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arXiv:2203.10318 · Physical Review Research

Exact rate analysis for quantum repeaters with imperfect memories and entanglement swapping as soon as possible

Lars Kamin, Evgeny V Shchukin, Frank Schmidt, Peter van Loock

Published May 10, 2023• 43 citations
arXivHTMLPDF

Abstract

We present an exact rate analysis for a secret key that can be shared among two parties employing a linear quantum repeater chain. One of our main motivations is to address the question whether simply placing quantum memories along a quantum communication channel can be beneficial in a realistic setting. The underlying model assumes deterministic entanglement swapping of single-spin quantum memories and it excludes probabilistic entanglement distillation, and thus two-way classical communication, on higher nesting levels. Within this framework, we identify the essential properties of any optimal repeater scheme: entanglement distribution in parallel, entanglement swapping as soon and parallel quantum storage as little as possible. While these features are obvious or trivial for the simplest repeater with one middle station, for more stations they cannot always be combined. We propose an optimal scheme including channel loss and memory dephasing, proving its optimality for the case of two stations and conjecturing it for the general case. In an even more realistic setting, we consider additional tools and parameters such as memory cutoffs, multiplexing, initial state and swapping gate fidelities, and finite link coupling efficiencies in order to identify potential regimes in memory-assisted quantum key distribution beyond one middle station that exceed the rates of the smallest quantum repeaters as well as those obtainable in all-optical schemes unassisted by stationary memory qubits and two-way classical communication. Our analytical treatment enables us to determine simultaneous trade-offs between various parameters, their scaling, and their influence on the performance ordering among different types of protocols, comparing two-photon interference after dual-rail qubit transmission with one-photon interference of single-rail qubits or, similarly, optical interference of coherent states. We find that for experimental parameter values that are highly demanding but not impossible (up to 10 s coherence time, about 80% link coupling, and state or gate infidelities in the regime of 1%--2%), one secret bit can be shared per second at a total channel loss budget of 157.6 dB, i.e. a total distance of 800 km for a fiber attenuation length of 22 km with repeater stations placed at every 100 km---a clear improvement over realistic twin-field or, much more pronouncedly, ideal point-to-point quantum key distribution at GHz clock rates.

DOI
10.1103/physrevresearch.5.023086

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We present an exact rate analysis for a secret key that can be shared among two parties employing a linear quantum repeater chain. One of our main motivations is to address the question whether simply placing quantum memories along a quantum communication channel can be beneficial in a realistic setting. The underlying model assumes deterministic entanglement swapping of single-spin quantum memories and it excludes probabilistic entanglement distillation, and thus two-way classical communication, on higher nesting levels. Within this framework, we identify the essential properties of any optimal repeater scheme: entanglement distribution in parallel, entanglement swapping as soon and parallel quantum storage as little as possible. While these features are obvious or trivial for the simplest repeater with one middle station, for more stations they cannot always be combined. We propose an optimal scheme including channel loss and memory dephasing, proving its optimality for the case of two stations and conjecturing it for the general case. In an even more realistic setting, we consider additional tools and parameters such as memory cutoffs, multiplexing, initial state and swapping gate fidelities, and finite link coupling efficiencies in order to identify potential regimes in memory-assisted quantum key distribution beyond one middle station that exceed the rates of the smallest quantum repeaters as well as those obtainable in all-optical schemes unassisted by stationary memory qubits and two-way classical communication. Our analytical treatment enables us to determine simultaneous trade-offs between various parameters, their scaling, and their influence on the performance ordering among different types of protocols, comparing two-photon interference after dual-rail qubit transmission with one-photon interference of single-rail qubits or, similarly, optical interference of coherent states. We find that for experimental parameter values that are highly demanding but not impossible (up to 10 s coherence time, about 80% link coupling, and state or gate infidelities in the regime of 1%--2%), one secret bit can be shared per second at a total channel loss budget of 157.6 dB, i.e. a total distance of 800 km for a fiber attenuation length of 22 km with repeater stations placed at every 100 km---a clear improvement over realistic twin-field or, much more pronouncedly, ideal point-to-point quantum key distribution at GHz clock rates.
We present a quantum repeater protocol for distributing entanglement over long distances, where a dedicated communication stage enables trial rates not limited by the travel time between repeater nodes. To accomplish this, each node contains several qubits that can couple to one single-photon emitter. Photons from the emitters in neighboring nodes generate heralded entanglement between qubits in these nodes. The protocol leaves the emitters disentangled from the rest of the system immediately after emitting the photons, thus allowing them to be reused to entangle other qubits without waiting for the repeater link round-trip time. This time multiplexing increases the protocol trial rate by up to an order of magnitude. The protocol is then combined with conventional deterministic entanglement swapping and heralded entanglement purification to extend the entanglement distance and reduce the entanglement error, respectively. We perform a complete protocol analysis by considering all relevant error sources, such as initialization, two-qubit gate, and qubit measurement errors, as well as the exponential decoherence of the qubits with time. The latter is particularly important since we analyze the protocol performance for a broad range of experimental parameters and obtain secret key rates ranging from 1 to 1000 Hz at a distance of 1000 km. Our results suggest that it is important to reach a qubit memory coherence time of around 1 s, and two-qubit gate and measurement errors on the order of 10 − 3 to obtain reasonable secret key rates over distances longer than achievable with direct transmission. While this work focuses on optimizing secret key rates, the protocol can also be used for Einstein-Podolsky-Rosen pair generation and is thus also relevant for, e.g., distributed quantum computing.
We propose a scheme that generalizes the loss scaling properties of twin-field or phase-matching quantum key distribution (QKD) related to a channel of transmission ${\ensuremath{\eta}}_{\mathrm{total}}$ from $\sqrt{{\ensuremath{\eta}}_{\mathrm{total}}}$ to $\sqrt[2n]{{\ensuremath{\eta}}_{\mathrm{total}}}$ by employing $n\ensuremath{-}1$ memory stations with spin qubits and nnn beam-splitter stations including optical detectors. Our scheme's resource states are similar to the coherent-state-based light-matter entangled states of a previous hybrid quantum repeater, but unlike the latter our scheme avoids the necessity of employing $2n\ensuremath{-}1$ memory stations and writing the transmitted optical states into the matter memory qubits. The full scaling advantage of this memory-assisted phase-matching QKD (MA-PM QKD) is obtainable with threshold detectors in a scenario with only channel loss. We mainly focus on the obtainable secret-key rates per channel use for up to n=4n=4n=4 including memory dephasing and for n=2n=2n=2 (i.e., $\sqrt[4]{{\ensuremath{\eta}}_{\mathrm{total}}}$-MA-PM QKD assisted by a single memory station) for error models including dark counts, memory dephasing and depolarization, and phase mismatch. By combining the twin-field concept of interfering phase-sensitive optical states with that of storing quantum states up to a cutoff memory time, distances well beyond 700 km with rates well above ${\ensuremath{\eta}}_{\mathrm{total}}$ can be reached for realistic, high-quality quantum memories (up to 1-s coherence time) and modest detector efficiencies. Similarly, the standard single-node quantum repeater, scaling as $\sqrt{{\ensuremath{\eta}}_{\mathrm{total}}}$, can be beaten when approaching perfect detectors and exceeding spin coherence times of 5 s; beating ideal twin-field QKD requires 1 s. As for further experimental simplifications, our treatment includes the notion of weak nonlinearities for the light-matter states, a discussion on the possibility of replacing the threshold by homodyne detectors, and a comparison between sequential and parallel entanglement distributions.
Abstract Efficient long-distance entanglement distribution is a key requirement for a global quantum internet. Quantum repeaters overcome photon loss in optical fibers, offering a route to long-distance entanglement distribution. Here, we study heralded quantum repeater architectures that are based on two-photon interference and integrate local or remote swapping. We derive an analytical expression for the entanglement distribution time as a function of transmission distance for both heralded local-swapping and remote-swapping schemes. This expression provides physical insight into how repeaters overcome channel loss in long-haul fibers. Considering a realistic quantum memory, we present theoretical curves showing the effect of finite memory lifetime on the entanglement distribution time. By comparing optimal distribution times over distances up to thousands of kilometers, we determine the optimal nesting architectures for the single-repeater channel in each scheme and compare the optimal distribution rates and entanglement fidelities of these schemes at various distances. Finally, by examining the limiting cases of zero channel loss, zero device loss, and lossless repeater systems, we find fundamental performance bounds for quantum repeaters. These results provide practical design principles for scalable quantum repeater networks and pave the way toward continent-scale quantum communication.
We address the challenge of incorporating encoded quantum memories into an exact secret key rate analysis for small and intermediate-scale quantum repeaters. To this end, we introduce the check matrix model and quantify the resilience of stabilizer codes of up to eleven qubits against Pauli noise, obtaining analytical expressions for effective logical error probabilities. Generally, we find that the five-qubit and Steane codes either outperform more complex, larger codes in the experimentally relevant parameter regimes or have a lower resource overhead. Subsequently, we apply our results to calculate lower bounds on the asymptotic secret key rate in memory-corrected quantum repeaters when using the five-qubit or Steane codes on the memory qubits. The five-qubit code drastically increases the effective memory coherence time, reducing a phase flip probability of 1 % to 0.001 % when employing an error syndrome identification adapted to the quantum noise channel. Furthermore, it mitigates the impact of faulty Bell state measurements and imperfect state preparation, lowering the minimally required depolarization parameter for nonzero secret key rates in an eight-segment repeater from 98.4 % to 96.4 % . As a result, the memory-corrected quantum repeater can often generate secret keys in experimental parameter regimes where the unencoded repeater fails to produce a secret key. In an eight-segment repeater, one can even achieve nonvanishing secret key rates up to distances of 2000 km for memory coherence times of t c = 10 s or less using multiplexing. Assuming a zero-distance link-coupling efficiency p 0 = 0.7 , a depolarization parameter μ = 0.99 , t c = 10 s, and an 800 km total repeater length, we obtain a secret key rate of 4.85 Hz, beating both the unencoded repeater that provides 1.25 Hz and ideal twin-field quantum key distribution with 0.71 Hz at GHz clock rates.
We evaluate the performance of two-way quantum repeater chains with sequential entanglement swapping. Within the analysis we consider memory decoherence, gate imperfections, and imperfect link-level entanglement generation. Our main results include closed-form expressions for the average entanglement fidelity of the generated end-to-end entangled states. We generalize previous findings for the one-shot fidelity analysis and study the case where repeater chains serve end-to-end requests continuously. We provide solutions to the continuous request scenario by combining results from quantum information theory and queuing theory. Finally, we apply the formulas obtained to analyze the impacts of hardware parameters, i.e., coherence times and gate fidelity, and distance on the entanglement fidelity and secret key rate of homogeneous quantum repeater chains.
We present a hybrid quantum repeater protocol for the long-distance distribution of atomic entangled states beyond qubits. In our scheme, imperfect noisy entangled pairs of two qudits, i.e., two discrete-variable ddd-level systems, each of, in principle, arbitrary dimension ddd, are initially shared between the intermediate stations of the channel. This is achieved via local, sufficiently strong light-matter interactions, involving optical coherent states and their transmission after these interactions, and optical measurements on the transmitted field modes, especially efficient continuous-variable homodyne detections (``hybrid'' here refers to the simultaneous exploitation of discrete- and continuous-variable degrees of freedom for the local processing and storage of entangled states as well as their nonlocal distribution, respectively) and unambiguous state discrimination. For qutrits we quantify the light-matter entanglement that can be effectively shared through an elementary lossy channel, and for a repeater spacing of up to 10 km we show that the realistic (lossy) qutrit entanglement is even larger than any ideal (loss-free) qubit entanglement. After including qudit entanglement purification and swapping procedures, we calculate the long-distance entangled-pair distribution rates and the final entangled-state fidelities for total communication distances of up to 1280 km. For example, employing unambiguous state discrimination, with three rounds of purification, entangled qudit pairs of near-unit fidelity can be distributed over 1280 km at an ideal maximal rate (assuming perfect gate operations) on the order of 100 Hz.