Statistical analysis of quantum-entangled-network generation

We develop techniques to analyze the statistics of completion times of nondeterministic elements in quantum-entanglement generation and how they affect the overall performance as measured by the secret key rate. By considering such processes as Markov chains, we show how to obtain exact expressions for the probability distributions over the number of errors that a network acquires, as well as the distribution of entanglement establishment times. We show how results from complex analysis can be used to analyze Markov matrices to extract information with a lower computational complexity than previous methods. We apply these techniques to the Briegel et al. quantum repeater protocol [H.-J. Briegel et al., Phys. Rev. Lett. 81, 5932 (1998)] and find that consideration of the effect of statistical fluctuations can tighten bounds on the secret key rate by three orders of magnitude, when compared to more simplistic analyses. We also use the theory of order statistics to derive tighter bounds on the minimum quantum memory lifetimes that are required in order to communicate securely.