber loss.
The characteristic function and the density matrix of the time-dependent
state are deduced strictly. We calculate the relative entropy of
entanglement for the truncated time-dependent state for N=1 case. The
phase measurement performance of the damped states is also studied.
The evolution of two-mode Gaussian state under complex
amplification, non-symmetric damping and thermal noise is studied.
The analytical time dependent solution of the state characteristic
function is obtained in some special cases. The separability
criterions are given for the final state of weak complex
amplification as well as strong complex amplification.
We in this paper calculate the relative entropy between the bipartite
Gaussian states. By evaluating the minimization of the relative entropy of an entangled Gaussian state with respect to the separable Gaussian states, we obtain the Gaussian relative entropy of entanglement of the state. The bipartite operations of squeezing and rotations on the separable states are also addressed. We prove that for bipartite symmetric Gaussian states the separable states we sought will be symmetric Gaussian states, and we get quite simple expressions of Gaussian relative entropy of entanglement for such states.
The entanglement of formation of bipartite quantum Gaussian state is investigated by means of local operation, which maps continuous variable state to a bipartite qubits system. A two parameters quantum Guassian state is introduced, the concurrence of its mapped qubits state is calculated to characterize the entanglement of the original state.