We study bidirectional teleportation while explicitly taking into account an environment. This environment initially causes pure dephasing decoherence of the Bell state which assists teleportation. We find that when teleportation is performed in one direction it is accompanied by a transfer of correlations into the post-teleportation state of qubit C, which results in decoherence of the state. In the other direction, if no new decoherence process occurs, we find that not only the state of the qubit but also its correlations with an environment are being teleported with unit Fidelity. These processes do not depend on the measurement outcome during teleportation and do not differentiate between classical and quantum correlations. If, on the other hand, the second teleportation step is preceded by decoherence of the Bell state then the situation is much more complicated. Teleportation and transfer of correlations occur simultaneously, yielding different teleported qubit-environment states for different measurement outcomes. These states can differ in the degree of coherence of the teleported qubit, but only for an entangling Bell-state-environment interaction in the first step of teleportation, can they have different amounts of qubit-environment entanglement. In the extreme case, one of the teleported qubit states can be entangled with the environment while the other is separable.
Transfer and teleportation of system-environment entanglement