Contrasting Effects of Control on Fidelity and Fidelity Deviation in Controlled Teleportation

Abstract: Teleportation performance is commonly characterized by the average teleportation fidelity, while its variation over input states provides additional information captured by the fidelity deviation. In controlled teleportation, the controller's measurement introduces an additional source of fidelity variation through its measurement outcomes. We investigate the fidelity deviation in controlled teleportation with three-qubit pure states. We derive a lower bound on the fidelity deviation and show that it is attainable together with the maximal average teleportation fidelity. Among the measurements attaining the maximal average fidelity, however, the fidelity deviation can vary depending on the controller's measurement. We further examine the effect of controller assistance by comparing controlled teleportation with direct teleportation using the reduced state. Unlike the maximal average fidelity, which cannot decrease with controller assistance, the minimal fidelity deviation is not necessarily reduced by controller assistance. In particular, for any W-class pure state, controller assistance cannot reduce the fidelity deviation. These results reveal contrasting effects of control on teleportation fidelity and its deviation in controlled teleportation.