Controlled cyclic assisted cloning of arbitrary unknown single-particle states

Making use of Hadamard-gate transformations and controlled-NOT transformations, we construct a seven-particle maximally entangled state, and obtain a (2N+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2N+1)$$\end{document}-particle (N>3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N>3$$\end{document}) maximally entangled state through this construction method. Then, using this seven-particle entangled state to serve as quantum channel, we suggest a three-party cyclic protocol for cloning three different unknown single-particle states with help of the state preparer and the permission of the controller. The first phase of this protocol needs a controlled cyclic quantu teleportation (CCQT), where Alice transmits an arbitrary unknown single-particle state to Bob, Bob teleports an arbitrary unknown single-particle state to Charlie, meanwhile, Charlie also convey an arbitrary unknown single-particle state to Alice under the consent of the controller. In the second phase, after receiving the three-particle measurement result from the preparer, three different unknown single-qubit states or their orthogonal complement states are cloned simultaneously and probabilistically at the positions of Alice, Bob, and Charlie respectively. Subsequently, we will extend the above three-party cyclic protocol to the case of (2N + 1)-party loops by exploiting the (2N + 1)-particle maximally entangled state act as quantum channel. Additionally, taking the controlled three-party cyclic protocol through non-maximally entangled channel as an example, we analyze the assisted cloning protocol for arbitrary unknown single-particle states from three perspectives: projective measurement, positive operator-value measurement (POVM), and generalized Bell-state measurement. We also point out that by increasing the number of state preparers or controllers, the above schemes can be promoted to meet the needs of future versatile quantum networks.