Concentrating Partial Entanglement by Local Operations

TL;DR

This work shows that entanglement in identical partly-entangled pure states can be concentrated into a smaller number of maximally entangled pairs using only local operations and classical communication, with asymptotic yield equal to the initial entanglement. It develops Schmidt projection and Procrustean methods, analyzes their efficiencies, and links entanglement concentration to quantum data compression, highlighting complementary regimes. It establishes asymptotic interconvertibility for pure bipartite states and discusses implications for mixed-state entanglement measures, including entanglement of formation and distillable entanglement, illustrated by singlet distillation and Werner states.

Abstract

If two separated observers are supplied with entanglement, in the form of $n$ pairs of particles in identical partly-entangled pure states, one member of each pair being given to each observer; they can, by local actions of each observer, concentrate this entanglement into a smaller number of maximally-entangled pairs of particles, for example Einstein-Podolsky-Rosen singlets, similarly shared between the two observers. The concentration process asymptotically conserves {\em entropy of entanglement}---the von Neumann entropy of the partial density matrix seen by either observer---with the yield of singlets approaching, for large $n$, the base-2 entropy of entanglement of the initial partly-entangled pure state. Conversely, any pure or mixed entangled state of two systems can be produced by two classically-communicating separated observers, drawing on a supply of singlets as their sole source of entanglement.