Practical long-distance quantum communication using concatenated entanglement swapping

TL;DR

This paper develops a theoretical framework for long-distance quantum communication using concatenated entanglement swapping, accounting for practical imperfections like loss and detector inefficiencies, and provides exact and numerical solutions for specific cases.

Contribution

It introduces a comprehensive model for long-distance quantum communication with entanglement swapping, including realistic imperfections, and derives exact and numerical results for the entangled states and their visibility.

Findings

Exact solution for N=2 in ideal conditions

Numerical results for N=2 with non-ideal conditions

Complexity analysis for the general case

Abstract

We construct a theory for long-distance quantum communication based on sharing entanglement through a linear chain of $N$ elementary swapping segments of length~$L=Nl$ where $l$ is the length of each elementary swap setup. Entanglement swapping is achieved by linear optics, photon counting and post-selection, and we include effects due to multi-photon sources, transmission loss and detector inefficiencies and dark counts. Specifically we calculate the resultant four-mode state shared by the two parties at the two ends of the chain, and we derive the two-photon coincidence rate expected for this state and thereby the visibility of this long-range entangled state. The expression is a nested sum with each sum extending from zero to infinite photons, and we solve the case $N=2$ exactly for the ideal case (zero dark counts, unit-efficiency detectors and no transmission loss) and numerically for $N=2$ in the non-ideal case with truncation at $n_\text{max}=3$ photons in each mode. For the general case, we show that the computational complexity for the numerical solution is $n_\text{max}^{12N}$.