Expand this Topic clickable element to expand a topic
Skip to content
Optica Publishing Group
Journal Home About Issues in Progress Current Issue All Issues Feature Issues

Indoor channel modeling for continuous variable quantum key distribution in the terahertz band

Yuqian He, Yiyu Mao, Duan Huang, Qin Liao, and Ying Guo

Open Access Open Access

Abstract

Continuous-variable quantum key distribution (CVQKD) in an indoor scenario can provide secure wireless access for practical short-distance communications with high rates. However, a suitable channel model for implementing the indoor CVQKD system has not been considered before. Here, we establish an indoor channel model to show the feasibility of CVQKD in terahertz (THz) band. We adopt both active and passive state preparation schemes to demonstrate the performance of the indoor CVQKD system involving multi-path propagation. We achieve the channel transmittance characterized by frequency, water-vapor density, antenna gain, reflection loss and the surrounding itself. The ray-tracing based numerical simulations show that the multi-path propagation can degrade the performance of the indoor CVQKD system. The maximum transmission distance is two meters at 410 GHz for both active and passive state preparations, and it can be extended to 35 and 20 meters respectively by using high gain antenna to combat the multi-path propagation.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Quantum key distribution (QKD) is a mature technology to share symmetric secret keys between distant parties over an untrusted channel, which may be controlled by malicious eavesdroppers [1–4]. Different from the classical key distribution whose security relies heavily on the complex computation in mathematical sense, QKD is guaranteed by the laws of quantum physics [5–7]. According to different implementation methods, QKD is mainly divided into two categories: discrete-variable quantum key distribution (DVQKD) [8–10] and continuous-variable quantum key distribution (CVQKD) [11–14]. Compared with DVQKD, CVQKD can be applied on off-the-shelf telecom networks with high secret key rates, avoiding the imperfection for using single photon counting [15]. So far, CVQKD has been proven to be secure against collective attacks in asymptotic limit, finite-size regime and composable security framework [16–18]. Meanwhile, its availability has been demonstrated in laboratories and in field environments [19,20].

Traditional CVQKD usually deals with information through optical communication systems [21–23]. Such an optical setup can be built upon high power laser, and may be impractical for a plethora of attractive low-power short-distance communications, such as Bluetooth and chip-to-chip communications [24]. As for the practical implementation, it is a technological preference to extend the application of QKD to indoor scenarios, as the portable personal devices and data security have been growing dramatically over the years. Recently, thermal states have been adopted for CVQKD on both microwave band and infrared band [25–27]. It has been found that a central broadcast CVQKD system can be implemented with low power devices [24], where a source emits thermal radiation through a broadcast channel on microwave regime. However, the afore-mentioned protocols didn’t show the actual secure distance, but only the hypothetical transmittance. Therefore, it is imperative to develop a suitable channel model to show how far quantum signals can reach in indoor environments.

It is known that indoor channel modeling in terahertz (THz) band [28–31] can offer emerging applications for ultra-high-speed links [32]. From the point of quantum communications, the THz band has great potentials for high speed CVQKD systems [33], as the unregulated spectrum can provide bandwidths even exceeding 50GHz [34]. On the other hand, the environmental thermal noise in THz band is relative small compared with microwave band [35], which reduces thermal noise and hence improves security threshold.

Inspired by the unregulated terahertz spectrum, in this paper, we propose an indoor channel model for high speed CVQKD systems with low power laser. We note that the indoor channel, which is influenced by the multi-path propagation, is characterized by the receiver, where Bob gets quantum signals from both line-of-sight (LOS) ray and non-line-of-sight (NLOS) ray [36,37]. We demonstrate the effect of channel modeling parameters on the performance of the CVQKD system in an indoor environment. We derive the mathematical framework of the indoor channel transmittance for LOS and NLOS links. We consider the active state preparation scheme and the passive state preparation scheme [38–40] for the indoor CVQKD system. Results show that for the tunable channel modeling parameters, such as frequency, water-vapor density and antenna gain, we can achieve the high key rate of the indoor CVQKD system.

This paper is organized as follows. In Sec. 2, we establish the indoor channel and obtain the channel transmittance for the indoor CVQKD system. In Sec. 3, we suggest the active and passive CVQKD systems in indoor environments, respectively. In Sec. 4, we conduct a coverage simulation using ray-tracing and demonstrate the factors affecting the secure region. Finally, we give a brief conclusion in Sec. 5.

2. Indoor channel for quantum communications

In an optical fiber channel, as shown in Fig. 1(a), the transmitter (Tx) sends signals from one side to the other side through an optical link, whereas in a wireless indoor channel the receiver (Rx) will get the signal from both LOS and NLOS links as shown in Fig. 1(b), which is in fact a multi-path propagation channel. The characteristics of the multi-path propagation channel are embodied in two-fold. On the one hand, it is influenced by the dimension of the surroundings, where a tiny change of indoor surroundings may have a strong impact on the multi-path propagation. For example, as shown in Fig. 1(c), when the table is removed, the blue NLOS link will not be connected. On the other, the multi-path propagation will cause a harmful effect on signal transmission, as it distributes the transmitted power over various paths rather than concentrates them on one ray like the optical fiber channel case [32,41].

 figure: Fig. 1.

Fig. 1. The channel model between the transmitter (TX) and the receiver (RX). (a) Optical fiber channel. (b) Indoor channel with a table. (c) Indoor channel without a table.

Download Full Size | PDF

As each LOS link and NOLS link gets different power loss and time delay, the indoor channel impulse response (CIR) in THz band can be described as [32]

$$\begin{aligned} h(\tau)=\alpha_{\textrm{LOS}}\delta(\tau-\tau_{\textrm{LOS}})+\sum_{k=1}^{N_{\textrm{NLOS}}}\alpha^{(k)}_{\textrm{NLOS}}\delta(\tau-\tau^{(k)}_{\textrm{NLOS}}), \end{aligned}$$
where $N_{\textrm{NLOS}}$ denotes the total number of NLOS rays, $\alpha _{\textrm {LOS}}$ and $\alpha ^{(k)}_{\textrm{NLOS}}$ are the attenuation coefficient of the LOS ray and the $k^{\textrm {th}}$ NLOS ray, respectively. $\tau _{\textrm {LOS}}$ and $\tau ^{(k)}_{\textrm{NLOS}}$ are the time delay. The delayed NLOS links are due to the reflection, scattering and diffraction resulting from indoor surroundings. The diffracted rays are usually negligible for indoor environment [30]. Slightly rough materials are assumed and hence the scatter ray can be excluded [34]. As a result, we only need to consider the effect of reflection on the indoor CVQKD system, so that the CIR in Eq. (1) can be transformed as
$$\begin{aligned} h(\tau)=\alpha_{\textrm{LOS}}\delta(\tau-\tau_{\textrm{LOS}})+\sum_{k=1}^{N_{\textrm{ref}}}\alpha^{(k)}_{\textrm{ref}}\delta(\tau-\tau^{(k)}_{\textrm{ref}}), \end{aligned}$$
where $\alpha ^{(k)}_{\textrm {ref}}$ refers to the attenuation and $\tau ^{(k)}_{\textrm {ref}}$ stands for the delay of the $k^{\textrm {th}}$ reflected rays. Notably, although the scatter rays are ignored, we still take the scattering phenomenon into account. As shown by the red circles in Fig. 1(b) and Fig. 1(c), the scattering phenomenon reduces the power of reflected rays in the specular direction [32].

In order to achieve the channel transmittance, we use the ray-tracing tool comsol to simulate propagation path and power loss of LOS and NLOS rays in a realistic indoor scenario in THz band (0.3-1 THz), as shown in Appendix A. The transmittance of the indoor channel is given by

$$\begin{aligned} T_{\textrm{Indoor}}=T_{\textrm{LOS}}+\sum_{k=1}^{N}T^{(k)}_{\textrm{NLOS}}, \end{aligned}$$
where $T_{\textrm {LOS}}$ is the channel transmittance of LOS link and $T^{(k)}_{\textrm{NLOS}}$ is the channel transmittance of the $k^{\textrm {th}}$ NLOS link. One can see more details about $T_{\textrm {LOS}}$ and $T_{\textrm{NLOS}}$ in Appendix B.

As shown in Fig. 2, the channel loss ($10\log T_{\textrm {LOS}}$) increases with both frequency and distance, which is independent of water-vapor density. Nevertheless, several peaks of attenuation can be found in Fig. 2(b) and Fig. 2(c) owing to the water-vapor absorption loss along the travel path. With the improvement of water-vapor density, the channel loss will be stronger. In practice, we can use a high gain antenna for combating the high channel loss.

 figure: Fig. 2.

Fig. 2. Channel loss in dB as a function of the frequency and the distance under three different water-vapor density (a) $\rho$=0 g/m$^3$, (b) $\rho$=7.5 g/m$^3$ and (c) $\rho$=40 g/m$^3$. The maximum values of channel loss have been cut off at 120 dB since only few values exceeds it.

Download Full Size | PDF

Similar to $T_{\textrm {LOS}}$, $T_{\textrm{NLOS}}$ is also influenced by frequency, distance and water-vapor density. However, the NLOS link may interact with obstacles, so $T_{\textrm{NLOS}}$ is additionally influenced by reflection loss. As shown in Fig. 3, the reflection loss increases monotonically with the incident angle. Meanwhile, the frequency and the roughness also play a role because the scale of wavelength relative to the roughness can influence the scattering phenomenon [29]. By comparing Fig. 3(a) and Fig. 3(b), we can find that the reflection loss is influenced by the type of material, as wood causes more reflection loss than plaster.

 figure: Fig. 3.

Fig. 3. The reflection loss (dB) versus the incident angle under different parameters $f$ and $\sigma$ for different materials (a) plaster and (b) wood. The solid lines show reflection loss for $\sigma$=0.05 mm while dashed line for $\sigma$=0.08 mm.

Download Full Size | PDF

3. Indoor CVQKD system

In this section, we propose the active and passive state preparation schemes for indoor CVQKD. We focus on direct reconciliation (DR) instead of reverse reconciliation (RR) since the thermal noise in THz band is extremely high and therefore DR is far beyond RR [26].

3.1 Active state preparation scheme

The prepare and measure scheme of the active CVQKD protocol is shown in Fig. 4(a). Alice uses the THz source to generate the mode $B_{0}$. Because the environment thermal noise in THz band is high, as shown in Fig. 4, mode $B_{0}$ is in fact a thermal state featured by thermal noise $V_{S}$ given by

$$\begin{aligned} V_{S}=2n_{0}+1, \end{aligned}$$
where $n_{0}$ is the average photon number for any wavelength [42]. We note that $n_{0}>0$ corresponds to the thermal state case and $n_{0}=0$ corresponds to the coherent state case. After that, Alice encodes Gaussian-distributed random numbers on mode $B_{0}$ by AM and PM, resulting in signal mode $B_{2}$. The variance of signal mode $B_{2}$ is given by
$$\begin{aligned} V=V_{S}+V_{M1}, \end{aligned}$$
where $V_{M1}$ is the modulation variance. Finally, Alice sends signal model $B_{2}$ to the receiver Bob through an indoor channel featured by $T_{\textrm{Indoor}}$.

 figure: Fig. 4.

Fig. 4. Schematic diagram of (a) active CVQKD and (b) passive CVQKD with an indoor channel. BS$_{1}$/BS$_{2}$, 50:50 beam splitter; HD, homodyne detector; Att., optical attenuator. AM/PM, amplitude and phase modulator. Beam splitters with a transmittance of $\eta _{H}$ are used to model the efficiency of the homodyne detectors.

Download Full Size | PDF

In the indoor channel, Eve will perform an entangling cloner attack [43–46] inside a cryostat in order to remove her from the environment thermal noise [26,27,47]. The variance of Eve’s modes can be characterized by $V_{E}$. Subsequently, Bob measures signal mode $B_{4}$ by homodyne detection. For simplicity, we assume all the devices on Bob’s side are perfect. Finally, the reconciliation ensures that Alice and Bob can remove the influence of Eve and extract the secret keys.

3.2 Passive state preparation scheme

In the passive CVQKD case, we employ an optical attenuator and homodyne detectors to encode information instead of directly using AM and PM, which simplifies the implementation and paves the way for future chip-scale CVQKD. Compared with CVQKD in optical band, CVQKD in THz band doesn’t need a particular device such as a fiber amplifier [38] to generate the thermal source, as the average photon number in THz band is already high. The passive state preparation scheme for CVQKD is depicted in Fig. 4(b) and can be described as follows.

  • Step1: Alice injects the thermal state $B_{0}$ into a beam splitter BS$_{1}$, which splits the mode $B_{0}$ into two spatial modes $B_{1}$ and $B_{3}$. The average output photon number of mode $B_{0}$ is $n_{0}$.
  • Step2: Mode $B_{3}$ is measured locally by performing conjugate homodyne detection. According to the measurement results $x_{B_{3}}$ and $p_{B_{3}}$, Alice can estimate the quadrature values of the output mode as $x_{E}$ and $p_{E}$.
  • Step3: Mode $B_{1}$ passes through an optical attenuator with a transmittance of $\eta$. Accordingly, the average photon number of mode $B_{1}$ changes from $n_{0}/2$ to $V_{M2}/2$, resulting in mode $B_{2}$, where $V_{M2}=n_{0}\eta$ is the desired modulation variance. Then, mode $B_{2}$ is sent to Bob through an indoor channel featured by $T_{\textrm{Indoor}}$.
  • Step4: The rest of the steps are identical to that in the active CVQKD, as the indoor channel and the devices on Bob’s side are almost same.
From Eve’s point of view, the quantum state $B_{2}$ sent to Bob is the same as the $B_{2}$ in the active CVQKD, i.e., they are both regarded as a thermal state. As a result, both the well-established security proofs and the calculation procedure of secret key rate can be transplanted into the passive CVQKD. As the modulation variance is featured by $V_{M2}$, the derivation of the secret key rate is only determined by how much additional noise will be introduced in the passive state preparation process.

Without loss of generality, we only consider the $X$ quadrature since the other quadrature $P$ can be analyzed in a similar way. The $X$ quadrature of the output state ${B_{2}}$ is described as

$$\begin{aligned} x_{B_{2}}=\sqrt{\frac{\eta}{2}}x_{B_{0}}+\sqrt{1-\frac{\eta}{2}}x_{v_{1}}, \end{aligned}$$
where $x_{B_{0}}$ is the $X$ quadrature of mode ${B_{0}}$ and $x_{v_{1}}$ represents the extra vacuum noise introduced by the combination of the beam splitter BS$_{1}$ and the optical attenuator. Similarly, Alice’s measurement result $x_{B_{3}}$ can be given by
$$\begin{aligned} x_{B_{3}}=\sqrt{\frac{\eta_{H}}{4}}x_{B_{0}}+\sqrt{1-\frac{\eta_{h}}{4}}x_{v_{2}}+\xi_{el}, \end{aligned}$$
where $x_{v_{2}}$ represents the extra vacuum noise caused by the balanced beam splitters and the detectors, and $\xi _{el}$ is the Gaussian noise of detectors featured by variance $v_{el}$. After that, Alice can estimate the numerical value of $x_{B_{2}}$ from her measurement result $x_{B_{3}}$ as $x_{E}=\sqrt {{2\eta }/{\eta _{H}}}x_{B_{3}}$. Then we obtain Alice’s uncertainty on $x_{B_{2}}$ given by
$$\begin{aligned} \epsilon=\frac{2\eta}{\eta_{H}}(1+v_{el}-\frac{\eta_{H}}{2})+1, \end{aligned}$$
therefore, the additional noise $V_{A}$ can be calculated as $V_{A}=\epsilon -1$.

We note that the modulation variance $V_{M1}$ in the active CVQKD and $V_{M2}$ in the passive CVQKD are different from each other, and so are the additional noises $V_{S}$ and $V_{A}$. For modulation variance, $V_{M1}$ is constrained by the performance of the modulator. The maximum value of $V_{M2}$, however, is restricted by average photon number $n_{0}$. In usual case, $V_{M1}$ is much lager than $V_{M2}$. For additional noise, $V_{S}$ depends on the frequency while $V_{A}$ is influenced by transmittance of attenuator. Besides, $V_{S}$ is regarded as trusted noise, i.e, preparation noise. Nevertheless, $V_{A}$ is regarded as untrusted noise, i.e, excess noise. We stress that this difference has an impact on the performance of the CVQKD system since the excess noise $V_{A}$ will change the channel noise from $V_{E}$ to $V_{E}+V_{A}{T_{\textrm{Indoor}}}/({1-T_{\textrm{Indoor}}})$, while $V_{S}$ will not.

4. Security analysis

In what follows, we show the effects of the indoor channels on the active CVQKD and the passive CVQKD. We calculate the secret key rate of the indoor CVQKD system in a realistic scenario (see Appendix A). Subsequently, we achieve the potential maximum distance by adopting a high gain antenna. Table 1 shows the parameters setting.

Tables Icon

Table 1. Parameter setting.

View Table

4.1 Performance of the indoor CVQKD

The performance of the active indoor CVQKD system will be influenced by frequency and multi-path propagation, as shown in Fig. 5(a) and Fig. 5(b). For $f=410$ GHz [32], the maximum transmission distance in indoor environment is only $2.1$ meters. Moreover, for $f=850$ GHz [32], the maximum transmission distance can be extended to $4.1$ meters, leading to an improvement of nearly 100$\%$ compared with the former one. In spite of this, the secret key rate in the case of $410$ GHz is higher than that in the case of $850$ GHz for a given same distance. Unfortunately, there are some irregular grids at the edge of the secure region, marked by ($\ast$), where the secret key rates decline rapidly. This phenomenon is in fact caused by the multi-path propagation since it reduces the total channel transmittance.

The untrusted noise in the passive CVQKD system will not affect the secure range. As shown in Fig. 5(c) and Fig. 5(d), the scale of the secure region is almost the same as the one in Fig. 5(a) and Fig. 5(b). After exceeding a certain distance, the channel transmittance will be too small to distill the secret keys due to the effects of the multi-path propagation. Therefore, the secure range is not sensitive to the untrusted noise. Similarly, the reduction of the channel noise from $5$ to $2$ also has no significant effect on the secure range. Whereas, the untrusted noise does reduce the secret key rates, as we can see the color depicted in Fig. 5(b) is deeper than that in Fig. 5(d). Given the passive CVQKD doesn’t have to integrate high extinction ratio modulators into a chip, thereby reducing the manufacturing time and cost for hand-hold devices, we argue that it is a better choice for indoor quantum communications.

In addition, the antenna can serve as a spatial filter for the multi-path propagation. As shown in Fig. 6, as the antenna gain rises up, the ratio $k=T_{\textrm {LOS}}/T_{\textrm{NLOS}}$ will be increased. The higher $k$ indicates the lower multi-path propagation. For the antenna gain $A$=12.5 cm$^{3}$ in points K$_1$ and K$_2$, $T_{\textrm {LOS}}$ is larger than $T_{\textrm{NLOS}}$, although the multi-path propagation exists.

Even if the antenna gain is large enough to make the free space path loss (FSPL) disappear, the secure range still can not stretch to the NLOS region, as the blank grids shown in Fig. 7(a). In an active CVQKD system without any thermal noise, the channel loss can not exceed 3 dB, otherwise, the secret key rates will decline to zero. However, in an active CVQKD system involving high thermal radiation, the reflection loss can even be up to tens of dB, as shown in Fig. 3. This high reflection loss makes it impossible to realize a secure NLOS region since the receiver can only get weak signal light. Hypothetically, if the reflection loss disappears, the secure region can be established successfully, as shown in Fig. 7(b). We note that the irregular color transition from LOS to NLOS region is due to the longer travel length of the NLOS links because they are reflected from the obstacles to Bob’s side.

4.2 Potential performance improvement

In what follows, we assume the antenna gain is high enough to make the multi-path propagation disappear. Therefore, Alice concentrates all the transmitted quantum signals on LOS link.

In Fig. 8, we show the secret key rates of the active CVQKD system involving tunable modulation variance $V_{M1}$ and water-vapor density $\rho$. We find the maximum transmission distance is improved when there is no restriction of the multi-path propagation. As shown in Fig. 8, the security threshold for 410 GHz and 850 GHz are 35 and 7.5 meters respectively, which are longer than 2.1 and 4.1 meters in Fig. 5. We stress that the maximum transmission distance for 410 GHz improves more rapidly than 850 GHz when multi-path propagation disappear, since the molecular absorption loss at 410 GHz is smaller. At the same time, we also find that modulation variance and water-vapor density play different roles. The maximum distance will remain unchanged when modulation variance $V_{M1}$ increases. However, with the increase of water-vapor density $\rho$, the maximum distance will be declined.

 figure: Fig. 5.

Fig. 5. The secret key rates of the indoor CVQKD system. (a) $f=410$ GHz for the active CVQKD. (b) $f=850$ GHz for the active CVQKD. (c) $f=410$ GHz for the passive CVQKD. (d) $f=850$ GHz for the passive CVQKD. In (c) and (d), the blue line divides coverage map into the origin region ($V_{E}$=5) and the enlarged region ($V_{E}$=2). Other parameters are given by $x_{t}$=4.875 m, $y_{t}$=0.125 m, $z_{t}$=2.3 m, $A$=12.5 cm$^{2}$, and $\rho$=7.5 g/m$^{3}$.

Download Full Size | PDF

 figure: Fig. 6.

Fig. 6. The ratio of channel transmittance $T_{\textrm {LOS}}$ to $T_{\textrm{NLOS}}$ as a function of antenna aperture in points $K_1$ and $K_2$ (see Fig. 5(a)).

Download Full Size | PDF

 figure: Fig. 7.

Fig. 7. The secret key rates of the indoor CVQKD system for height of $z$=1 m for (a) With reflection loss and (b) Without reflection loss. Other parameters are given by $x_{t}$=4.875 m, $y_{t}$=0.125 m, $z_{t}$=2.3 m, $f$=850 GHz, $A$=12.5 cm$^{2}$, and $\rho$=7.5 g/m$^{3}$.

Download Full Size | PDF

 figure: Fig. 8.

Fig. 8. The secret key rate of the active CVQKD system without the multi-path propagation for (a) $f$=410 GHz and (b) $f$=850 GHz. The solid, dashed and dash-dotted lines denote $\rho$=7.5 g/m$^{3}$, 10 g/m$^{3}$, and 20 g/m$^{3}$, respectively.

Download Full Size | PDF

In Fig. 9, we show the performance of the passive CVQKD system and find its potential performance depends on both transmittance of optical attenuator $\eta$ and water-vapor density $\rho$. As shown in Fig. 9(a), when $\eta$ increases from $0.01$ to $0.1$ for $\rho$=7.5 g/cm$^{3}$ (brown line), the maximum transmission distance will be declined from $20$ meters to $18$ meters. Given $V_{M2}=n_{0}\eta$, we find that the tunable modulation $V_{M2}$ has an impact on the maximum transmission distance, which is different from the active CVQKD system. Compared with the active CVQKD system, the maximum transmission distance of the passive CVQKD system is shorter (20 meters versus 35 meters) due to the effects of the additional untrusted noise. Besides, the increase of water-vapor density $\rho$ can reduce the maximum transmission distance. Therefore, one way to improve the maximum transmission distance is to increase the average photon number. As shown in Fig. 9(b), the more average photon number means the longer the maximum transmission distance. When the average photo number rises to 1000$n_{0}$, the secure region can even reach nearly $35$ meters, which is the upper limit of the active CVQKD system at $410$ GHz.

 figure: Fig. 9.

Fig. 9. (a) The secret key rate of the passive CVQKD system. The red, green and blue lines denote $\rho$=7.5 g/m$^3$, 10 g/m$^3$ and 20 g/m$^3$ respectively. (b) The secret key rates of the passive CVQKD system with the average photo numbers $n_{0}$, 10$n_{0}$, 100$n_{0}$, and 1000$n_{0}$.

Download Full Size | PDF

In Fig. 10, we show the maximum transmission distance of the passive CVQKD system as a function of the frequency $f$ and the transmittance of the optical attenuator $\eta$. As mentioned above, increasing the average photo number $n_{0}$ will improve the maximum transmission distance. Although decreasing frequency $f$ can increase the average photo number $n_{0}$, the maximum transmission distance does not decrease monotonically with the frequency $f$. Instead, a wave curve including five peaks could be observed in Fig. 10. The reason is that decreasing the frequency may additionally change the molecular absorption loss (see Fig. 2), which will also influence the maximum transmission distance. In addition, we find the maximum transmission distance can reach the peak value of $75$ meters. We note that although $75$ meters is a small secure region compared to the typical long-range QKD system, it could be suitable for some practical applications such as Blue-tooth ($\sim 10$ meters) and Wifi ($\sim 75$ meters).

 figure: Fig. 10.

Fig. 10. The maximum transmission distance of the passive CVQKD system as a function of frequency $f$ and transmittance of optical attenuator $\eta$. There are total 5 peaks corresponding to $300$ GHz, $340$ GHz, $410$ GHz, $460$ GHz, and $480$ GHz respectively. The simulation results over 550 GHz and the impact of reconciliation efficiency are both shown in Appendix C.

Download Full Size | PDF

5. Conclusion

We have suggested an indoor channel model to demonstrate the performance of the active and passive CVQKD system. We have found it is feasible to implement short-distance CVQKD system in THz band with a suitable indoor setting. However, the multi-path propagation is a factor that restricts the maximum transmission distance. Without the multi-path propagation, the maximum transmission distance can stretch to $35$ meters at $410$ GHz. In addition, the secure range will also be affected by other factors such as frequency, water-vapor density, modulation variance and the indoor surrounding itself. For the realization of the indoor CVQKD system, it is necessary to employ a high gain antenna. Numerical simulations show the performance of the active and passive CVQKD system in indoor environments, which are compared with each other. We have found that the active CVQKD performs better than the passive CVQKD owing to the untrusted noise. Fortunately, these gaps can be closed by increasing the average photo numbers.

Appendix A: ray tracing in realistic scenarios

The propagation path and power loss of rays can be simulated by ray-tracing, which is a popular way for designing indoor communication systems when an in-depth environment information is available [31,32]. It utilizes geometric optics to trace the rays that interact with surroundings and bounce back to the receiver. It is demonstrated that measured and ray-tracing simulated results show high consistency at THz band [49]. Because of the lack of experimental data such as refractive index to calculate the reflection loss at high THz frequencies, i.e., 1–10 THz, the ray-tracing can only be conducted on the lower THz band, i.e., 0.3 to 1 THz. Notably, once the experimental data about material are made available, the investigated frequency range can be extended to the higher terahertz frequencies immediately. Here, we employ a ray-tracing tool comsol for the indoor channel modeling.

The indoor channel, which involves multi-path propagation, is characterized by line-of-sight ray (LOS) and non-line-of-sight (NLOS) ray [36,37].

  • • LOS link. The attenuation of THz wave mainly comes from oscillations of atmospheric gas or water molecules. This absorption in THz band is much stronger than that in optical case (usually 0.2 dB/km in optical fiber). Besides, the free space path loss (FSPL) is an another cause that even without any additional molecule absorption, the received power will be much lower than the transmitted power.
  • • NLOS link. When the LOS link is blocked by obstacle, the received signals will come from the NLOS links such as reflected, scattered and diffracted rays. The NLOS link may suffer from the same power loss as the LOS link does (molecule absorption loss, FSPL). Besides, a part of power will be absorbed when they reflect from walls, floor and ceiling.

Because the multipath propagation channel changes rapidly with the surroundings, a fixed indoor scenario [29] can be established, as shown in Fig. 11. This room (5 m$\times$2.75 m$\times$2.5 m) is equipped with two tables (1.6 m$\times$0.8 m$\times$0.7 m) and a screen (1.5 m$\times$1.8 m). The screen is used to divide the room into LOS region and NLOS region so that reflection loss can be investigated. As we can see in Fig. 11, the NLOS region is shadowed by the screen that only the NLOS links can be set there and all the LOS links are blocked. The transmitter is placed in the corner of this room (4.75 m,0.25 m, 2.3 m) to achieve a high coverage. The receiver is placed in 220 test points, from $x_{r}=0.125$ m and $y_{r}=0.125$ m to $x_{r}=4.875$ m and $y_{r}=2.625$ m with step size of $0.25$ cm and height of $1$ m.

 figure: Fig. 11.

Fig. 11. The 2D top view of the investigated indoor scenario. The transmitter is located at the position $x$=4.875 m, $y$=0.125 m, $z$=2.3 m and the NLOS region is marked by shaded area.

Download Full Size | PDF

This homogeneous room consists of walls, screen, ceiling, tables and floor. The term homogeneous refers to the walls, screen and ceiling are all made of plaster with the same refractive index. The tables are made of pine wood. The reflection of the floor is ignored owing to the high power absorption of the carpet, which is supposed to cover the entire floor.

Appendix B: derivation of $T_{\textrm {LOS}}$ and $T_{\textrm{NLOS}}$

Propagation of the THz wave can cause vibration of atmospheric gas or water molecules. As a result of this vibration, part of energy of the transmitted signal is converted into kinetic energy, or, from the communication perspective, simply lost [28]. This molecular absorption loss in clear atmosphere is usually stronger than that in optical fiber. At $550$ GHz, the molecular absorption loss can even approach to 4000 dB/km. The channel transmittance $T_{M}$ that involves the molecular absorption is given by

$$\begin{aligned} T_{M}=10^{\frac{-\alpha d_{\textrm{LOS}}}{10}}, \end{aligned}$$
where $d_{\textrm {LOS}}$ is the path length of the LOS ray, and $\alpha$ is the medium absorption coefficient (dB/km) influenced by water-vapor density.

Apart from the molecular absorption loss, the transmitted power may be lost due to the expansion of the THz wave in atmosphere, i.e., FSPL. The channel transmittance $T_{F}$ that involves FSPL can be calculated with the Friis equation [50]

$$\begin{aligned} T_{F}=(\frac{\chi}{4\pi d_{\textrm{LOS}}})^{2}, \end{aligned}$$
where $\lambda$ is the wave length of the THz wave. Therefore, the total channel transmittance $T_{\textrm {LOS}}$ is given by
$$\begin{aligned} T_{\textrm{LOS}}=T_{F}T_{M}=(\frac{\chi}{4\pi d_{\textrm{LOS}}})^{2}10^{\frac{-\alpha d_{\textrm{LOS}}}{10}}. \end{aligned}$$
In practical implementation, an antenna can be adopted to combat the FSPL. Consequently, Eq. (11) can be modified as
$$\begin{aligned} T_{\textrm{LOS}}=(\frac{\chi}{4\pi d_{\textrm{LOS}}})^{2}10^{\frac{-\alpha d_{\textrm{LOS}}}{10}}G_{TX}G_{RX}, \end{aligned}$$
where $G$ denotes the antenna gain given by
$$\begin{aligned} G_{TX}=G_{RX}=\frac{4\pi \delta A}{\lambda^2}, \end{aligned}$$
where $\delta$ is the aperture efficiency and $A$ is the antenna aperture.

Similar to the LOS link, the power of the NLOS link will also be influenced by the molecular absorption loss and the FSPL. However, the NLOS link may additionally interact with environment obstacles, hence quite a part of power will be absorbed, i.e. the reflection loss.

The channel transmittance of the NLOS links which are composed of $N$ paths in an indoor channel can be mathematically described as

$$\begin{aligned} T_{\textrm{NLOS}}=\sum_{k=1}^{N}\psi r^{(k)}_{\textrm{NLOS}}(\frac{\chi}{4\pi d^{(k)}_{\textrm{NLOS}}})^{2}10^{\frac{-\alpha d^{(k)}_{\textrm{NLOS}}}{10}}G_{TX}G_{RX}, \end{aligned}$$
where $\psi$ is the Rayleigh roughness factor, $r^{(k)}_{\textrm{NLOS}}$ is the Fresnel reflection coefficient for a smooth surface, and $d^{(k)}_{\textrm{NLOS}}$ is the shortest path length for the $k^{\textrm {th}}$ link. The reflection coefficient $r_{\textrm{NLOS}}$ on a smooth surface is given by
$$\begin{aligned} r_{\textrm{NLOS}}=\frac{Z\cos\theta_{i}-Z_{0}\theta_{t}}{Z\cos\theta_{i}+Z_{0}\theta_{t}}, \end{aligned}$$
where $\theta _{t}$ is the angle of refraction given by $\theta _{t}=\arcsin (\sin \theta _{i}{Z}/{Z_{0}})$. Here $\theta _{i}$ is the angle of incidence ray and can be calculated by locations of the transmitter, reflection point and the receiver. More concretely, $\theta _{i}=\frac {1}{2}\arcsin (({r^{2}_{a}+r^{2}_{b}-r^{2}})/({2r_{a}r_{b}}))$, where $r$ is the linear distance between transmitter and receiver, $r_{a}$ is the linear distance between transmitter and reflection point and $r_{b}$ is the linear distance between reflection point and receiver. The terms $Z_{0}\approx 377$ $\Omega$ is the free space wave impedance and $Z$ is the wave impedance of the reflecting material given by
$$\begin{aligned} Z\approx\frac{377}{n}, \end{aligned}$$
where $n$ refers to the refractive index, which varies with the frequency and medium material [48,51]. In a word, the reflection coefficient $r_{\textrm{NLOS}}$ can be simply expressed as
$$\begin{aligned} r_{\textrm{NLOS}}\approx-\exp(\frac{-2\cos\theta_{i}}{\sqrt{n^{2}-1}}). \end{aligned}$$
To account for the scattering phenomenon happened during reflection, $r_{\textrm{NLOS}}$ must be multiplied with $\psi =e^{-\frac {\mu }{2}}$, where the parameter $\mu$ is given by
$$\begin{aligned} \mu=(\frac{4\pi \sigma \cos\theta_{i}}{\lambda})^{2}, \end{aligned}$$
where $\sigma$ is a statistical parameter representing the rough surface height standard deviation, which is usually considered to be Gaussian-distributed.

Appendix C: maximum transmission distance in different cases

In Fig. 12, we show the maximum transmission distance in different conditions, which can be seen as a supplementary part of Fig. 10. We find that changing the reconciliation efficiency from 1 to 0.98 can not change the peak value significantly. As shown in Fig. 12(a), the transmission distance is 73 meters, which is almost the same to the case in Fig. 10. In practice, frequency band below 550 GHz is preferred to frequency band over 550 GHz. This is not only because of the molecular absorption loss as the transmission distance of 73 meters in Fig. 12(a) while only 6 meters in Fig. 12(b), but also because of the device design as the frequency band below 550 GHz is more realistic of enabling technologies such as integrated circuits and semiconductor electronic devices [52].

 figure: Fig. 12.

Fig. 12. The maximum transmission distance of the passive CVQKD system as a function of frequency $f$ and transmittance of optical attenuator $\eta$. (a) 300-550 GHz, $\beta$=0.98. (b) 550-1000 GHz, $\beta$=1. For (a), there are total 5 peaks corresponding to 300 GHz, 340 GHz, 410GHz, 460 GHz, and 480 GHz respectively while for (b) 3 peaks corresponding to 670 GHz, 850 GHz and 930 GHz respectively.

Download Full Size | PDF

Funding

Fundamental Research Funds for the Central Universities (531118010371); Natural Science Foundation of Hunan Province (2020JJ5088); National Natural Science Foundation of China (11964013, 61871407).

Disclosures

The authors declare no conflicts of interest.

References

1. F. Grosshans, G. Van Assche, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, “Quantum key distribution using gaussian-modulated coherent states,” Nature 421(6920), 238–241 (2003). [CrossRef]  

2. C. Weedbrook, A. M. Lance, W. P. Bowen, T. Symul, T. C. Ralph, and P. K. Lam, “Quantum cryptography without switching,” Phys. Rev. Lett. 93(17), 170504 (2004). [CrossRef]  

3. S. Pirandola, U. L. Andersen, L. Banchi, M. Berta, D. Bunandar, R. Colbeck, D. Englund, T. Gehring, C. Lupo, C. Ottaviani, J. Pereira, M. Razavi, J. S. Shaari, M. Tomamichel, V. C. Usenko, G. Vallone, P. Villoresi, and P. Wallden, “Advances in quantum cryptography,” arXiv:1906.01645 (2019).

4. V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81(3), 1301–1350 (2009). [CrossRef]  

5. P. W. Shor and J. Preskill, “Simple proof of security of the BB84 quantum key distribution protocol,” Phys. Rev. Lett. 85(2), 441–444 (2000). [CrossRef]  

6. H. Lo, M. Curty, and K. Tamaki, “Secure quantum key distribution,” Nat. Photonics 8(8), 595–604 (2014). [CrossRef]  

7. J. Y. Bang and M. S. Berger, “Quantum mechanics and the generalized uncertainty principle,” Phys. Rev. D 74(12), 125012 (2006). [CrossRef]  

8. S. Takeda, M. Fuwa, P. van Loock, and A. Furusawa, “Entanglement swapping between discrete and continuous variables,” Phys. Rev. Lett. 114(10), 100501 (2015). [CrossRef]  

9. S. Pirandola, C. Ottaviani, G. Spedalieri, C. Weedbrook, S. L. Braunstein, S. Lloyd, T. Gehring, C. S. Jacobsen, and U. L. Andersen, “Reply to’discrete and continuous variables for measurement-device-independent quantum cryptography’,” Nat. Photonics 9(12), 773–775 (2015). [CrossRef]  

10. J. Niu, Y. Sun, Y. Zhang, and Y. Ji, “Noise-suppressing channel allocation in dynamic DWDM-QKD networks using lightgbm,” Opt. Express 27(22), 31741–31756 (2019). [CrossRef]  

11. S. L. Braunstein and A. K. Pati, “Quantum information with continuous variables,” Rev. Mod. Phys. 77(2), 513–577 (2005). [CrossRef]  

12. Q. Liao, G. Xiao, C.-G. Xu, Y. Xu, and Y. Guo, “Discretely modulated continuous-variable quantum key distribution with an untrusted entanglement source,” Phys. Rev. A 102(3), 032604 (2020). [CrossRef]  

13. S. Fossier, E. Diamanti, T. Debuisschert, A. Villing, R. Tualle-Brouri, and P. Grangier, “Field test of a continuous-variable quantum key distribution prototype,” New J. Phys. 11(4), 045023 (2009). [CrossRef]  

14. M. Li and M. Cvijetic, “Continuous-variable quantum key distribution with self-reference detection and discrete modulation,” IEEE J. Quantum Electron. 54(5), 1–8 (2018). [CrossRef]  

15. C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84(2), 621–669 (2012). [CrossRef]  

16. A. Leverrier, F. Grosshans, and P. Grangier, “Finite-size analysis of a continuous-variable quantum key distribution,” Phys. Rev. A 81(6), 062343 (2010). [CrossRef]  

17. F. Grosshans, “Collective attacks and unconditional security in continuous variable quantum key distribution,” Phys. Rev. Lett. 94(2), 020504 (2005). [CrossRef]  

18. A. Leverrier, “Composable security proof for continuous-variable quantum key distribution with coherent states,” Phys. Rev. Lett. 114(7), 070501 (2015). [CrossRef]  

19. Y. Zhang, Z. Li, Z. Chen, C. Weedbrook, Y. Zhao, X. Wang, Y. Huang, C. Xu, X. Zhang, Z. Wang, M. Li, X. Zhang, Z. Ziyong, C. Binjie, X. Gao, W. Meng, N. Cai, Z. Wang, Z. Wang, G. Wang, S. Yu, and H. Guo, “Continuous-variable qkd over 50 km commercial fiber,” Quantum Sci. Technol. 4(3), 035006 (2019). [CrossRef]  

20. B. Qi, L.-L. Huang, L. Qian, and H.-K. Lo, “Experimental study on the gaussian-modulated coherent-state quantum key distribution over standard telecommunication fibers,” Phys. Rev. A 76(5), 052323 (2007). [CrossRef]  

21. Y. Wang, Y. Mao, W. Huang, D. Huang, and Y. Guo, “Optical frequency comb-based multichannel parallel continuous-variable quantum key distribution,” Opt. Express 27(18), 25314–25329 (2019). [CrossRef]  

22. T. Wang, P. Huang, Y. Zhou, W. Liu, H. Ma, S. Wang, and G. Zeng, “High key rate continuous-variable quantum key distribution with a real local oscillator,” Opt. Express 26(3), 2794–2806 (2018). [CrossRef]  

23. Q. Liao, Y. Wang, D. Huang, and Y. Guo, “Dual-phase-modulated plug-and-play measurement-device-independent continuous-variable quantum key distribution,” Opt. Express 26(16), 19907–19920 (2018). [CrossRef]  

24. E. Newton, A. Ghesquière, F. L. Wilson, B. T. Varcoe, and M. Moseley, “Quantum secrecy in thermal states,” J. Phys. B: At., Mol. Opt. Phys. 52(12), 125501 (2019). [CrossRef]  

25. C. Weedbrook, S. Pirandola, S. Lloyd, and T. C. Ralph, “Quantum cryptography approaching the classical limit,” Phys. Rev. Lett. 105(11), 110501 (2010). [CrossRef]  

26. C. Weedbrook, S. Pirandola, and T. C. Ralph, “Continuous-variable quantum key distribution using thermal states,” Phys. Rev. A 86(2), 022318 (2012). [CrossRef]  

27. C. Weedbrook, C. Ottaviani, and S. Pirandola, “Two-way quantum cryptography at different wavelengths,” Phys. Rev. A 89(1), 012309 (2014). [CrossRef]  

28. J. M. Jornet and I. F. Akyildiz, “Channel modeling and capacity analysis for electromagnetic wireless nanonetworks in the terahertz band,” IEEE Trans. Wirel. Commun. 10(10), 3211–3221 (2011). [CrossRef]  

29. C. Jansen, S. Priebe, C. Moller, M. Jacob, H. Dierke, M. Koch, and T. Kurner, “Diffuse scattering from rough surfaces in THz communication channels,” IEEE Trans. Terahertz Sci. Technol. 1(2), 462–472 (2011). [CrossRef]  

30. C. Han, A. O. Bicen, and I. F. Akyildiz, “Multi-ray channel modeling and wideband characterization for wireless communications in the terahertz band,” IEEE Trans. Wirel. Commun. 14(5), 2402–2412 (2015). [CrossRef]  

31. C. Han and Y. Chen, “Propagation modeling for wireless communications in the terahertz band,” IEEE Commun. Mag. 56(6), 96–101 (2018). [CrossRef]  

32. T. Kaiser, F. Sheikh, and N. Zarifeh, “Terahertz band: Channel modelling for short-range wireless communications in the spectral windows,” IET Microw. Antennas Propag. 10(13), 1435–1444 (2016). [CrossRef]  

33. C. Ottaviani, M. J. Woolley, M. Erementchouk, J. F. Federici, P. Mazumder, S. Pirandola, and C. Weedbrook, “Terahertz quantum cryptography,” IEEE J. Sel. Areas Commun. 38(3), 483–495 (2020). [CrossRef]  

34. S. Priebe and T. Kurner, “Stochastic modeling of THz indoor radio channels,” IEEE Trans. Wireless Commun. 12(9), 4445–4455 (2013). [CrossRef]  

35. P. Panagiotis, O. Carlo, and P. Stefano, “Gaussian one-way thermal quantum cryptography with finite-size effects,” Phys. Rev. A 98(3), 032314 (2018). [CrossRef]  

36. T. Kurner and S. Priebe, “Towards THz communications - status in research, standardization and regulation,” J. Infrared, Millimeter, Terahertz Waves 35(1), 53–62 (2014). [CrossRef]  

37. Y. Choi, J. Choi, and J. M. Cioffi, “A geometric-statistic channel model for THz indoor communications,” J. Infrared, Millimeter, Terahertz Waves 34(7-8), 456–467 (2013). [CrossRef]  

38. B. Qi, P. G. Evans, and W. P. Grice, “Passive state preparation in the gaussian-modulated coherent-states quantum key distribution,” Phys. Rev. A 97(1), 012317 (2018). [CrossRef]  

39. X. Wu, Y. Wang, S. Li, W. Zhang, D. Huang, and Y. Guo, “Security analysis of passive measurement-device-independent continuous-variable quantum key distribution with almost no public communication,” Quantum Inf. Process. 18(12), 372 (2019). [CrossRef]  

40. X. Wu, Y. Wang, and D. Huang, “Passive continuous-variable quantum secret sharing using a thermal source,” Phys. Rev. A 101(2), 022301 (2020). [CrossRef]  

41. Y. Zheng, P. Huang, A. Huang, J. Peng, and G. Zeng, “Security analysis of practical continuous-variable quantum key distribution systems under laser seeding attack,” Opt. Express 27(19), 27369–27384 (2019). [CrossRef]  

42. C. C. Gerry, P. L. Knight, and M. Beck, “Introductory quantum optics,” Am. J. Phys. 73(12), 1197–1198 (2005). [CrossRef]  

43. F. Grosshans, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, “Virtual entanglement and reconciliation protocols for quantum cryptography with continuous variables,” Quantum Inform. Comput. 3, 535–552 (2003).

44. X. Wu, Y. Wang, H. Zhong, Q. Liao, and Y. Guo, “Plug-and-play dual-phase-modulated continuous-variable quantum key distribution with photon subtraction,” Front. Phys. 14(4), 41501 (2019). [CrossRef]  

45. W. Liu, J. Peng, P. Huang, D. Huang, and G. Zeng, “Monitoring of continuous-variable quantum key distribution system in real environment,” Opt. Express 25(16), 19429–19443 (2017). [CrossRef]  

46. W. Ye, H. Zhong, Q. Liao, D. Huang, L. Hu, and Y. Guo, “Improvement of self-referenced continuous-variable quantum key distribution with quantum photon catalysis,” Opt. Express 27(12), 17186–17198 (2019). [CrossRef]  

47. T. Wang, L. Li, M. Li, X. Wang, A. Lu, D. Zhang, and Z. Zhang, “Unidimensional continuous-variable quantum key distribution with noisy source,” Quantum Inf. Process. 19(1), 3 (2020). [CrossRef]  

48. R. Piesiewicz, C. Jansen, S. Wietzke, D. M. Mittleman, M. Koch, and T. Kurner, “Properties of building and plastic materials in the THz range,” J. Infrared, Millimeter, Terahertz Waves 28(5), 363–371 (2007). [CrossRef]  

49. R. Piesiewicz, T. Kleineostmann, N. Krumbholz, D. M. Mittleman, M. Koch, J. Schoebel, and T. Kurner, “Short-range ultra-broadband terahertz communications: concepts and perspectives,” IEEE Antennas Propag. Mag. 49(6), 24–39 (2007). [CrossRef]  

50. X. Liu, C. Zhu, N. Chen, and C. Pei, “Practical aspects of terahertz wireless quantum key distribution in indoor environments,” Quantum Inf. Process. 17(11), 304 (2018). [CrossRef]  

51. R. Piesiewicz, T. Kleineostmann, N. Krumbholz, D. M. Mittleman, M. Koch, and T. Kurner, “Terahertz characterisation of building materials,” Electron. Lett. 41(18), 1002–1004 (2005). [CrossRef]  

52. T. Nagatsuma, “Terahertz technologies: present and future,” IEICE Electron. Express 8(14), 1127–1142 (2011). [CrossRef]  

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (12)
Fig. 1.
Fig. 1. The channel model between the transmitter (TX) and the receiver (RX). (a) Optical fiber channel. (b) Indoor channel with a table. (c) Indoor channel without a table.

View in Article | Download Full Size | PDF

Fig. 2.
Fig. 2. Channel loss in dB as a function of the frequency and the distance under three different water-vapor density (a) $\rho$=0 g/m$^3$, (b) $\rho$=7.5 g/m$^3$ and (c) $\rho$=40 g/m$^3$. The maximum values of channel loss have been cut off at 120 dB since only few values exceeds it.

View in Article | Download Full Size | PDF

Fig. 3.
Fig. 3. The reflection loss (dB) versus the incident angle under different parameters $f$ and $\sigma$ for different materials (a) plaster and (b) wood. The solid lines show reflection loss for $\sigma$=0.05 mm while dashed line for $\sigma$=0.08 mm.

View in Article | Download Full Size | PDF

Fig. 4.
Fig. 4. Schematic diagram of (a) active CVQKD and (b) passive CVQKD with an indoor channel. BS$_{1}$/BS$_{2}$, 50:50 beam splitter; HD, homodyne detector; Att., optical attenuator. AM/PM, amplitude and phase modulator. Beam splitters with a transmittance of $\eta _{H}$ are used to model the efficiency of the homodyne detectors.

View in Article | Download Full Size | PDF

Fig. 5.
Fig. 5. The secret key rates of the indoor CVQKD system. (a) $f=410$ GHz for the active CVQKD. (b) $f=850$ GHz for the active CVQKD. (c) $f=410$ GHz for the passive CVQKD. (d) $f=850$ GHz for the passive CVQKD. In (c) and (d), the blue line divides coverage map into the origin region ($V_{E}$=5) and the enlarged region ($V_{E}$=2). Other parameters are given by $x_{t}$=4.875 m, $y_{t}$=0.125 m, $z_{t}$=2.3 m, $A$=12.5 cm$^{2}$, and $\rho$=7.5 g/m$^{3}$.

View in Article | Download Full Size | PDF

Fig. 6.
Fig. 6. The ratio of channel transmittance $T_{\textrm {LOS}}$ to $T_{\textrm{NLOS}}$ as a function of antenna aperture in points $K_1$ and $K_2$ (see Fig. 5(a)).

View in Article | Download Full Size | PDF

Fig. 7.
Fig. 7. The secret key rates of the indoor CVQKD system for height of $z$=1 m for (a) With reflection loss and (b) Without reflection loss. Other parameters are given by $x_{t}$=4.875 m, $y_{t}$=0.125 m, $z_{t}$=2.3 m, $f$=850 GHz, $A$=12.5 cm$^{2}$, and $\rho$=7.5 g/m$^{3}$.

View in Article | Download Full Size | PDF

Fig. 8.
Fig. 8. The secret key rate of the active CVQKD system without the multi-path propagation for (a) $f$=410 GHz and (b) $f$=850 GHz. The solid, dashed and dash-dotted lines denote $\rho$=7.5 g/m$^{3}$, 10 g/m$^{3}$, and 20 g/m$^{3}$, respectively.

View in Article | Download Full Size | PDF

Fig. 9.
Fig. 9. (a) The secret key rate of the passive CVQKD system. The red, green and blue lines denote $\rho$=7.5 g/m$^3$, 10 g/m$^3$ and 20 g/m$^3$ respectively. (b) The secret key rates of the passive CVQKD system with the average photo numbers $n_{0}$, 10$n_{0}$, 100$n_{0}$, and 1000$n_{0}$.

View in Article | Download Full Size | PDF

Fig. 10.
Fig. 10. The maximum transmission distance of the passive CVQKD system as a function of frequency $f$ and transmittance of optical attenuator $\eta$. There are total 5 peaks corresponding to $300$ GHz, $340$ GHz, $410$ GHz, $460$ GHz, and $480$ GHz respectively. The simulation results over 550 GHz and the impact of reconciliation efficiency are both shown in Appendix C.

View in Article | Download Full Size | PDF

Fig. 11.
Fig. 11. The 2D top view of the investigated indoor scenario. The transmitter is located at the position $x$=4.875 m, $y$=0.125 m, $z$=2.3 m and the NLOS region is marked by shaded area.

View in Article | Download Full Size | PDF

Fig. 12.
Fig. 12. The maximum transmission distance of the passive CVQKD system as a function of frequency $f$ and transmittance of optical attenuator $\eta$. (a) 300-550 GHz, $\beta$=0.98. (b) 550-1000 GHz, $\beta$=1. For (a), there are total 5 peaks corresponding to 300 GHz, 340 GHz, 410GHz, 460 GHz, and 480 GHz respectively while for (b) 3 peaks corresponding to 670 GHz, 850 GHz and 930 GHz respectively.

View in Article | Download Full Size | PDF

Tables (1)

Tables Icon

Table 1. Parameter setting.

View Table

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

h ( τ ) = α LOS δ ( τ − τ LOS ) + ∑ k = 1 N NLOS α NLOS ( k ) δ ( τ − τ NLOS ( k ) ) ,
h ( τ ) = α LOS δ ( τ − τ LOS ) + ∑ k = 1 N ref α ref ( k ) δ ( τ − τ ref ( k ) ) ,
T Indoor = T LOS + ∑ k = 1 N T NLOS ( k ) ,
V S = 2 n 0 + 1 ,
V = V S + V M 1 ,
x B 2 = η 2 x B 0 + 1 − η 2 x v 1 ,
x B 3 = η H 4 x B 0 + 1 − η h 4 x v 2 + ξ e l ,
ϵ = 2 η η H ( 1 + v e l − η H 2 ) + 1 ,
T M = 10 − α d LOS 10 ,
T F = ( χ 4 π d LOS ) 2 ,
T LOS = T F T M = ( χ 4 π d LOS ) 2 10 − α d LOS 10 .
T LOS = ( χ 4 π d LOS ) 2 10 − α d LOS 10 G T X G R X ,
G T X = G R X = 4 π δ A λ 2 ,
T NLOS = ∑ k = 1 N ψ r NLOS ( k ) ( χ 4 π d NLOS ( k ) ) 2 10 − α d NLOS ( k ) 10 G T X G R X ,
r NLOS = Z cos ⁡ θ i − Z 0 θ t Z cos ⁡ θ i + Z 0 θ t ,
Z ≈ 377 n ,
r NLOS ≈ − exp ⁡ ( − 2 cos ⁡ θ i n 2 − 1 ) .
μ = ( 4 π σ cos ⁡ θ i λ ) 2 ,
Select as filters
Select Topics Cancel
Top
       
© Copyright 2024 | Optica Publishing Group. All rights reserved, including rights for text and data mining and training of artificial technologies or similar technologies.
Privacy | Terms of Use