Quantum teleportation in vacuum using only Unruh-DeWitt detectors

We consider entanglement extraction into two two-level Unruh-DeWitt detectors from a vacuum of a neutral massless quantum scalar field in a four-dimensional spacetime, where the general monopole coupling to the scalar field is assumed. Based on the reduced density matrix of the two detectors derived within the perturbation theory, we show that the single copy of the entangled pair of the detectors can be utilized in quantum teleportation even when the detectors are separated acausally, while we observe no violation of the Bell-CHSH inequality. In the case of the Minkowski vacuum, in particular, we find that entanglement usable in quantum teleportation is extracted due to the special relativistic effect when the detectors are in a relative inertial motion, while it is not when they are comoving inertially and the switching of the detectors is executed adiabatically at infinite past and future.
Comment: 23 pages, 3 figures; ver.2: 21 pages, format and appearance of Fig. 1 improved, reference added, along with minor corrections; ver.3 (final version): title and appearance of figures improved, note added in proof and a new reference added along with minor corrections