The Generation Cost of Bipartite Quantum States under LOCC

We consider a realistic setting of quantum tasks that generate shared bipartite quantum states. Suppose \alice and \bob are located at different places and need to produce a target shared quantum state $\rho$. In order to save quantum communication, they can choose to share a proper smaller quantum state $\sigma$ first, and then turn $\sigma$ to $\rho$ by performing only local quantum operations and classical communications (LOCC). We hope $\sigma$ is the optimal such that the quantum communication needed is as little as possible, which is called the generation cost of $\rho$. In this paper, for an arbitrary bipartite $\rho$, we characterize its generation cost completely by proving that it is exactly equivalent to the logarithm of the Schmidt number of $\rho$. Similar quantum schemes where classical communication is not allowed have actually been considered. By comparing the two settings, we are able to look into the role that classical communication plays in these fundamental tasks, where we exhibit some instances in which classical communication is not helpful completely.