0 Teleportation of the Relativistic Quantum Field

The process of teleportation of a completely unknown one-particle state of a free relativistic quantum field is considered. In contrast to the non-relativistic quantum mechanics, the telepor-tation of an unknown state of the quantum field cannot be in principle described in terms of a measurement in a tensor product of two Hilbert spaces to which the unknown state and the state of the EPR-pair belong. The reason is of the existence of a cyclic (vacuum) state common to both the unknown state and the EPR-pair. Due to the common vacuum vector and the microcausality principle (commutation relations for the field operators), the teleportation amplitude contains inevitably contributions which are irrelevant to the teleportation process. Hence in the relativistic theory the teleportation in the sense it is understood in the non-relativistic quantum mechanics proves to be impossible because of the impossibility of the realization of the appropriate measurement as a tensor product of the measurements related to the individual subsystems so that one can only speak of the amplitude of the propagation of the field as a whole. One of the fundamental results of the non-relativistic quantum information theory consists in the possibility of the teleportation of an unknown quantum state by means of a quantum communication channel realized by a non-local entangled state (an EPR-pair [1]) used together with a classical communication channel [2]. When the unknown quantum state |ψ s belongs to a finite-dimensional Hilbert space (dimH s < ∞), the teleportation can be preformed ideally and with the unit probability employing an EPR-pair with finite energy. On the other hand, if the state space of the teleported system is infinite-dimensional, (dimH s = ∞), the ideal teleportation formally requires an EPR-pair with infinite energy [3,4]. However, the teleported state can be made arbitrarily close to the input unknown state with the probability arbitrarily close to unit by increasing the energy of the EPR-pair and thus making the EPR correlations more and more close the ideal ones. The non-relativistic quantum mechanics yields only an approximate description of the reality. A more correct and complete description is provided by quantum field theory (since the relativistic quantum mechanics does not allow any sensible physical interpretation, the relativistic theory arises from the very beginning as a quantum field theory). Therefore, it is interesting to consider the possibility of teleportation of a completely unknown state of the relativistic quantum field. In addition, although …