A comparative study to analyze efficiencies of $$(N+2)$$-qubit partially entangled states in real conditions from the perspective of N controllers

Abstract

We address the efficiencies of $$(N+2)$$ -qubit partially entangled states for the involvement of N controllers in a noisy environment from the perspective of controller’s authority and average fidelity of a controlled teleportation protocol. For this, we design a generalized circuit using single- and two-qubit gates and study different cases of two sets of partially entangled multiqubit states. The analysis shows that for a particular set of partially entangled $$(N+2)$$ -qubit states average fidelity is independent of the state parameter and measurements performed by $$N-1$$ controllers in ideal conditions and measurements performed by $$N-1$$ controllers in noisy conditions with or without applications of weak measurements and its reversal operations. The results thus facilitate the experimental setups to worry about less number of parameters in the protocol. We further compare the efficiencies of these states from a controller’s perspective to increase his/her authority in the protocol. In addition, we also analyze dense coding protocol with and without the involvement of controllers to demonstrate the usefulness of partially entangled states.