Concentrating entanglement by local actions - beyond mean values

Abstract

Previous investigations of entanglement manipulations have focused on the average properties of the outcomes and little is known about the actual probability distribution. Here we go beyond the average properties. We show that, for a pure entangled state shared between two separated persons Alice and Bob, the mathematical interchange symmetry of the Schmidt decomposition can be promoted into a physical symmetry between the actions of Alice and Bob. Consequently, the most general (multi-step two-way-communications) strategy of entanglement manipulation of a pure state is, in fact, equivalent to a strategy involving only a single (generalized) measurement by Alice followed by one-way communications of its result to Bob. One important question is whether coherent manipulations in quantum mechanics can enhance the probability of large deviations from the average behaviour. We answer this question in the negative by showing that, given n pairs of identical partly entangled pure states |Ψ〉 with entropy of entanglement E(|Ψ〉), the probability of getting nK (K>E(|Ψ〉)) singlets out of entanglement concentration tends to zero as n tends to infinity.