Statistical analysis of quantum-entangled-network generation

Abstract

We develop techniques to analyse the statistics of completion times of non-deterministic 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 analyse Markov matrices to extract information with a lower computational complexity than previous methods. We apply these techniques to the Innsbruck quantum repeater protocol, and find that consideration of the effect of statistical fluctuations tightens bounds on the secret key rate by 3 orders of magnitude. 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.

Efficient Computation of the Waiting Time and Fidelity in Quantum Repeater Chains

“…Existing analytical work is mostly aimed at estimating the mean waiting time or fidelity (see also [22], [42], [43] for other figures of merit). Some of this work builds on an approximation of the mean waiting time under the small-probability assumption [17], [23], [31], [44], while for a small number of segments or for some protocols it is possible to compute the waiting time probability distribution exactly [13], [30], [43], [45], [46]. However, depending on the application different statistics become relevant.…”

“…Its runtime scales with the number of vertices in the Markov chain, which grows exponentially with the number of repeater segments. In more recent work, Vinay and Kok show how to improve the runtime using results from complex analysis [46]. However, this method still remains exponential in the number arXiv:1912.07688v1 [quant-ph] 16 Dec 2019 of repeater segments.…”

Efficient computation of the waiting time and fidelity in quantum repeater chains

“…Its runtime scales with the number of vertices in the Markov chain, which grows exponentially with the number of repeater segments. In more recent work, Vinay and Kok show how to improve the runtime using results from complex analysis [46]. However, this method still remains exponential in the number of repeater segments.…”

Policies for elementary links in a quantum network

“…One of the goals of this work is to explicitly formalize the approaches taken in the aforementioned works within the context of decision processes, because this allows us to systematically study different policies and calculate quantities that are relevant for quantum networks, such as entanglement distribution rates and fidelities of the quantum states of the links. This work is complementary to prior work that uses Markov chains to analyze waiting times and entanglement distribution rates for a chain of quantum repeaters [65,[130][131][132]; we also refer to the work on entanglement switches in [80][81][82][83], which use both discrete-time and continuous-time Markov chains. This work is also complementary to prior work that analyzes the quantum state in a quantum repeater chain with noisy quantum memories [133][134][135][136][137].…”