Application of Non-Equilibrium Thermo Field Dynamics to quantum teleportation under the environment

Quantum teleportation for continuous variables is treated by Non-Equilibrium Thermo Field Dynamics (NETFD), a canonical operator formalism for dissipative quantum systems, in order to study the effect of imperfect quantum entanglement on quantum communication. We used an entangled state constructed by two squeezed states. The entangled state is imperfect due to two reasons, i.e., one is the finiteness of the squeezing parameter r and the other comes from the process that the squeezed states are created under the dissipative interaction with the environment. We derive the expressions for one-shot fidelity (OSF), probability density function (PDF) associated with OSF and (averaged) fidelity by making full use of the algebraic manipulation of operator algebra within NETFD. We found that OSF and PDF are given by Gaussian forms with its peak at the original information α to be teleported, and that for r≫1 the variances of these quantities blow up to infinity for κ/χ≤1, while they approach to finite values for κ/χ>1. Here, χ represents the intensity of a degenerate parametric process, and κ the relaxation rate due to the interaction with the environment. The blow-up of the variances for OSF and PDF guarantees higher security against eavesdropping. With the blow-up of the variances, the height of PDF reduces to small because of the normalization of probability, while the height of OSF approaches to 1 indicating a higher performance of the quantum teleportation. We also found that in the limit κ/χ≫1 the variances of both OSF and PDF for any value of r (>0) reduce to 1 which is the same value as the case r=0, i.e., no entanglement.