We investigate quantum repeaters in the context of quantum key distribution. We optimize the secret key rate per memory per second with respect to different distillation protocols and distillation strategies. For this purpose, we also derive an analytical expression for the average number of entangled pairs created by the quantum repeater, including classical communication times for entanglement swapping and entanglement distillation. We investigate the impact of this classical communication time on the secret key rate. Finally, we study the effect of the detector efficiency on the secret key rate.
We investigate quantum repeaters in the context of quantum key distribution. We optimize the secret key rate per memory per second with respect to different distillation protocols and distillation strategies. For this purpose, we also derive an analytical expression for the average number of entangled pairs created by the quantum repeater, including classical communication times for entanglement swapping and entanglement distillation. We investigate the impact of this classical communication time on the secret key rate. Finally, we study the effect of the detector efficiency on the secret key rate.
We analyze various prominent quantum repeater protocols in the context of long-distance quantum key distribution. These protocols are the original quantum repeater proposal by Briegel, D\"ur, Cirac and Zoller, the so-called hybrid quantum repeater using optical coherent states dispersively interacting with atomic spin qubits, and the Duan-Lukin-Cirac-Zoller-type repeater using atomic ensembles together with linear optics and, in its most recent extension, heralded qubit amplifiers. For our analysis, we investigate the most important experimental parameters of every repeater component and find their minimally required values for obtaining a nonzero secret key. Additionally, we examine in detail the impact of device imperfections on the final secret key rate and on the optimal number of rounds of distillation when the entangled states are purified right after their initial distribution.
We investigate secret key rates for the quantum repeater using encoding [L. Jiang et al., Phys. Rev. A 79, 032325 (2009)] and compare them to the standard repeater scheme by Briegel, D\"ur, Cirac, and Zoller. The former scheme has the advantage of a minimal consumption of classical communication. We analyze the trade-off in the secret key rate between the communication time and the required resources. For this purpose we introduce an error model for the repeater using encoding which allows for input Bell states with a fidelity smaller than one, in contrast to the model given by L. Jiang et al. [Phys. Rev. A 79, 032325 (2009)]. We show that one can correct additional errors in the encoded connection procedure of this repeater and develop a suitable decoding algorithm. Furthermore, we derive the rate of producing entangled pairs for the quantum repeater using encoding and give the minimal parameter values (gate quality and initial fidelity) for establishing a nonzero secret key. We find that the generic quantum repeater is optimal regarding the secret key rate per memory per second and show that the encoded quantum repeater using the simple three-qubit repetition code can even have an advantage with respect to the resources compared to other recent quantum repeater schemes with encoding.
We propose a postselection technique, based on quantum error detection, for quantum key distribution (QKD) systems that run over quantum repeaters with encoding. In such repeaters, quantum error-correction techniques are used for entanglement distillation. By developing an analytical approach to study such quantum repeaters, we show that, in the context of QKD, it is often more efficient to use the error-detection, rather than the error-correction, capability of the underlying code to sift out cases where an error has been detected. We implement our technique for three-qubit repetition codes by modeling different sources of error in crucial components of the system. We then investigate in detail the impact of such imperfections on the secret key generation rate of the QKD system, and how one can use the information obtained during entanglement swapping and decoding stages to maximize the rate. For benchmarking purposes, we specify the maximum allowed error rates in different components of the setup below which positive key rates can be obtained.
We propose a postselection technique, based on quantum error detection, for quantum key distribution (QKD) systems that run over quantum repeaters with encoding. In such repeaters, quantum error-correction techniques are used for entanglement distillation. By developing an analytical approach to study such quantum repeaters, we show that, in the context of QKD, it is often more efficient to use the error-detection, rather than the error-correction, capability of the underlying code to sift out cases where an error has been detected. We implement our technique for three-qubit repetition codes by modeling different sources of error in crucial components of the system. We then investigate in detail the impact of such imperfections on the secret key generation rate of the QKD system, and how one can use the information obtained during entanglement swapping and decoding stages to maximize the rate. For benchmarking purposes, we specify the maximum allowed error rates in different components of the setup below which positive key rates can be obtained.
Efficiently distributing secret keys over long distances remains a critical challenge in the development of quantum networks. "First-generation" quantum repeater chains distribute entanglement by executing protocols composed of probabilistic entanglement generation, swapping and distillation operations. However, finding the protocol that maximizes the secret-key rate is difficult for two reasons. First, calculating the secret-key rate for a given protocol is non-trivial due to experimental imperfections and the probabilistic nature of the operations. Second, the protocol space rapidly grows with the number of nodes, and lacks any clear structure for efficient exploration. To address the first challenge, we build upon the efficient machinery developed by Li et al. [1] and we extend it, enabling numerical calculation of the secret-key rate for heterogeneous repeater chains with an arbitrary number of nodes. For navigating the large, unstructured space of repeater protocols, we implement a Bayesian optimization algorithm, which we find consistently returns the optimal result. Whenever comparisons are feasible, we validate its accuracy against results obtained through brute-force methods. Further, we use our framework to extract insight on how to maximize the efficiency of repeater protocols across varying node configurations and hardware conditions. Our results highlight the effectiveness of Bayesian optimization in exploring the potential of near-term quantum repeater chains.
An important contribution to the understanding of quantum key distribution has been the discovery of entangled states from which secret bits, but no maximally entangled states, can be extracted [Horodecki et al., Phys. Rev. Lett. 94, 200501 (2005)PRLTAO0031-900710.1103/PhysRevLett.94.200501]. The construction of those states was based on an intuition that the quantum mechanical phenomena of data hiding and privacy might be related. In this Letter we firmly connect these two phenomena and highlight three aspects of this result. First, we simplify the definition of the secret key rate. Second, we give a formula for the one-way distillable entanglement of certain private states. Third, we consider the problem of extending the distance of quantum key distribution with help of intermediate stations, a setting called the quantum key repeater. We show that for protocols that first distill private states, it is essentially optimal to use the standard quantum repeater protocol based on entanglement distillation and entanglement swapping.