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arXiv:0810.5334 · Physical Review A

Quantum repeaters with imperfect memories: Cost and scalability

Mohsen Razavi, Marco Piani, Norbert Lütkenhaus

Published Sep 1, 2009• 124 citations
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Abstract

Memory dephasing and its impact on the rate of entanglement generation in quantum repeaters is addressed. For systems that rely on probabilistic schemes for entanglement distribution and connection, we estimate the maximum achievable rate per employed memory for our optimized partial nesting protocol, when a large number of memories are being used in each node. The above rate scales polynomially with distance, $L$, if quantum memories with infinitely long coherence times are available or if we employ a fully fault-tolerant scheme. For memories with finite coherence times and no fault-tolerant protection, the above rate optimistically degrades exponentially in $\sqrt{L}$, regardless of the employed purification scheme. It decays, at best, exponentially in $L$ if no purification is used.

DOI
10.1103/physreva.80.032301

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Memory dephasing and its impact on the rate of entanglement generation in quantum repeaters is addressed. For systems that rely on probabilistic schemes for entanglement distribution and connection, we estimate the maximum achievable rate per employed memory for our optimized partial nesting protocol, when a large number of memories are being used in each node. The above rate scales polynomially with distance, LLL, if quantum memories with infinitely long coherence times are available or if we employ a fully fault-tolerant scheme. For memories with finite coherence times and no fault-tolerant protection, the above rate optimistically degrades exponentially in L\sqrt{L}L​, regardless of the employed purification scheme. It decays, at best, exponentially in LLL if no purification is used.
Quantum repeater networks that allow long-distance entanglement distribution will be the backbone of distributed quantum information processing. In this paper we explore entanglement distribution using quantum repeaters with optimized buffer time, equipped with noisy quantum memories and performing imperfect entanglement purification and swapping. We observe that increasing the number of memories on end nodes leads to a higher entanglement distribution rate per memory and higher probability of high-fidelity entanglement distribution, at least for the case with perfect operations. When imperfect operations are considered, however, we make the surprising observation that the permemory entanglement rate decreases with increasing number of memories. Our results suggest that building quantum repeaters that perform well under realistic conditions requires careful modeling and design that takes into consideration the operations and resources that are finite and imperfect.
Quantum key distribution over probabilistic quantum repeaters is addressed. We compare, under practical assumptions, two such schemes in terms of their secret key generation rates per quantum memory. The two schemes under investigation are the one proposed by Duan et al. [Nature (London) 414, 413 (2001)] and that of Sangouard et al. [Phys. Rev. A 76, 050301 (2007)]. We consider various sources of imperfection in both protocols, such as nonzero double-photon probabilities at the sources, dark counts in detectors, and inefficiencies in the channel, photodetectors, and memories. We also consider memory decay and dephasing processes in our analysis. For the latter system, we determine the maximum value of the double-photon probability beyond which secret key distillation is not possible. We also find crossover distances for one nesting level to its subsequent one. We finally compare the two protocols in terms of their achievable secret key generation rates at their optimal settings. Our results specify regimes of operation where one system outperforms the other.
Quantum repeaters enable us to distribute entanglement between remote parties by relying on a network of quantum memory units that exhibit efficient coupling to light, scalability, and long coherence times. Entanglement is initially distributed between nearest neighbors and then extended to the far-end nodes using entanglement swapping techniques. For real-time applications, such as quantum key distribution, the above tasks must be repeated successively, according to a proper protocol, to generate entangled states at a certain rate. This paper studies a number of such protocols and the interplay between the rate of entanglement generation, the number of employed memories, and the coherence time of memory units.
We develop entanglement swapping protocols and memory allocation methods for quantum repeater chains. Unlike most of the existing studies, the memory size of each quantum repeater in this work is a parameter that can be optimized. Based on Markov chain modeling of the entanglement distribution process, we determine the trade-off between the entanglement distribution rate and the memory size for temporal multiplexing techniques. We then propose three memory allocation methods that achieve entanglement distribution rates decaying polynomially with respect to the distance while using constant average memory slots per node. We also quantify the average number of memory slots required due to classical communication delay, as well as the delay of entanglement distribution. Our results show that a moderate memory size suffices to achieve a polynomial decay of entanglement distribution rate with respect to the distance, which is the scaling achieved by the optimal protocol even with infinite memory size at each node.
We present a detailed rate analysis for a hybrid quantum repeater assuming perfect memories and using optimal probabilistic entanglement generation and deterministic swapping routines. The hybrid quantum repeater protocol is based on atomic qubit-entanglement distribution through optical coherent-state communication. An exact, analytical formula for the rates of entanglement generation in quantum repeaters is derived, including a study on the impacts of entanglement purification and multiplexing strategies. More specifically, we consider scenarios with as little purification as possible and we show that for sufficiently low local losses, such purifications are still more powerful than multiplexing. In a possible experimental scenario, our hybrid system can create near-maximally entangled (F=0.98F=0.98F=0.98) pairs over a distance of 1280 km at rates of the order of 100 Hz.
At the core of the quantum Internet lie quantum repeaters that enable remote end-to-end entanglement generation. Fundamentally, the entanglement generation rate and fidelity of quantum repeaters constitute the bottleneck for end-to-end performance. To achieve high rates, quantum repeaters employ quantum memory multiplexing. In a high-rate standard repeater, each memory sequentially generates an entanglement with its neighboring nodes and then applies entanglement swapping. This, however, results in low fidelity due to decoherence of the first-formed entanglement in the sequential generation process. By allocating different numbers of memories to simultaneously form entanglements with the left and right adjacent nodes, quantum repeaters reduce high waiting times and achieve high fidelity. In such a repeater, a mismatch problem arises due to the difference between the probabilistic number of generated entanglements on both sides. Consequently, some entanglements remain stored until opposite entanglements are available. The mismatch problem reduces the repeater rate and particularly the entanglement fidelity. In this paper, we consider the mismatch problem in an asymmetric repeater with different distances to its adjacent nodes. To mitigate the mismatch problem, we derive a dynamic optimal memory allocation. Under the optimal allocation, we derive statistical lower bounds on the achievable rate and fidelity. We demonstrate that the optimal allocation significantly improves the fidelity while maintaining a comparable rate to the standard repeater. In contrast, our results show that fixed memory allocation may be detrimental to the fidelity.