  It is commonly believed that the fidelity of quantum teleportation using
localized quantum objects with one party or both accelerated in vacuum would be
degraded due to the heat-up by the Unruh effect. In this paper we point out
that the Unruh effect is not the whole story in accounting for all the
relativistic effects in quantum teleportation. First, there could be
degradation of fidelity by a common field environment even when both quantum
objects are in inertial motion. Second, relativistic effects entering the
description of the dynamics such as frame dependence, time dilation, and
Doppler shift, already existent in inertial motion, can compete with or even
overwhelm the effect due to uniform acceleration in a quantum field. We show it
is not true that larger acceleration of an object would necessarily lead to a
faster degradation of fidelity. These claims are based on four cases of quantum
teleportation we studied using two Unruh-DeWitt detectors coupled via a common
quantum field initially in the Minkowski vacuum. We find the quantum
entanglement evaluated around the light cone, rather than the conventional ones
evaluated on the Minkowski time-slices, is the necessary condition for the
averaged fidelity of quantum teleportation beating the classical one. These
results are useful as a guide to making judicious choices of states and
parameter ranges and estimation of the efficiency of quantum teleportation in
relativistic quantum systems under environmental influences.
