2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/90147Entanglement 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)).10 pages, 2 figures, RevTeX. v2 improved presentation, added remarksQuantum Physicstext