Quantum state rotation: Circularly transferring quantum states of multiple users

Quantum state exchange is a quantum communication task for two users in which the users faithfully exchange their respective parts of an initial state under the asymptotic scenario. In this work, we generalize the quantum state exchange task to a quantum communication task for $M$ users in which the users circularly transfer their respective parts of an initial state. We assume that every pair of users may share entanglement resources, and they use local operations and classical communication in order to perform the task. We call this generalized task the (asymptotic) quantum state rotation. First of all, we formally define the quantum state rotation task and its optimal entanglement cost, which means the least amount of total entanglement required to carry out the task. We then present lower and upper bounds on the optimal entanglement cost, and provide conditions for zero optimal entanglement cost. Based on these results, we find out a difference between the quantum state rotation task for three or more users and the quantum state exchange task.