Olivier Lalonde

We study the following task: Alice is given a classical description of a rank-k projector P on C^d, and Alice and Bob want to prepare the state P/k on Bob's side using shared entanglement and classical communication. The general form of this task is known as remote state preparation (RSP). We give nearly-matching lower and upper bounds for the entanglement cost and communication cost for RSP of the states P/k. Ours are the first nearly matching upper and lower bounds for RSP of mixed states, and in the special case of pure states, our lower bound outperforms the best previously known lower bound. Our results show that any pure entangled state that can be used to do RSP of these states with o(d) bits of communication, can distill log d ebits of entanglement, and conversely, any state that can distill log d ebits of entanglement can be used to do RSP of these states efficiently. As an application of our results, we give a new entanglement-assisted communication protocol for the equality function that uses 1/2 log n + O(1) many ebits, and O(1) communication.