Irreversibility of entanglement manipulations: vagueness of the entanglement of cost and entanglement of distillation

We show that the entanglement of cost and entanglement of distillation can be vague when we consider a more general form of entanglement manipulation in which we collectively deal with not only states of our concern but also other states. We introduce the most general entanglement manipulation in which the formation and distillation can be simultaneously performed. We show that in a certain case entanglement manipulations are reversible with respect to the most general entanglement manipulation. This broadens our scope of vision on the irreversibility of entanglement manipulations. (C) 2003 Elsevier Science B.V. All rights reserved.