Linear programs for entanglement and key distribution in the quantum internet

Abstract

Quantum networks will allow to implement communication tasks beyond the reach of their classical counterparts. A pressing and necessary issue for the design of quantum network protocols is the quantification of the rates at which these tasks can be performed. Here, we propose a simple recipe that yields efficiently computable lower and upper bounds for network capacities. For this we make use of the max-flow min-cut theorem and its generalization to multi-commodity flows to obtain linear programs (LPs). We exemplify our recipe deriving the LPs for bipartite settings, settings where multi-pairs of users obtain entanglement in parallel as well as multipartite settings, covering almost all known situations. We also make use of a generalization of the concept of paths between user pairs in a network to Steiner trees spanning the group of users wishing to establish GHZ states. Contents 26A. Proof of Lemma 2 26 B. Proof of Lemma 3 27 C. Proof of Lemma 4 28 arXiv:1809.03120v2 [quant-ph] 4 Jan 2019 total/worst case network and P total/worst case networkas the maximum rate achievable by means of an adaptive protocol, as well as (3) the maximization over a weighted sum of rates over all user pairs. From [13,14], we can obtain upper bounds on the private capacities that involve (1) a minimization over multicuts, i.e. sets of edges the removal of which connects all pairs, or (2) the so-called minimum cut ratio [17]. Using the respective results of [17,19], we can, up to a factor of order O(log r), upper bound the capacities for both cases (1) and (2) by a maximization over concurrent multi-commodity flows, i.e. flows between several user pairs that can be achieved in parallel. In the first case we maximize the sum of flows for all user pairs, whereas in the second case we maximize the worst case flow that is guaranteed for every pair. Both multi-commodity flow maximizations can be cast into LPs. Applying the aggregated repeater protocol [12] to multiple user pairs we also obtain lower bounds in terms of the maximum concurrent multi-commodity flows, providing us with the following efficiently computable bounds: f total/worst case Q ↔ ≤ Q total/worst case network ≤ P total/worst case network ≤ O(log r)f total/worst case Esq/E

Advances in quantum cryptography

“…In the information-theoretic setting and considering the general case of adaptive protocols, upper bounds on this key rate have been studied in Refs. [39,40] and later in a series of other papers [41][42][43][44][45][46]. Here we report the tightest-known bounds that establish the end-toend capacities in chains and networks connected by fundamental channels (subsections XII B and XII C).…”

“…All seems well except for the fact that the value of i in equations (46) and (47) runs from 0 to infinite! This literally means that to have a precise value for Y 1 and e 1 , Alice should in principle, use and infinite number of decoy states.…”

“…In this way, we may achieve better long-distance scalings and further boost the rates by resorting to more complex routing strategies. The study of quantum repeaters and secure QKD networks is one of the hottest topics today [39][40][41][42][43][44][45][46][47][48].…”

Bounds for multi-end communication over quantum networks

“…For instance, for R i = 0 and R j =i = 0, we get the unicast bounds of Eq. (26). For R i = 0, R j =i = 0 and R k =i,j = 0 we get the double-unicast bounds of Eq.…”

“…In recent literature, several studies have appeared where the performance of quantum communication networks have been studied using different configurations and methods [17][18][19][20][21][22][23][24][25][26]. In fact, once understood that the use of relays or repeaters [27][28][29] can overcome the PLOB bound, it is also important to understand the ultimate limits achievable by repeater-assisted quantum communications.…”