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He Yu-Qian, Mao Yun, Zhong Hai, Huang Duang, Guo Ying. Hybrid linear amplifier-involved detection for continuous variable quantum key distribution with thermal states. Chinese Physics B, 2020, 29(5): 050309  
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Hybrid linear amplifier-involved detection for continuous variable quantum key distribution with thermal states
He Yu-Qian1, Mao Yun1, †, Zhong Hai1, Huang Duang2, Guo Ying1
School of Automation, Central South University, Changsha 410083, China
School of Computer Science and Engineering, Central South University, Changsha 410083, China

 

† Corresponding author. E-mail: mail: maocsu@sina.com

Project supported by the National Natural Science Foundation of China (Grant Nos. 61572529 and 61871407).

Abstract

Continuous-variable quantum key distribution (CVQKD) can be integrated with thermal states for short-distance wireless quantum communications. However, its performance is usually restricted with the practical thermal noise. We propose a method to improve the security threshold of thermal-state (TS) CVQKD by employing a heralded hybrid linear amplifier (HLA) at the receiver. We find the effect of thermal noise on the HLA-involved scheme in near-and-mid infrared band or terahertz band for direct and reverse reconciliation. Numerical simulations show that the HLA-involved scheme can compensate for the detriment of thermal noise and hence increase the security threshold of TS-CVQKD. In near-and-mid infrared band, security threshold can be extended by 2.1 dB in channel loss for direct reconciliation and 1.6 dB for reverse reconciliation, whereas in terahertz band, security threshold can be slightly enhanced for the gain parameter less than 1 due to the rise in thermal noise.

PACS: ;03.67.-a;;03.67.Dd;;03.67.Hk;
Keyword:;continuous-variable quantum key distribution;;thermal state;;hybrid linear amplifier;
1. Introduction

Continuous-variable quantum key distribution (CVQKD) allows two distant parties, Alice and Bob, to share random secret keys over an untrusted environment controlled by an eavesdropper, Eve.[1–4] Compared with discrete-variable quantum key distribution (DVQKD),[5–9] CVQKD can be applied on off-the-shelf optical fiber system with relative higher secret key rate, avoiding the imperfection for using single photon counting.[10,11] In a typical optical fiber system,[12–14] Alice encodes Gaussian-distributed random numbers on weak coherent states and then sends them to Bob through an optical fiber. However, such infrastructure is built on high power laser, may well be impractical for a plethora of attractive low-power ultra-high-speed quantum communications, such as WiFi, Bluetooth and chip-to-chip communications.[15] For example, customers in a supermarket may use mobile phone to scan the two-dimensional code wirelessly, in order to exchange secret keys with cash registers.

Unlike an optical fiber system that usually works in near infrared band,[16] modern wireless communication works in microwave and far infrared band.[17,18] In the above-mentioned frequency band, Alice may send thermal states to Bob instead of traditional coherent states because environment thermal noise is extremely high. Therefore, it is inevitable to deploy thermal-state CVQKD (TS-CVQKD) if we want to extend the application of quantum communication to a common wireless system. On the other hand, as the transmission rate of CVQKD has been growing dramatically over the years,[20,19] preparing quantum state precisely becomes challenging. Using TS-CVQKD will loosen the stringent requirements for optical amplitude and phase modulators since it needs no exact Guassian modulation.[21] Although TS-CVQKD is suitable to wireless communication and low cost to implement, thermal noise would dramatically reduce the actual performance of TS-CVQKD.[22,23] Thus, how to improve the performance of TS-CVQKD is a significance issue.

Currently, an amplifier has been used for combating the channel loss and other decoherence, resulting in the performance improvement of CVQKD.[24–28] The noiseless linear amplifier (NLA) is the most typical one.[24,29,30] It can increase the performance in a nondeterministic manner theoretically.[31,32] However, the actual success rate of a physical realization is far below the theoretical prediction owing to experimental limitation.[33] Alternative to the physical counterpart, the measurement-based noiseless linear amplifier (MB-NLA) is a virtualization version of NLA by simple data processing.[25] The MB-NLA offers a fully tunable cutoffs between probability and fidelity and hence more flexible. At the same time, a deterministic linear amplifier (DLA) can compensate for the imperfection of detector at receiver’s side, leading to an improvement of maximal transmission distance.[26] However, the DLA cannot increase the signal-to-noise ratio (SNR) owing to the bosonic nature of photons.[34] Interestingly, a combination of DLA and MB-NLA, called the hybrid linear amplifier (HLA), was proposed and realized in experiments.[27] It inherits advantages of both MB-NLA and DLA and hence sets effective gain ge free from success rate, which means one could fix ge when the probability changes. It can be used to enhance the secret key rate and the transmission distance of the traditional CVQKD system.[35]

In this paper, we consider the deploy of HLA on Bob’s side for the performance improvement of TS-CVQKD in near-and-mid infrared band or terahertz band. We find that in near-and-mid infrared band, the HLA-involved scheme can bring performance improvement of TS-CVQKD in direct reconciliation (DR), although the performance of the HLA-involved TS-CVQKD in reverse reconciliation (RR) is vulnerable to thermal noise. As the thermal radiation increases, the improved channel loss will decrease rapidly. In terahertz band, we find that DR is vastly superior to RR, where gDLA plays a more important role than gNLA. In addition, we find that the optimized gNLA will lie in the area gNLA < 1 if we want to extend security threshold.

The rest of this paper is organized as follows. In Section 2, we propose the HLA-involved scheme for performance improvement of the TS-CVQKD system and demonstrate the interaction of gDLA and gNLA. In Section 3, we derive the secret key rate and security threshold in near-and-mid infrared band and terahertz band. Meanwhile, we achieve the suitable parameters gDLA and gNLA for maximizing security threshold of channel loss. Finally, we conclude in Section 4.

2. The HLA-involved TS-CVQKD
2.1. The scheme design of TS-CVQKD

The prepare and measure (PM) scheme of TS-CVQKD is shown in Fig. 1. Alice displaces thermal states with a bi-dimensional Gaussian distribution and sends signal model B0 to Bob. The signal mode B0 can be given by

where xm without a hat is the classical variable for the coded information and with a hat is an operator that denotes uncertain quantum noise. The vectorial operator involves both quadratures and . The entire variance of signal state can be given by

where VM is modulated variance obeying Gaussian distribution and VS is the shot noise variance, as shown in Fig. 1(c). We assume that VS is equal to 1 for unit vacuum shot-noise (SNU), as the yellow circular region shown in Fig. 1(c). As frequency band moves down from near infrared regime to microwave regime, however, the shot noise VS will rise up and adapt to the environment noise. The environment noise over different frequencies is given by

where is the average photon number for any wavelength given by[16]

where h is Planck’s constant, f is the frequency of quantum signals, kB is Boltzmann’s constant, and TK is the temperature. Even if without environment thermal fluctuations, thermal noise will still exist due to the imperfect modulation. Consequently, the shot noise VS becomes VS = 1 + μ, where μ > 0 is for the thermal state case and μ = 0 is for the coherent state case.

Fig. 1. (a) Schematic diagram of the standrad TS-CVQKD. (b)Schematic diagram of the HLA-involved TS-CVQKD. (c) Phase space representation of thermal state. As the frequency goes down from infrared regime to microwave regime, thermal noise would be enhanced. (d) Schematic of HLA achieved with a feed-forward loop. EOM, electro-optic modulator.

Subsequently, signal mode B0 goes through a noisy and lossy channel controlled by Eve, as depicted in Fig. 1. With the existing environment thermal noise, Eve has to first cool down her thermal mode E0 so as to produce pure vacuum mode and then add known noise to the pure vacuum mode by modulating E0. This added known noise should match the level of environment noise N0 in order to cover her tracks.[16] Then, Eve performs an entangling cloner attack[36–38] by a beam splitter with transmission T and interacts one of her ancilla modes E1 with B0, resulting in B1. E3 and E2 are stored in a quantum memory.

Finally, Bob performs homodyne detection to recover information. We model Bob’s detection efficiency by a beam splitter with transmission efficiency η and detector electronics with vel.[39] After collecting correlated strings from Alice, the postprocessing begins, which can be either DR or RR. For DR, the previous experiment result shows that TS-CVQKD is robust against thermal noise.[16] However, a channel loss of no more than 1.5 dB is possible,[23] which is not far enough for metropolitan transport networks. For RR, the performance would become terrible since the possible channel loss will rapidly decline to zero as thermal noise rises up.[23]

2.2. The HLA-involved scheme for enhancing TS-CVQKD

In what follows, we propose an approach for improving TS-CVQKD by inserting an HLA before Bob’s detector, as shown in Fig. 1(b). Figure1(d) shows the practical implementation of HLA. When the signal mode enters the HLA, it will first pass through a beam splitter with transmission constant Tg given by[27]

The reflected mode is measured and selected via the MB-NLA, which virtualizes the physical NLA by Gaussian post-selection. It consists of a filter function and a rescaling factor whose probabilistic gain is .[40] The effect of the MB-NLA can be described as a covariance matrix (CM) transformation given by[31]

where γin and γout are covariance matrices of input and output respectively, and gNLA = Diag(gNLA, gNLA) is the amplifying matrix for the individual mode. For gNLA > 1, the success probability can be given by[25]

where is the measurement outcome and xc is the cutoff parameter which truncates the original post-selection filter in an appropriate amplitude and hence makes a trade-off between probability and fidelity. For gNLA < 1, the success probability is given by

where Vin is the variance of input mode. After signal mode goes through the MB-NLA, it is further amplified by times deterministically. Finally, the signal mode will be injected into the electro-optic modulator (EOM) to fulfill the displacement operation. The CM of output mode B3 is given by

with VB1 = TV + (1 – T)VE. By solving for the singularity for the output Eq. (10), we can obtain the restriction of gNLA given by

In order to show how the HLA improves the performance of TS-CVQKD protocol, we compares the signal-to-noise ratio (SNR) of mode B4 with and without HLA as shown in Fig. 1(b). When a thermal state passes through the HLA, the improvement of SNR at B4 can be expressed as

Figure 2 illustrates ΔSNR for the given parameters VM = 1, VS = 1.2, vel = 0.05 and η = 0.6. We find the larger the gNLA or gDLA means the more improvement of the SNR. In addition, the parameters gNLA and gDLA may have interacted with each other. For example, for gDLA = 2, a unit increase of gNLA will give birth to the improved SNR by ΔSNR = 2.8. For gDLA = 4.5, ΔSNR becomes 3. That is to say, gNLA (gDLA) may exhibit different effect when the other gain factor changes. Furthermore, we should point out the same effective gain ge = gNLA gDLA does not signify the same improvement of SNR.

Fig. 2. The ΔSNR as functions of gNLA and gDLA. The blank area indicates unphysical region of gNLA > gDLA. The two white dashed lines represent ΔSNR when gNLA changes from 1 to 2. The white solid line represents contour line of ge = gNLA gDLA. Modulation variance VM = 1, shot noise VS = 1.2, transmission efficiency η = 0.6 and detector electronics vel = 0.05.

Actually, we note the SNR improvement does not mean the increased secret key rates because it ignores the influence of Eve. The most typical example is the so-called fighting noise with noise,[41] where noise is added to the reference side to increase secret key rates when Eve’s attack is strong. However, it may enhance the performance of TS-CVQKD for the tunable parameters in suitable ranges.

3. Security analysis

We demonstrate the performance improvement of the HLA-involved TS-CVQKD with the numerical simulations in near-and-mid infrared band (6 THz–400 THz) or terahertz band (60 GHz–1 THz). The secret key rates of TS-CVQKD with and without HLA are derived in Appendix B. Table 1 shows the parameters setting. In near-and-mid infrared band, the average photon number is very low and hence thermal noise is small, whereas thermal noise in terahertz band is extremely high. Consequently, the HLA-involved scheme makes a difference in terms of the secret key rates. In addition, the emerging terahertz band is identified as one of the promising spectrum bands to enable ultra-high-speed quantum communications.

Table 1.

Parameter setting. The N–M represents near-and-mid infrared and T represents terahertz.

.
3.1. Near-and-mid infrared band

In near-and-mid infrared band, the performance of the HLA-involved TS-CVQKD is compared with standard TS-CVQKD in numerical simulations. We take into account the configuration of homodyne detection in both the DR and RR cases. As shown in Fig. 3, the thermal noise in near infrared band comes from imperfect modulation. Once the frequency exceeds a certain threshold in mid infrared regime, thermal noise will rise up quickly.

Fig. 3. (a) Near-and-mid infrared band and (b) terahertz band.

We find that the HLA can improve standard TS-CVQKD for both DR and RR in terms of security threshold as shown in Fig. 4. For DR, the added security threshold maintains at 1.8 dB as VS changes, corresponding to an improvement over 125 %. For RR, the added security threshold will decrease from 1.3 dB to 0.2 dB as VS changes, leading to an improvement of no more than 6 %. Then, we compare the performance of HLA-involved TS-CVQKD (dashed line) and LOCM-involved TS-CVQKD (dash-dotted line) in homodyne detection. It is shown that inserting an HLA on Bob’s side is better than inserting an LOCM no matter in DR or RR. For DR, LOCM even decreases the security threshold as compared to the standard protocol.

Fig. 4. Secret key rate as a function of channel loss (dB) using (a) Direct reconciliation (b) reverse reconciliation. The solid line is the standard TS-CVQKD, while the dashed line represents the HLA-involved TS-CVQKD with gNLA = 1.2, in DR and gNLA = 1.5, in RR. The dash-dotted line represents the LOCM-involved TS-CVQKD with τ = 0.9 and λ = 1.02 in both the cases.

The performance of the HLA-involved TS-CVQKD will also be influenced by frequency f in DR. In Fig. 5, it shows that the security threshold can be extended by 2.1 dB for f = 120 THz, resulting in an improvement of 117 % compared with the standard TS-CVQKD. However, when the frequency moves down to 6 THz, as shown in Fig. 5(b), the maximum value of gNLA will be shrunk from 1.43 to 1.34, and hence the improved security threshold declines to 1.6 dB. It is noteworthy that although the security threshold increases monotonically with both gNLA and gDLA, the amplification effect of gNLA on TS-CVQKD is more than that of gDLA no matter what the frequency is, as shown by the dashed line in Fig. 5.

Fig. 5. Secure threshold of channel loss as functions of gDLA and gNLA in DR for (a) f = 120 THz and (b) f = 6 THz.

In Fig. 6, we find that the security threshold becomes the peak value (black circle) in RR for instead of , which means that decreasing SNR of signal model can, at least within a certain range, improve security threshold. This situation is the so-called fighting noise with noise,[41] i.e., noise enhances the performance of TS-CVQKD. The same phenomenon can be seen for the parameter gNLA as the peak value located at gNLA = 1.54. At the same time, we also find HLA-involved TS-CVQKD in DR is superior to that in RR when the frequency drops quickly. In RR, the peak value fades from 7.52 dB to 0.8 dB as the frequency goes down from f = 1200 THz to f = 6 THz. In DR, however, the peak value only slightly fades from 4 dB to 3.5 dB.

Fig. 6. Secure threshold of channel loss as functions of gDLA and gNLA in RR for (a) f = 120 THz and (b) f = 6 THz.
3.2. Terahertz band

It is known that thermal noise in terahertz band is extraordinary high. Even so, the terahertz band communication has some advantages for future wireless communication since the terahertz wave can integrate advantages of microwave and optical wave. Compared with the microwave communication, the terahertz communication possesses larger capacity and better direction. Compared with the optical wave, the terahertz wave holds advantages of high energy efficiency and good penetrability.[43] In addition, it allows to transfer huge messages very quickly and expects to offer applications for high speed data links.[44,45]

As shown in Fig. 3(b), the terahertz band is the bridge of microwave band and infrared band through boundary at 300 GHz. In near-and-mid infrared band, thermal noise is very tiny. However, it can be up to 200 in terahertz band at frequency of 60 GHz. Consequently, the security threshold is extremely small compared with that of near-and-mid infrared regime. Here we let TK = 15 °C and f ∈ {1 THz, 300 GHz } to demonstrate the amplification effect of the HLA-involved scheme. We focus on DR instead of RR since thermal noise here is extremely high and therefore DR is far beyond RR.

Bob’s electronic noise vel will change when it comes to terahertz band. This electronic noise vel is basically due to thermal noise introduced by the intrinsic amplifier circuit posterior to photon detection (not HLA) and it will still exist even when no signal light enters homodyne detector.[39] This thermal noise can be influenced by temperature, which means that one may use cryogenic amplifier circuit to suppress thermal fluctuations.[46] However, cryogenic amplifier circuit is too expensive for the practical implementation and hence we do not apply this setup in the proposed scheme. As a result, the original noise should adopt to terahertz band and change to velN0.

In Fig. 7, we achieve the optimal value of gNLA to maximize the secret key rate at f = 300 GHz and f = 1 THz. The parameter gNLA must satisfy the restriction in Eq. (11). Thus, the gNLA in terahertz band nearly cannot exceed 1 and thus the signal mode cannot be amplified. Fortunately, it has been pointed out that noiseless attenuation is also able to increase secret key rate and secure range of the protocol in the high-noise low-loss regime.[25] This inspires us because it happens to suit our case for high-noise and low-loss. Consequently, we consider the range of gNLA < 1 and obtain the optimal gNLA for various channel loss. As shown in Fig. 7(b), the secret key rates reach the highest value for the optimized parameter gNLA = 0.96 in a fixed secure interval. As channel loss has been decreasing from 0.2 dB to 0.14 dB, the optimized parameter gNLA will get closer to one and hence the fixed secure interval would become wider.

Fig. 7. The optical parameter gNLA for the tunable channel loss at different frequencies: (a) channel loss fixed at {0.04,0.05,0.06}, (b) channel loss fixed at {0.16,0.18,0.20}.

As shown in Fig. 8, the performance of the HLA-involved TS-CVQKD can be improved in 300 GHZ and 1 THz. In Fig. 8, when gDLA increases from 1 to , the imperfection of a practical detector will be compensated and gNLA will further enhance the secret key rate. Although the HLA-involved scheme can still improve TS-CVQKD, which is similar to the counterpart in near-and-mid infrared band, there are different results. The parameter gDLA plays a much more important role than gNLA. An ideal DLA can compensate for imperfections of Bob’s practical detector in order to approach perfect detection apparatus. In near-and-mid infrared band, the electronic noise vel is relative small so that gDLA will not help too much. However, the electronic noise is much higher in terahertz band and therefore sensitive to gDLA.

Fig. 8. The secret key rates versus channel loss in terahertz band with the tunable parameters gNLA and gDLA for (a) f = 300 GHz and (b) f = 1 THz. The three dashed lines show the secret key rates as gDLA is increased from 1 to without amplification effect of gNLA. The purple lines signify the secret key rates when gNLA = 0.96 is used on the basis of perfect detector.

In addition, controlling channel noise is another useful approach for enhancing the secret key rate. In the above analysis, we demonstrate that the HLA-involved scheme is relative robust against thermal noise in DR. However, we should point out that channel noise VE, unlike thermal noise, can dominate the HLA-involved scheme in the very limited range of security. The smaller the channel noise VE, the greater the secure distance. In Fig. 9, we compare the secret key rates with and without HLA for VE ∈ {5,12,20}. When VE decreases from 20 (green dashed line) to 5 (red dashed line), the maximal transmission distance of the HLA-involved TS-CVQKD will be extended from 15 cm to 56 cm. We note that 56 cm, despite being a small secure region, is enough for some specific scenarios in near field communication (NFC, 10 cm), such as mobile payment and access control system.

Fig. 9. How channel noise VE influences the security of CVQKD with (dashed line, , gNLA v = 0.96) and without HLA (solid line) at f = 1 THz. The green, blue, and red line denote VE = 5, 12, 20 respectively. The attenuation is around 850 dB/km. A decrease in channel noise VE will enhance the performance of TS-CVQKD protocol rapidly.
4. Conclusion

We have proposed an HLA-involved scheme for performance improvement of TS-CVQKD. Numerical simulations show that inserting an HLA before Bob’s detector can enhance the security of TS-CVQKD no matter in near-and-mid infrared band or terahertz band. The direction of reconciliation and the frequency will influence performance of TS-CVQKD. We find that at 300 GHz and 1 THz, the optimal parameter gNLA lies in the area of gNLA < 1, which is not only confined to 300 GHz and 1 THz but also suitable for all frequency band of high thermal radiation. In addition, the HLA-involved scheme can extend the maximal transmission distance at 1 THz from 15 cm to 56 cm, which indicates a potential application for not only NFC but also other wireless system requiring longer secure distance. Furthermore, future work may extend this scheme deeply to microwave band (300 MHz–300 GHz) which seems more attractive in the microwave regime.[47,48]

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