Classical-communication cost in distributed quantum-information processing: A generalization of quantum-communication complexity

We study the amount of classical communication needed for distributed quantum-information processing. In particular, we introduce the concept of “remote preparation” of a quantum state. Given an ensemble of states, Alice’s task is to help Bob in a distant laboratory to prepare a state of her choice. We find several examples of an ensemble with an entropy S where the remote preparation can be done with a communication cost lower than the amount $\mathrm{}\left(2S\right)\mathrm{}$ required by standard teleportation. We conjecture that, for an arbitrary N-dimensional pure state, its remote preparation requires $2{\log }_{2}N$ bits of classical communication, as in standard teleportation.