How the quality of a received EPR pair depends on the distances from an EPR source?

Abstract

Let an EPR source which generates maximally entangled pairs be located so that it has distances $L_1$ and $L_2$ to two users. After taking into account various effects like loss of photons, deficiencies in the source and detectors, an entangled pair traveling through the channel may loose its perfect correlation due to errors in the channel. How the entanglement of the received pair depends on the above distances and the local properties of the channels used for this transmission? What is the best location of the source if we want to achieve the highest fidelity? What is the threshold distance beyond which the entanglement of the pair vanishes and becomes useless for using in teleportation. We discuss these problems for the Pauli channel which simulates the effect of optical fibers and possibly the atmosphere on the polarization-entangled photons.