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Bulletin of the Russian Academy of Sciences Physics

Performance of Quantum Repeaters Using Multimode Schrödinger Cat States

R. Goncharov, Alexei D. Kiselev, V. I. Egorov

Published Jun 1, 2024• 0 citations
Publisher

Abstract

Abstract We analyze the performance of quantum repeaters where the protocols of entanglement generation and swapping are based on multimode optical Schrödinger cat states produced using electro-optic modulation of single-mode optical cats. We employ both the mean-only approximation and exact results to calculate the distance dependencies of the mean waiting time and the repeater rate depending on the number of elementary links. The results are compared with the rates of quantum repeater schemes based on photon pairs generated via spontaneous parametric down-conversion.

DOI
10.1134/s1062873824706809

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Abstract We analyze the performance of quantum repeaters where the protocols of entanglement generation and swapping are based on multimode optical Schrödinger cat states produced using electro-optic modulation of single-mode optical cats. We employ both the mean-only approximation and exact results to calculate the distance dependencies of the mean waiting time and the repeater rate depending on the number of elementary links. The results are compared with the rates of quantum repeater schemes based on photon pairs generated via spontaneous parametric down-conversion.
We present a scheme of quantum repeater that uses entangled multimode coherent states that are obtained by electro-optic modulation of symmetric and antisymmetric Schr\"odinger cat states. Part of the generated entangled frequency modes is sent to a symmetric beam splitter at the central node, while the remaining modes are stored locally in quantum memories. The entangled coherent states between remote quantum memories are conditionally prepared by photon-counting measurements at the output channels of the beam splitter. We study how the effects of decoherence in the quantum channel affect statistics of photocounts and, for the heralding outcomes determined by the parity of photocounts, evaluate the probability of success and the fidelity of the prepared entanglement depending on the symmetry of input cat states. It is demonstrated that the generated entanglement can be employed for teleportation of the phase information from the modulated states that are utilized in quantum key distributions with subcarrier wave encoding.
Entangled coherent states can be prepared remotely by subtracting nonlocally a single photon from two quantum superpositions of coherent states, the so-called “Schrödinger’s cat” state. Such entanglement can further be distributed over longer distances by successive entanglement swapping operations using linear optics and photon-number resolving detectors. The aim of this paper is to evaluate the performance of this approach to quantum repeaters for long-distance quantum communications. Despite many attractive features at first sight, we show that, when using state-of-the-art photon counters and quantum memories, they do not achieve higher entanglement generation rates than repeaters based on single-photon entanglement. We discuss potential developments, which may take better advantage of the richness of entanglement based on continuous variables, including in particular efficient parity measurements.
A numerical model of a quantum repeater operating with Schrödinger cat states is constructed. The model describes the performance of such a system in the presence of decoherence effects, namely, noise in the quantum channel and the efficiency of the photon-number-resolving detector. In the framework of the numerical model, a theoretical analysis of the system functioning is carried out for the elementary link by calculating its performance characteristics. Namely, we calculate photodetector click probabilities and fidelity for various sets of decoherence parameters. These estimates are necessary in the context of further experimental research at the junction with other branches of quantum communications, so that to use the entanglement distribution when it comes to operating quantum teleportation and quantum key distribution protocols based on entanglement. The model will be developed further as a versatile drag-and-drop software simulating the full-fledged entanglement swapping protocol operation.
Long distance entanglement generation at a high rate is a major quantum technological goal yet to be fully realized, with the promise of many interesting applications, such as secure quantum computing on remote servers and quantum cryptography. One possible implementation is using a variant of the DLCZ-scheme by combining atomic-ensemble memories and linear optics with spontaneous parametric down conversion (SPDC) sources. As we edge closer to the realization of such a technology, the complete details of the underlying components become crucial. In this paper we consider the impact of the multimode emission from the SPDC source on quantum repeaters based on the DLCZ-scheme. We consider two cases, driving the SPDC using short Gaussian pulses and continuously. For pulsed driving, we find that the use of very narrow laser pulses to drive SPDC source is crucial to obtain high fidelity end-to-end entangled states but this puts demands on the peak intensity. By introducing a maximally allowed laser intensity, we find optimal pulse widths for each swap depth. For continuous driving, we find the temporal acceptance window of clicks relative to the heralding time to be a crucial parameter, and we can similarly optimize the acceptance window for each swap depth. For both cases, we thus identify optimal parameters given experimental limitations and aims. We have thus provided helpful knowledge towards the realization of long distance entanglement generation using the DLCZ-scheme.
The standard approach to realize a quantum repeater relies upon probabilistic but heralded entangled state manipulations and the storage of quantum states while waiting for successful events. In the literature on this class of repeaters, calculating repeater rates has typically depended on approximations assuming sufficiently small probabilities. Here we propose an exact and systematic approach including an algorithm based on Markov chain theory to compute the average waiting time (and hence the transmission rates) of quantum repeaters with arbitrary numbers of links. For up to four repeater segments, we explicitly give the exact rate formulas for arbitrary entanglement swapping probabilities. Starting with three segments, we explore schemes with arbitrary (not only doubling) and dynamical (not only predetermined) connections. The effect of finite memory times is also considered and the relative influence of the classical communication (of heralded signals) is shown to grow significantly for larger probabilities. Conversely, we demonstrate that for small swapping probabilities the statistical behavior of the waiting time in a quantum repeater cannot be characterized by its average value alone and additional statistical quantifiers are needed. For large repeater systems, we propose a recursive approach based on exactly but still efficiently computable waiting times of sufficiently small subrepeaters. This approach leads to better lower bounds on repeater rates compared to existing schemes.
Satellite optical links may be able to extend quantum communication range beyond that which is possible through optical fibres which are constrained by exponential loss scaling. Quantum repeaters have been proposed to mitigate high channel losses in both fibre and satellite links. Although satellite quantum repeaters could improve entanglement distribution efficiency, the attempt rate is constrained by classical communication latency in the entanglement swapping step. Direct entanglement distribution does not suffer this latency and can therefore compensate for high dual-downlink losses by increasing the source rate. A systematic comparative analysis of repeater-assisted and direct entanglement distribution rates using a satellite is absent from the literature. In addition to entanglement distribution rates, a detailed analysis of the fidelity of entanglement distributed by repeater-assisted satellite links has yet to be performed. Here, we develop an in-depth framework for describing a space quantum link consisting of a single satellite and two ground stations. We compare the entanglement distribution rates of direct and repeater-assisted satellite entanglement distribution, taking into account variations in overpass geometry and quantum memory capacity. The long-term entanglement distribution performance to different pairs of major cities is also considered, and the altitudinal dependence is explored. Furthermore, we develop a Monte Carlo model of the repeater satellite to explore the effect that different memory management policies and capacities have on the waiting time statistics of memory qubits. From these statistics we calculate the fidelity distributions of the entangled pairs established between the ground stations. These results will inform mission design for future space-borne quantum repeater nodes, as well as requirements on space-based memory platforms.