In this work, we invoke several stochastic models to investigate the behavior of quantum networks. Motivated by the increasing desire for secure communication, a special focus lies on how to secure such a network in the sense that a message can be sent from an arbitrary node to any other without being disclosed.
Due to quantum key distribution (QKD) that we shortly review in this work, a quantum network is known to have perfectly secure links from an information theoretic point of view. Hence, our analysis concentrates on the nodes of the network, as this is where a potential eavesdropper has to focus on as well.
Among the quantities that characterize a quantum network is its capacity. We here consider the situation where a quantum network is given and analyzed rather then how to design a new network. A simple criterion is given when viewing a network as a queue where messages arrive to be encrypted. Further, we derive the probability of getting stock when sending a message from one node to another as well as we give a lower bound for the probability to get delayed during a transmission over the entire network.
A possible way to extend the security of the links to the entire quantum network is multipath transmission. Still, this approach depends on the assumption that traffic runs across the network as intended. We imagine an indirect eavesdropper that causes traffic to be redirected towards the nodes under his control and hence getting more information then estimated from the user of the network. With a Markov model at hand, we give necessary and sufficient conditions that ensure perfect security of a network in case of such an attack for both a passive and an active adversary.
Finally, we investigate the relationship between two quantities that describe the quality of the QKD-key used to secure the links of the network, namely between the quantum bit error rate (QBER) and the keyrate. Copula models allow a separate treatment of marginal behavior and dependence and are hence introduced and illustrated. For a subclass of copulas we introduce a novel estimator that overcomes some limitations of current estimators. When applying this method to the quantum network we also determine a critical value whose exceedence indicates the presence of an eavesdropper.