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arXiv:1205.6933 · Physical Review A

Gaussian postselection and virtual noiseless amplification in continuous-variable quantum key distribution

Jaromı́r Fiurášek, Nicolas J. Cerf

Published Dec 26, 2012• 127 citations
arXivHTMLPDF

Abstract

Noiseless amplification or attenuation are two heralded filtering operations that enable amplifying or de-amplifying a quantum state of light with no added noise, at the cost of a small success probability. We show that inserting such noiseless operations in a transmission line improves the performances of continuous-variable quantum key distribution over this line. Remarkably, these noiseless operations do not need to be physically implemented but can simply be simulated in the classical data postprocessing stage. Hence, virtual noiseless amplification or attenuation amounts to performing a Gaussian postselection, which enhances the secure range or tolerable excess noise while keeping the benefits of Gaussian security proofs.

DOI
10.1103/physreva.86.060302

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Noiseless amplification or attenuation are two heralded filtering operations that enable amplifying or de-amplifying a quantum state of light with no added noise, at the cost of a small success probability. We show that inserting such noiseless operations in a transmission line improves the performances of continuous-variable quantum key distribution over this line. Remarkably, these noiseless operations do not need to be physically implemented but can simply be simulated in the classical data postprocessing stage. Hence, virtual noiseless amplification or attenuation amounts to performing a Gaussian postselection, which enhances the secure range or tolerable excess noise while keeping the benefits of Gaussian security proofs.
In a continuous-variable quantum key distribution (CV-QKD) protocol, which is based on heterodyne detection at the receiver, the application of a noiseless linear amplifier (NLA) on the received signal before the detection can be emulated by the postselection of the detection outcome. Such a postselection, which is also called a measurement-based NLA, requires a cutoff on the amplitude of the heterodyne-detection outcome to produce a normalizable filter function. Increasing the cutoff with respect to the received signals results in a more faithful emulation of the NLA and nearly Gaussian output statistics at the cost of discarding more data. While recent works have shown the benefits of postselection via an asymptotic security analysis, we undertake an investigation of such a postselection utilizing a composable security proof in the realistic finite-size regime, where this tradeoff is extremely relevant. We show that this form of postselection offers only a small fraction of the asymptotic improvement in the finite-size regime. This postselection can improve the secure range of a CV-QKD over lossy thermal channels if the finite block size is sufficiently large and the optimal value for the filter cutoff is typically in the non-Gaussian regime. The relatively modest improvement in the finite-size regime as compared to the asymptotic case highlights the need for new tools to prove the security of non-Gaussian cryptographic protocols. These results also represent a quantitative assessment of a measurement-based NLA with an entangled-state input in both the Gaussian and non-Gaussian regime.
Photon subtraction can enhance the performance of continuous-variable quantum key distribution (CV QKD). However, the enhancement effect will be reduced by the imperfections of practical devices, especially the limited efficiency of a single-photon detector. In this paper, we propose a non-Gaussian postselection method to emulate the photon substraction used in coherent-state CV QKD protocols. The virtual photon subtraction not only can avoid the complexity and imperfections of a practical photon-subtraction operation, which extends the secure transmission distance as the ideal case does, but also can be adjusted flexibly according to the channel parameters to optimize the performance. Furthermore, our preliminary tests on the information reconciliation suggest that in the low signal-to-noise ratio regime, the performance of reconciliating the postselected non-Gaussian data is better than that of the Gaussian data, which implies the feasibility of implementing this method practically.
Abstract We propose a method to improve transmission distance of continuous-variable quantum key distribution via a heralded hybrid linear amplifier (HLA), involving an ideal deterministic linear amplifier and a probabilistic noiseless linear amplifier. The phase-insensitive amplification of quantum light states may experience a degradation of signal-to-noise ratio (SNR) of 3dB which can be surmounted by the probabilistic amplification process. The new hybrid amplifier allows one to tune between the regimes of high gain or high noise reduction and control the trade-off between the amplification coefficient and a finite heralding probability flexibly. Our equipment could clone unknown coherent states with fidelity surpassing the no-cloning limit by abandoning determinism and using probabilistic methods. The setup has a better compatibility with continuous variable protocols as it only uses Gaussian elements and a post-selection algorithm. The hybrid linear amplifier enhances the SNR and hence the transmission distance of the CVQKD protocol is extended. Both the asymptotic secret key rate and the secret key rate in a finite-size scenario are analyzed in this paper.
Quantum catalysis is a feasible approach to increase the performance of continuous-variable quantum key distribution (CVQKD), involving the special zero-photon catalysis (ZPC) operation. However, in the practical point of view, the improvement effect of this operation will be limited by the imperfection of the photon detector. In this paper, we show that the ZPC operation at the sender can be simulated by a post-selection method without implementing it in practical devices. While performing this virtual version of ZPC in CVQKD, we can not only reach the ideal case of its practical implementation with minimal hardware requirement, but also keep the benefit of Gaussian security proofs. Based on Gaussian modulated coherent state protocols with achievable parameters, we enhance the security of the proposed scheme from the asymptotical case to the finite-size scenario and composable framework. Simulation results show that similar to the asymptotical case, both the maximal transmission distance and the tolerable excess noise of virtual ZPC-involved CVQKD outperform the original scheme and the scheme using virtual photon subtraction while considering finite-size effect and composable security. In addition, the virtual ZPC-involved CVQKD can tolerate a higher imperfection of the detector, enabling its practical implementation of the CVQKD system with state-of-the-art technology.
We show that the maximum transmission distance of continuous-variable quantum key distribution in presence of a Gaussian noisy lossy channel can be arbitrarily increased using a heralded noiseless linear amplifier. We explicitly consider a protocol using amplitude- and phase-modulated coherent states with reverse reconciliation. Assuming that the secret key rate drops to zero for a line transmittance Tlim{T}_{\mathrm{lim}}Tlim​, we find that a noiseless amplifier with amplitude gain ggg can improve this value to Tlim/g2{T}_{\mathrm{lim}}/{g}^{2}Tlim​/g2, corresponding to an increase in distance proportional to logg\mathrm{log}glogg.
We present a protocol based on continuous-variable quantum teleportation and Gaussian postselection that can be used to correct errors introduced by a lossy channel. We first show that the global transformation enacted by the protocol is equivalent to an effective system composed of a noiseless amplification (or attenuation), and an effective quantum channel, which can in theory have no loss and an amount of thermal noise arbitrarily small, hence tending to an identity channel. An application of our protocol is the probabilistic purification of quantum non-Gaussian states using only Gaussian operations.