A tight lower bound on the classical communication cost of entanglement   dilution
  Entanglement concentration requires no classical communication, but the best prior art result for diluting to N copies of a partially entangled state requires an amount of communication on the order of sqrt(N) bits. Our main result is to prove this prior art result optimal up to a constant factor; any procedure for creating N partially entangled states from singlets requires Omega(sqrt(N)) bits of classical communication. Previously not even a constant bound was known for approximate entanglement transforms.   We also prove a lower bound on the inefficiency of the process: to dilute singlets to N copies of a partially entangled state, the entropy of entanglement must decrease by Omega(sqrt(N)). 
Aram Harrow and Hoi-Kwong Lo