Improved analytical bounds on delivery times of long-distance entanglement

Abstract

The ability to distribute high-quality entanglement between remote parties is a necessary primitive for many quantum communication applications. A large range of schemes for realizing the long-distance delivery of remote entanglement has been proposed, for both bipartite and multipartite entanglement. For assessing the viability of these schemes, knowledge of the time at which entanglement is delivered is crucial. Specifically, if the communication task requires multiple remote-entangled quantum states and these states are generated at different times by the scheme, the earlier states will need to wait and thus their quality will decrease while being stored in an (imperfect) memory. For the remote-entanglement delivery schemes which are closest to experimental reach, this time assessment is challenging, as they consist of nondeterministic components such as probabilistic entanglement swaps. For many such protocols even the average time at which entanglement can be distributed is not known exactly, in particular when they consist of feedback loops and forced restarts. In this work, we provide improved analytical bounds on the average and on the quantiles of the completion time of entanglement distribution protocols in the case that all network components have success probabilities lower bounded by a constant. A canonical example of such a protocol is a nested quantum repeater scheme which consists of heralded entanglement generation and entanglement swaps. For this scheme specifically, our results imply that a common approximation to the mean entanglement distribution time, the 3-over-2 formula, is in essence an upper bound to the real time. Our results rely on a novel connection with reliability theory.

Analysis of Multipartite Entanglement Distribution using a Central Quantum-Network Node

“…This is then exactly the same scenario as for the factory node, only the target state is not a GHZ state but a bipartite state. For the rate of such a repeater chain, analytical results similar to ours already exist [34,73,74].…”

“…where H N is the N th harmonic number, is valid up to leading order in q link . This is a well-known result [34,73,74]. Additionally, Eq.…”

“…( 9) to obtain an exact expression for the rate. However, we also report here a known leading-order expression [34,73,74],…”

“…A paper that does derive results for an entanglement switch of general n = k with only a single qubit of buffer memory is [34]. The authors provide analytical tools for understanding and bounding the rate, but do not consider the fidelity.…”

“…( 14) more clearly shows how the rate scales with q link , N and q BSM , thereby providing more intuition. We additionally note that there exists an upper bound [34,75],…”