High-dimensional bidirectional controlled teleportation based on network coding

Two explicit schemes are designed to illustrate bidirectional controlled teleportation in a fair and practical manner. One is a symmetric scheme for single-qutrit states. The other is an asymmetric scheme for single- and two-qutrit states. Then, we extend three-dimensional system to high-dimensional system and come up with a universal scheme for arbitrary n1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_1$$\end{document}- and n2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_2$$\end{document}-qudit states by using a (2n1+2n2+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2n_1+2n_2+1)$$\end{document}-qudit entangled state as the quantum channel, where n1≤n2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_1\le n_2$$\end{document}. The senders and the controller need to perform general Bell basis measurement and Z basis measurement, respectively. According to their measurement results, the receivers can reestablish the initial states simultaneously by performing single-qudit recovery operations which are derived by a general formula. It is worth mentioning that 2n1+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2n_1+1$$\end{document} dit classical communication cost is saved at the controller’s broadcast channel with the aid of network coding. Moreover, we consider the effect of two-type high-dimensional decoherence noises: dit-flip noise and d-phase-flip noise.