In this paper, we revisit the entanglement concentration protocol of Bennett et al. and the entanglement dilution protocol of Lo and Popescu via the method of types and provide a simple upper bound on the coefficient of the [Formula: see text] bound on the classical communication cost of the Lo-Popescu protocol.
We study the amount of communication needed for two parties to transform some given joint pure state into another one, either exactly or with some fidelity. Specifically, we present a method to lower bound this communication cost even when the amount of entanglement does not increase. Moreover, the bound applies even if the initial state is supplemented with unlimited entanglement in the form of EPR (Einstein-Podolsky-Rosen) pairs and the communication is allowed to be quantum mechanical. We then apply the method to the determination of the communication cost of asymptotic entanglement concentration and dilution. While concentration is known to require no communication whatsoever, the best known protocol for dilution, discovered by H.-K. Lo and S. Popescu [Phys. Rev. Lett. 83, 1459 (1999)], requires exchange of a number of bits that is of the order of the square root of the number of EPR pairs. Here we prove a matching lower bound of the same asymptotic order, demonstrating the optimality of the Lo-Popescu protocol up to a constant factor and establishing the existence of a fundamental asymmetry between the concentration and dilution tasks. We also discuss states for which the minimal communication cost is proportional to their entanglement, such as the states recently introduced in the context of ``embezzling entanglement'' (W. van Dam and P. Hayden, e-print quant-ph/0201041).
We propose a protocol for realizing two entanglement concentration scheme with cross-Kerr nonlinearity and double cross-phase modulation method. In our protocols, two remote parties do not require the accurate information of the less entangled state and the double cross-phase modulation is introduced to overcome the errors caused by the imperfect Kerr interaction. By this way, a single-photon entanglement is concentrated, and a three-photon polarized GHZ state is also achieved. Furthermore this protocol can be scaled to multiphoton polarized GHZ state concentration and the giant Kerr media is not needed, which makes our protocols more feasible in experiment.