Repeaters for continuous-variable quantum communication

Fabian Furrer and William J. Munro

Phys. Rev. A 98, 032335 – Published 28 September 2018

Abstract

Optical telecommunication is at the heart of today's internet and is currently enabled by the transmission of intense optical signals between remote locations. As we look to the future of telecommunication, quantum mechanics promise new ways to be able to transmit and process that information. Demonstrations of quantum key distribution and quantum teleportation using multiphoton states have been performed, but only over ranges limited to a few hundred kilometers. To go beyond this, we need repeaters that are compatible with these quantum multiphoton continuous-variable pulses. Here we present a design for continuous-variable quantum repeaters that can distribute entangled and pure two-mode squeezed states over arbitrarily long distances with a success probability that scales only polynomially with distance. The proposed quantum repeater is composed from several basic known building blocks such as non-Gaussian operations for entanglement distillation and an iterative Gaussification protocol (for retaining the Gaussian character of the final state), but complemented with a heralded non-Gaussian entanglement swapping protocol, which allows us to avoid extensive iterations of quantum Gaussification. We characterize the performance of this scheme in terms of key rates for quantum key distribution and show a secure key can be generated over thousands of kilometers.

Received 16 January 2018
Revised 13 August 2018

DOI:https://doi.org/10.1103/PhysRevA.98.032335

Physics Subject Headings (PhySH)

Quantum channels Quantum communication Quantum protocols Quantum repeaters

Quantum Information, Science & TechnologyGeneral Physics

Authors & Affiliations

Fabian Furrer^1,* and William J. Munro^1,2,†

¹NTT Basic Research Laboratories & Research Center for Theoretical Quantum Physics, NTT Corporation, 3-1 Morinosato-Wakamiya, Atsugi, Kanagawa 243-0198, Japan
²National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan

^*furrer@eve.phys.s.u-tokyo.ac.jp
^†bill.munro@me.com

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Vol. 98, Iss. 3 — September 2018

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Figure 1
Schematic illustration of a traditional first generation quantum repeater scheme [15] between Alice $A$ and Bob $B$ , where that communication distance is divided into shorter distances by adding repeater nodes (stations) in between (three nodes in this case). The repeater scheme begins by establishing entanglement between the adjacent nodes. This entanglement is stored in quantum memories present within each repeater node until we require its use. Then, entanglement distillation (Distillation 1) is performed to distill a smaller number of highly purified and entangled states (number of copies is illustrated by the thickness). Subsequently, entanglement swapping is performed until one established the required entangled state between Alice and Bob. Distillation 2 and Distillation 3 might be necessary to compensate the loss of purity and entanglement during swapping. Instead of using distillation protocols one can utilize a quantum error correction code for the distribution of the entanglement and for the errors induced by the swapping operation.
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Figure 2
Schematic illustration of the various CV repeater components. In (a) the two-mode squeezed vacuum entanglement source $| χ_{λ} 〉 = \sqrt{1 - λ^{2}} \sum_{k = 0}^{\infty} λ^{k} | k, k 〉$ is depicted. In the important Gaussification protocol $s$ is shown which transforms non-Gaussian states back to Gaussian ones. On both nodes, the two modes are mixed with a balanced 50:50 BS. The operation succeeds conditioned on vacuum port detection (a Gaussian filtering operation) at both nodes. This procedure can be iteratively executed with the obtained state as the new input state until one is close enough to the desired Gaussian state. By choosing different filtering operations [31] different Gaussification protocols are obtained. In (c) the symmetric PR distillation protocol is depicted where both modes are mixed with a single photon using a BS with transmissivity $η$ . The operation is successful if a single photon is measured at the respective outcome of the BS. Panel (d) shows the operations $S$ and $D (q)$ used in the purifying distillation protocol. Here $S$ is a photon replacement with a balanced BS, while $D (q)$ is a probabilistic operation consisting of a Mach-Zehnder interferometer with $S$ placed in each path and conditioning on $| ξ (q) 〉$ on one of the output ports. Next (e) illustrates the purifying distillation operation in which $D (q)$ is applied to two initial states $ρ$ . Heralding is applied on the successful application of $D (q)$ on both sides. The usual Gaussian entanglement swapping protocol $S$ is shown in (f), where the two modes are mixed with a balanced BS and homodyne detection (HD) is used on both output ports to measure $X$ and $P$ . A correction operation in the form of a displacement depending on the measurement outcomes is applied on the left and right mode. Finally (g) shows a NG entanglement swapping protocol. The operation is heralded upon success of the operation $D (q)$ and measuring the state $| ξ (- q) 〉$ .
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Figure 3
Plot of the Gauss parameter for the transmissivity of the output state $τ_{\infty} (\tilde{ρ})$ and the corresponding success probability $p_{succ}$ depending on the Gauss parameter for the transmissivity of the input state $τ_{\infty} (ρ)$ . In (a),(b) we illustrate the purifying distillation situation where our projective measurement $| ξ (q) 〉$ has the free parameter $q$ depending on the choice of state on which we wish to condition. We can tune $q$ such that the Gaussian parameter for the input squeezing $λ_{\infty}^{in} = λ_{\infty} (ρ)$ and output squeezing $λ_{\infty}^{out} = λ_{\infty} (\tilde{ρ})$ are equal but $ɛ (\tilde{ρ}) < ɛ (ρ)$ , that is, $\tilde{ρ}$ is more pure than $ρ$ . In (c),(d) we consider the non-Gaussian entanglement swapping situation. The Gaussian parameters for the input squeezing $λ_{\infty}^{in} = λ_{\infty} (ρ)$ and output squeezing $λ_{\infty}^{out} = λ_{\infty} (\tilde{ρ})$ are shown. In order to illustrate the decrease of the Gauss output transmissivity parameter (c), we also plotted the input transmissivity (dotted line). We see that there exists a minimal initial transmissivity for the protocol to conserve the desired structure of the input state (i.e., the Gaussification protocol converges). The success probability increases and the threshold for the initial transmissivity gets reduced for larger squeezing $λ_{\infty}^{in} = λ_{\infty}^{out}$ . The threshold for the initial transmissivity is also reduced if the output squeezing $λ_{\infty}^{out}$ is larger than initial squeezing $λ_{\infty}^{in}$ , but the success probability decreases.
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Figure 4
(a) Plot showing the characteristic behavior of the normalized key rate per pulse $r_{Q R} (σ)$ obtained with a CV QR based on Gaussian swapping. The rates are (over)estimated and computed under the simplification that three iterations of the Gaussification protocol would lead to a perfect Gaussian state. (b) Plot showing the performance of the CV QR based on NG entanglement swapping. Shown is the comparison of the estimate key rate or pulse for the CV QR (straight) with the key rate for direct transmission (dashed) using a CV reverse reconciliation scheme [47] and a tight upper direct transmission bound (dotted) [48]. We see that the crossing point is slightly above 350 km (500 kn) for the Gaussian swapping (NG swapping) and the gap increases fast with the distance. (c) The lower plot compares the key rate or pulse of the CV QR without (straight) and with (dashed) additional purifying entanglement distillation. We observe that, while for low distances the key rate without additional purifying distance is significantly larger, the gap closes for large distances above $10 000$ km. The dots correspond to actual simulation points and the curves are interpolations.
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