Distributed Feedback Control of Quantum Networks**The Japanese version of this work was presented in Takeuchi (2015); Takeuchi & Tsumura (2015). This work is supported in part by Grant-in-Aid for Scientific Research (B) (25289127, 16H04382), Japan Society for the Promotion of Science.

In this paper, we deal with a distributed feedback control of quantum networks called a quantum consensus algorithm (QCA) with local quantum observation and feedback proposed by Kamon & Ohki (2013, 2014) and prove strictly that QCA makes quantum states converge to a quantum state called symmetric state consensus (SSC) with probability one from arbitrary initial states keeping purity The difficulty of the proof is from that the objective system is stochastic and non-linear, and we solve it by employing the stochastic Lyapunov stability analysis. We also show that QCA can generate a desirable W-state, which is known as an important entangled quantum state and utilized for many applications of quantum information technology.