Controlled remote implementation of operations via graph states

We propose protocols for controlled remote implementation of operations with convincing control power. Sharing a $(2N+1)$-partite graph state, $2N$ participants collaborate to prepare the stator and realize the operation $\otimes_{j=1}^N\exp{[i\alpha_j\sigma_{n_{O_j}}]}$ on $N$ unknown states for distributed systems $O_j$, with the permission of a controller. All the implementation requirements of our protocol can be satisfied by means of local operations and classical communications, and the experimental feasibility is presented according to current techniques. We characterize the entanglement requirement of our protocol in terms of geometric measure of entanglement. It turns out to be economic to realize the control function from the perspective of entanglement cost. Further we show that the control power of our protocol is reliable by positive operator valued measurement.