Cyclic quantum teleportation

We propose a scheme of cyclic quantum teleportation for three unknown qubits using six-qubit maximally entangled state as the quantum channel. Suppose there are three observers Alice, Bob and Charlie, each of them has been given a quantum system such as a photon or spin-12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{2}$$\end{document} particle, prepared in state unknown to them. We show how to implement the cyclic quantum teleportation where Alice can transfer her single-qubit state of qubit a to Bob, Bob can transfer his single-qubit state of qubit b to Charlie and Charlie can also transfer his single-qubit state of qubit c to Alice. We can also implement the cyclic quantum teleportation with N⩾3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\geqslant 3$$\end{document} observers by constructing a 2N-qubit maximally entangled state as the quantum channel. By changing the quantum channel, we can change the direction of teleportation. Therefore, our scheme can realize teleportation in quantum information networks with N observers in different directions, and the security of our scheme is also investigated at the end of the paper.