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Abstract
Continuous-variable quantum key distribution (CVQKD) can be effectively compatible with off-the-shelf communication systems and has been proven to be the security against collective attacks in the finite-size regime and composability. In this paper, we classify three different trust levels for the loss and noise experienced by the sender and receiver. Based on these trust levels, we derive the composable finite-size security bounds of inter-satellite CVQKD in the terahertz (THz) band. We also show how these trust levels can nontrivially increase the composable secret key rates of THz-CVQKD and tolerate higher loss. Furthermore, the numerical simulations strongly support the feasibility of inter-satellite THz-CVQKD even in the worst trust level. This work provides an efficient path for building an inter-satellite quantum communication network.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Quantum key distribution (QKD) [1–6] based on the fundamental laws of quantum mechanics can share unconditional secure keys between two authenticated parties (Alice and Bob) over an insecure channel controlled by a malicious eavesdropper (Eve).
Generally, QKD is further divided into two categories according to different implementation approaches, i.e., continuous-variable (CV) QKD [7–13] and discrete-variable (DV) QKD [14–22]. In DVQKD schemes, they typically use single-photon degrees of freedom to encode key information and recover it by single-photon detectors. As an alternative of its DVQKD counterpart, CVQKD encodes key information on the field quadratures of light and can be easily integrated with off-the-shelf optical devices such as homodyne detectors, avoiding the imperfection of employing single-photon counting [9,13]. At present, the theoretical security of CVQKD against collective attacks has been demonstrated in the case of the asymptotic limit, finite-size regime and composable security framework (i.e., there is a associated error in each step of the protocol, which adds to the total “epsilo”-security) [23–29]. The composable security of QKD means that if the protocol is $\varepsilon$-correctness ($\varepsilon _\mathrm {cor}$), $\varepsilon$-secrecy ($\varepsilon _\mathrm {sec}$), and $\varepsilon$-robustness ($\varepsilon _\mathrm {rob}$) then it is $\varepsilon$-security, for $\varepsilon =\varepsilon _\mathrm {cor}+\varepsilon _\mathrm {sec}+\varepsilon _\mathrm {rob}$[28]. The first composable security proof for CVQKD was proposed by Leverrier [29], and some subsequent related studies further improved and developed the composable security of CVQKD [30–34].
However, some realistic assumptions in experiments may elude the assessment of composable security. For instance, we may assume that Eve does not control the internal noise of the receiver, that is, she cannot actively tamper with the receiver; thus, the electronic noise of the detector can be considered as trusted, which is reasonable assumption often made by experimentalists. Moreover, Eve may be subject to some realistic constraints, e.g., it is considered as a passive attack in the case of the line-of-sight free-space implementations (Eve is unlikely able to tamper with the channel since the transmitter and receiver are in line-of-sight). As a result, it is necessary to consider various possible trust levels to address these situations.
Recently, Pirandola [34] proposed a general framework for composable security CVQKD under different assumptions of Eve’s ability. Based on this framework, the security can be classified into three trust levels for these assumptions according to the main sources of noise in the communication scenario. The first one is the most trustful scenario where Eve is assumed to be able to attack only the external channel, that is, she is excluded from side-channels to the receiver; thus, in this case, the loss and noise of the detector are considered as trusted. Then the second one is the more general case where the noise of the detector is trusted rather than its loss, which corresponds to Eve’s ability to collect leakage from the receiver but cannot control the receiver. Finally, the third one is the worst-case (traditional) scenario, where none of the loss and noise in Bob’s receiver setup is trusted; this means that in addition to attacking the external channel, Eve may also perform an active side-channel attack on the receiver. For different trust levels, the composable secret key rates (SKRs) of CVQKD will be nontrivially increased compared with the traditional scenario and can tolerate higher loss.
Traditional CVQKD usually processes information through optical communication or microwave communication systems. In other words, so far, most CVQKD protocols take advantage of optical or microwave photons to carry the key information over telecom fiber or free space channels for transmission. With the rapid development of wireless communication, wireless spectrum resources appear increasingly scarce. It is known that terahertz (THz) communication [35–43] due to the large availability of its bandwidth is considered to have great potentials to meet the needs of high-speed data transmission. THz communication can apply to the indoor channel for high-speed wireless communication [35–38,40]; but the THz band is easily absorbed by the pervasive atmospheric water molecules, so the transmission distance is greatly limited [36,44]. Fortunately, it has an enormous potentiality for inter-satellite communication [39,45] since the concentration of water molecules in the vacuum can be negligible and some previous works have studied the feasibility [38,39]. Moreover, compared with the microwave band [46], the environmental thermal noise of the THz band is relatively lower, meaning that the security threshold of the THz band is more higher. Thus, it is imperative to investigate the THz communication to build a high-speed inter-satellite quantum communication network.
Inspired by Pirandola’s general framework [34] which contains various assumptions of Eve’s ability, we classify the security into three different trust levels according to the loss and noise experienced by the sender and receiver and analyze the composable finite-size security of inter-satellite CVQKD in the THz band based on these trust levels. Moreover, we show how these trust levels can nontrivially increase the composable SKRs and tolerate higher dBs. We also verify the feasibility of the long-distance inter-satellite THz-CVQKD even in the worst trust level.
This paper is organized as follows. In Section 2, we introduce details of THz-CVQKD under three different trust levels. In Section 3, we analyze the various sources of noise in inter-satellite links. Section 4 analyzes the SKR of THz-CVQKD. The performance of the protocol under three different trust levels in the composable finite-size regime is analyzed in Section 5. Finally, in Section 6, we draw the conclusions.
2. THz continuous-variable quantum key distribution and three different trust levels in inter-satellite links
2.1 Schematic setup of the THz-CVQKD system and three different trust levels
The schematic setup of the THz-CVQKD system under three different trust levels is shown in Fig. 1. Here, a typical Gaussian-modulated thermal-state (GMTS) protocol [10] between the transmitter (Alice) and the receiver (Bob) is considered in the THz band. Note that the local oscillator (LO) is not shown in Fig. 1, since two different cases of LO configuration are taken into account (see Section 3.2 for details). The detailed process of the protocol is shown as follows.
At Alice, she uses the THz source to generate the thermal state in the mode $\hat {x_0}$ and then performs the Gaussian modulation with quadrature components $q$ and $p$ on $\hat {x_0}$, resulting in signal mode $\hat {x}=\hat {q}$ or $\hat {p}$. The mode $\hat {x}$ can be given by
where $x_a=q$ or $p$ denotes the real variable for the encoded information, $\hat {x}_{\mathrm {0}}$ denotes the quantum noise related to the thermal mode. The overall variance of mode $\hat {x}$ can be given by where $V_A$ is the modulation variance obeying the zero-centered Gaussian distribution. $V_0$ is the shot noise given by [10] where where $f$ denotes the carrier frequency, $h$ denotes Planck’s constant, $k_\tau$ denotes the temperature in Kelvin, and $k_b$ denotes Boltzmann’s constant. The shot noise derives from environment thermal fluctuations and imperfect modulation. As a rule, the shot noise $V_0$ becomes $V_0=1+\rho$, where $\rho =0$ is for the coherent state case and $\rho >0$ is for the thermal state case.The signal mode $\hat {x}$ is sent through lossy and noisy channel controlled by Eve. The channel can be characterized by transmittance $T$ and average number of photons $\bar {n}_{c}$ with the variance $W=2\bar {n}_{c} +1$. Note that the evaluation of $\bar {n}_{c}$ is associated with environmental thermal noise $\bar {n}_{E}$ under different trust levels. Based on different assumptions for Eve, we model three different trust levels for the practical application scenarios. The corresponding eavesdroppers are Eve${_1}$, Eve${_2}$, and Eve${_3}$ as shown in Fig. 1. The details are as follows.
1. Trusted-Loss and Trusted-Noise (TLTN) detector. Eve is assumed to be unable to control Bob’s receiver and can only attack the external channel. In this case, the loss and noise are considered as trusted in Bob’s receiver. Thus, the collective Gaussian attacks performed by Eve are considered to be a purification by taking advantage of an environmental beam splitter (BS) with transmittance $T$. The number of injected photons $\bar {n}_{c_1}$ can be regarded as part of Eve’s two-mode squeezed vacuum (TMSV) state given by
2. Trusted-Noise (TN) detector. In this more general case, it is further assumed that Eve can passively collect the leakages from Bob’s receiver but is unable to actively control the internal noise. Therefore, the noise of receiver can be considered to be trusted which is a feasible assumption often made for the electronic noise of the detector. In this scenario, Bob’s detection efficiency $\eta$ together with channel transmittance $T$ will become the transmittance $\mu =T \eta$ of Eve’s environmental BS. The number of injected photons $\bar {n}_{c_2}$ can be given by 3. Untrusted-Noise (UN) detector. In this pessimistic case, Eve not only can collect the leakages but also actively control the extra photons $\bar {n}_{\mathrm {ex}}$ in Bob’s receiver. Therefore, the extra photons $\bar {n}_{\mathrm {ex}}$ will become untrusted. Eve actually performs an active side-channel attack and injects the photons $\bar {n}_{c_3}$ given byAt Bob, he performs the homodyne or heterodyne detection on the received quantum signals. Bob’s setup can be characterized by detection efficiency $\eta$ and extra noise variance (phase errors, electronic noise, etc.) $V_{\mathrm {ex}}=2\bar {n}_{\mathrm {ex}}$, where $\bar {n}_{\mathrm {ex}}$ denotes the equivalent number of the photons caused by imperfections of the setup.
From an overall point of view, we can take advantage of the classical input-output relation to describe the communication scenario between Alice and Bob. In both homodyne and heterodyne protocols, Alice inputs the signal $x$, after being transmitted through the channel, Bob recovers the signal $y$. The initial average photons $\bar {n}_{T}$ at Alice are attenuated by an overall factor $\mu =T\eta$, so the overall average photons $\bar {n}_{R}$ observed by Bob’s detector can be expressed as
where $\bar {n}=\eta \bar {n}_{E} + \bar {n}_{\mathrm {ex}}$ is the average photons associated with total excess noise generated by the various imperfect sources. For Bob’s detector, it will randomly perform homodyne detection [7], measuring the quadrature $\hat {q}$ or $\hat {p}$, or heterodyne detection [47], measuring both the quadratures $\hat {q}$ and $\hat {p}$. In both cases, Bob can recover an outcome $y$ corresponding to Alice’s input $x$. The relation for the input $x$ to the output $y$ is given by where $t$ is the total noise variable given by2.2 Channel model
Let us consider a free-space inter-satellite link between Alice and Bob. As we mentioned in the introduction, THz waves will be impaired by atmospheric absorption, turbulence, scattering, etc, when they are propagated through a free-space channel. These atmospheric effects (especially the absorption of atmospheric water molecules) seriously reduce the communication performance of THz waves. However, in the vacuum, the atmospheric absorption is almost insignificant and the beam drift effect can be practically neglected. This allows us to approximatively characterize a diffraction-only channel model as a fixed attenuation. The loss derived from diffraction effects is caused only by the size of the diffracted beam at the receiver aperture. Therefore, the transmittance $T$ in a free-space inter-satellite link is given by [39]
where $a{_R}$ denotes the radius of the receiver aperture. $d$ denotes the transmission distance of the beam. $\mathcal {L}(d)$ denotes the beam radius at transmission distance $d$. Taking advantage of the Gaussian approximation, $\mathcal {L}(d)$ is given by where $w_0$ denotes the radius of the beam-waist, and $\lambda$ denotes the wavelength. Note that the receiver aperture and the beam waist are impacted by the limitations of the realistic hardware. In this channel model, we set both the radius of the receiver aperture $a{_R}$ and the beam-waist radius $w_0$ to 60 cm and the temperature of the environment to 30 K [48].3. Noise analysis for THz-CVQKD inter-satellite links
The noises from various imperfect implementations have a significant impact on the performance of THz-CVQKD. Therefore, a detailed analysis of the noise sources is necessary. It is remarkable that we focus on the environmental thermal noise and receiver noise here.
3.1 Environmental thermal noise
In this section, we analyze the basic theoretical model of environmental thermal noise $\bar {n}_{E}$ affecting inter-satellite links. The quantum communication scenario is depicted in Fig. 1, where the channel transmittance $T$ is primarily determined by free-space diffraction and the environmental thermal noise $\bar {n}_{E}$ needs to be evaluated carefully according to the sky brightness. Then, with the trust levels $i = 1, 2, 3$, we can naturally assume standard security, in which Eve’s interaction is defined by various effective beam-splitters with different amounts of input thermal photons $\bar {n}_{c_i}$.
The evaluated value of $\bar {n}_{E}$ is influenced by sky brightness $H_{\lambda }^{\text{sky }}$ and features of Bob’s receiver. Assume the receiver possesses a detector with spectral filter $\mathrm {\Delta }_\lambda$ and bandwidth $W_d$. Assume the receiver has a shielding device so that the field of view is not directly exposed to the bright sources. Then, the average number of environmental thermal photons per mode received by Bob can be expressed as [49,50]
where where $\mathrm {\Theta }_{\mathrm {fov}}$ is the receiver angular field of view, $\mathrm {\Delta }_v$ is the filter bandwidth and $c$ is the propagation speed of the light in the vacuum. We set the typical value of $H_{\lambda }^{\text{sky }}$ to $1.5\times 10^{-6}$ $\mathrm {W} \mathrm {m}^{-2} \mathrm {~nm}^{-1} \mathrm {sr}^{-1}$ for clear night [49]. The typical value of the angular field of view in long-range satellite communications is $\mathrm {\Theta }_{\mathrm {fov}} \simeq 10^{-10}$ $\mathrm {sr}$[51]. The other parameters are listed in Table 1.
Table 1. Protocol parameters.
3.2 Local oscillator and receiver noise
In CVQKD, LO contains the phase information which is essential for the parties to take advantage of the two quadratures of the mode. That is to say, the phase-locking is necessary for the rotating reference frames of the parties so that Bob can measure the same quadratures of the state sent by Alice. There are two techniques to accomplish this goal, one is the transmitted LO (TLO) which is the simplest solution and the other is the local LO (LLO) which is the more challenging but more secure. The average number of photons associated with thermal noise in Bob’s receiver can be expressed as
where $\bar {n}_{\mathrm {phase}}$ is the average number of photons generated by the LO phase noise, $\bar {n}_{\mathrm {e}}$ is the average number of photons related to electronic noise of detector, and $\bar {n}_{\text{r }}$ is the average number of photons generated by the other imperfection noise (here neglected).Generally, the estimation of $\bar {n}_{\mathrm {ex}}$ is directly related to the chosen LO. For a TLO, the phase noise can be ignored, but since the LO is attenuated during the transmission process, the electronic noise of the detector becomes nontrivial. On the contrary, for an LLO, since the brightness of the LO is independent of the transmittance, the electronic noise of the detector is lower; however, some nonzero phase noise will be generated by the imperfect restoration of the rotating reference frame in Bob’s receiver. The average number of photons for LLO phase noise can be given by [52]
where $W_{\mathrm {line}}$ denotes the linewidth of the laser and $C$ is the clock of the system. The estimation for the electronic noise of the detector strictly depends on the type of LO given byTherefore, the total number of photons for the thermal noise in the receiver can be given by
4. Analysis of the secret key rate
In this section, we analyze the asymptotic SKR and the composable finite-size SKR of THz-CVQKD respectively according to the three trust levels described in Section 2. First, we consider the most optimistic situation, that is, the noise and loss are trusted (Eve$_{1}$ in Fig. 1). Then the more general situation is considered, i.e., trusted-noise detector (Eve$_{2}$ in Fig. 1). Finally, we analyze the worst-case scenario where all excess noise is untrusted (Eve$_{3}$ in Fig. 1).
4.1 Asymptotic secret key rate with a TLTN detector
As described in Section 2, we assume Eve cannot collect the leakage from Bob’s receiver but can only attack the external channel, which makes the noise and the loss of the receiver be trusted. In this case, the transmittance of BS in Eve’s attack is $T$ and the input thermal photons is $\bar {n}_{c_1}=\frac {\bar {n}_{E}}{1-T}$. For reverse reconciliation, the asymptotic SKR with a TLTN detector can be calculated by
where $I_{A B}$ is Alice and Bob’s mutual information, $\beta \in [0,1]$ is the efficiency of reverse reconciliation, and $\chi _{B E}$ is the Holevo bound between Bob and Eve.The mutual information $I_{A B}$ can be given by [13]
The expression of $\chi _{B E}$ is defined as [10]
where $S$ stands for the von Neumann entropy which is calculated from the symplectic eigenvalues of the corresponding covariance matrices $\gamma _{E}$ and $\gamma _{({E|B})}$. The expression of $S$ is given by whereTo calculate the symplectic eigenvalues, we need to derive the covariance matrices $\gamma _{E}$ and $\gamma _{({E|B})}$ which can be calculated by
The symplectic eigenvalues of $\gamma _{E}$ and $\gamma _{({E|B})}$ are easily calculated by the symplectic diagonalization [53]. Therefore, $\chi _{BE}$ can be calculated by using Eq. (27) and Eq. (30)–(36) and then the asymptotic SKR with a TLTN detector can be obtained.
4.2 Asymptotic secret key rate with a TN detector
In the case of TN detector, Eve can collect the leakage from Bob’s receiver which actually performs a passive side-channel attack. The number of the injected thermal photons becomes $\bar {n}_{c_2}=\frac {\eta \bar {n}_{E}}{1-\mu }$ with the total transmittance $\mu =T \eta$. The asymptotic SKR for reverse reconciliation can also be calculated by using Eq. (25). The mutual information $I_{A B}$ can be estimated by using Eq. (26). Based on the simple modification of the previous derivation, the Holevo bound $\chi _{BE}$ can be calculated by
4.3 Asymptotic secret key rate with an UN detector
In this case, since Eve can actively control the extra photons $\bar {n}_{\mathrm {ex}}$ in the receiver, $\bar {n}_{ex}$ becomes untrusted, resulting in all excess noise is now considered to be untrusted. The number of the injected photons is sufficient to replace $\bar {n}_{c_2}=\frac {\eta \bar {n}_{E}}{1-\mu }$ $\rightarrow$ $\frac {\eta \bar {n}_{E} + \bar {n}_{\mathrm {ex}}}{1-\mu }=\bar {n}_{c_3}$ in Eq. (38) and the other elements of the covariance matrix $\gamma _{E}$ will be implicitly modified in Eq. (37). The expression of $W$ is given by
the rest of the expressions and calculation steps are the same as the asymptotic SKR with a TN detector. However, in this worst-case, Eve is assumed to hold the purification of the state $\rho _{AB}$ (Alice and Bob’s shared Gaussian state) with the covariance matrixIn order to calculate the value of $\chi _{BE}$, we need to derive covariance matrix corresponding to $S_{AB}$ and $S_{({E|B})}$. The former is given by Eq. (43) and the latter is expressed as
4.4 Composable finite-size secret key rate
The composable finite-size SKR of the protocol with reverse reconciliation is given by [52]
The expression of $\mathrm {\Delta }_{A E P}\left (\varepsilon _{s},|\mathcal {H}|\right )$ and $\Theta$ are given by
All these parameters provide the overall security parameter [52]:
5. Performance analysis
In this section, we analyze the performance of THz-CVQKD under three different trust levels in inter-satellites links. Note that the asymptotic SKR takes advantage of the unlimited data length which is impossible to be achieved in practice. Thus, we also analyze the composable finite-size SKR which is more practical than that obtained in the asymptotic limit. The protocol parameters are presented in Table 1.
The plot of Fig. 2 shows the composable finite-size SKR as a function of the frequency and the transmission for three different trusted levels. We can obtain that as the frequency increases, the maximum transmission distance increases dramatically. Assume that the transmission distance between satellites is at least 200 km; from Fig. 2(a) and Fig. 2(c), it is observed that in the case of UN-detector, the frequency needs to attain 26.7 THz in the heterodyne protocol and 17.4 THz in the homodyne protocol for a 200 km transmission distance. Thus, it is necessary to extend the traditional THz frequency (0.1 THz-10 THz) to the higher frequency bands (mid-infrared and far-infrared bands) to study the performance of the system. However, for a practical inter-satellite communication system, the higher the frequency, the smaller the beam radius. In other words, the high-frequency system needs a high-precision acquisition, pointing, and tracking subsystem to communicate with the receiver. In this work, we will extend the THz frequency to a feasible band (up to 50 THz) to analyze the performance of the protocol.
Fig. 2. The composable finite-size SKR as a function of the frequency and the transmission for three different trusted levels with LLO. (a) and (b) for the heterodyne protocol. (c) and (d) for the homodyne protocol. Simulation parameters are $V_A=10$, $\eta =0.7$, $\beta = 0.98$, and the block size $N=10^9$.
In view of Fig. 2, we can also observe that within a certain distance, the SKR of the heterodyne protocol is higher than that of the homodyne protocol, but the maximum transmission distance of the homodyne protocol is higher than that of the heterodyne protocol. This is because the heterodyne protocol introduces more receiver noise ($\xi _{\mathrm {ex}}$) as the transmittance decreases (See Fig. 3). Compared with the UN-detector, the SKR and the transmission distance of TLTN and TN-detector have been significantly improved, which further increases the practicability in communication systems. For both protocols, the maximum transmission distance in the cases of TLTN and TN-detector can exceed 1000 km. Interestingly, from Fig. 3(b) and Fig. 3(d), we can find that for the worst case (UN-detector), the maximum transmission distance is almost 600 km for the heterodyne protocol and is nearly 800 km for the homodyne protocol. The results shown here strongly support the feasibility of inter-satellite THz quantum communication under different trust levels.
Fig. 3. The relationship between the receiver noise $\xi _{\mathrm {ex}}$ and the loss by using LLO and TLO for the heterodyne or homodyne protocol. Simulation parameters are $V_A=10$, $\eta =0.7$, and $f= 50$ THz.
Fig. 3 illustrates the relationship between the receiver noise $\xi _{\mathrm {ex}}$ and the loss by using LLO and TLO in the homodyne or heterodyne protocol. It is clear that as the loss increases, the receiver noise using TLO increases rapidly. In inter-satellites links, the receiver noise generated by using LLO is smaller. This is because the receiver noise using TLO is positively correlated with $1 /\mu$, which will generate more noise in long-distance transmission. We can also obtain that the noise using the homodyne protocol is smaller than that of the heterodyne protocol.
To investigate the effects of the detection efficiency $\eta$ on the receiver noise $\xi _{\mathrm {ex}}$ with LLO, we make the frequency $f=50$ THz as an example and the other frequency values can be analyzed in the same way. As depicted in Fig. 4, we can find that as the modulation variance and transmission distance increase, the receiver noise increases. The higher detector efficiency makes receiver noise lower. The homodyne protocol has better noise immunity in long-distance transmission than that of the heterodyne protocol. Thus, for inter-satellites links, the homodyne protocol using LLO may optimally reduce the receiver noise.
Fig. 4. The receiver noise $\xi _{\mathrm {ex}}$ as a function of the transmission distance and the modulation variance with LLO and $f=50$ THz.
After the analysis of the receiver noise, we add the environmental thermal noise $\bar {n}_{E}$ on the system to further analyze the effects of the total excess noise, as shown in Fig. 5. It is observed that the homodyne protocol has also lower total noise in long-distance transmission than that of the heterodyne protocol. Note that when the detector efficiency is changed from 0.7 to 0.4, the total excess noise increases dramatically as the distance increases. It is worth mentioning that when $d=1000$ km, the total excess noise of the homodyne protocol is increased from 0.12 ($\eta =0.7$) to 0.2 ($\eta =0.4$) . Thus, the detector efficiency is sensitive to the influence of the total excess noise transmitted over a long distance.
Fig. 5. The relationship between the total excess noise and the transmission distance with LLO, $f=50$ THz and $V_A=10$.
In the studies below, we plot the relationship between the composable finite-size SKR and the transmission distance with different block sizes as illustrated in Fig. 6. It is clear that in the finite-size-UN case, the maximum transmission distance increases as the block size $N$ increases and gradually approaches the asymptotic SKR-UN case. From Fig. 6, we obtain that the maximum transmission distance for the heterodyne protocol in the case of asymptotic SKR-UN is 660 km [shown in Fig. 6(a)] and is 900 km for the homodyne protocol [shown in Fig. 6(b)]. Increasingly, we observe that with the increase of transmission distance, the SKR of finite-size-TLTN gradually approaches that of finite-size-TN. This is because the number of photons for the thermal noise $\bar {n}_{\mathrm {ex}}^{\mathrm {LLO}}$ decreases as the transmittance $\mu$ decreases, making $\bar {n}_{c_3} \simeq \bar {n}_{c_2}$. It is remarkable that we also plot the Pirandola-Laurenza-Ottaviani-Banchi (PLOB) bound [54] to make a detailed comparison. This bound illustrates the ultimate limit of repeater-less quantum communication.
Fig. 6. The relationship between the composable finite-size SKR and the transmission distance with different block size. (a) LLO for the heterodyne protocol. (b) LLO for the homodyne protocol. Simulation parameters are $V_A=10$, $\eta =0.7$, $\beta = 0.98$ and $f=50$ THz.
In addition, the relationship between the composable finite-size SKR and the transmission distance with different modulation variances is depicted in Fig. 7. It is observed that the modulation variance $V_A$ is sensitive to the performance of the system. The maximum transmission distance decreases dramatically as the modulation variance increases. For the case of finite-size-UN, when $V_A$ is changed from 10 to 100, the maximum transmission distance decreases dramatically from 606.3 to 71.9 km for heterodyne protocol [shown in Fig. 7(a)] and decreases from 787.3 to 135.9 km for homodyne protocol [shown in Fig. 7(b)]. It is worth mentioning that for the finite-size-TLTN and TN cases, our protocol can still achieve a reasonable transmission distance even in a relatively high level of the modulation variance ($V_A=100$).
Fig. 7. The relationship between the composable finite-size SKR and the transmission distance with different modulation variance. (a) LLO for the heterodyne protocol. (b) LLO for the homodyne protocol. Simulation parameters are $\eta =0.7$, $\beta = 0.98$ , $N=10^9$ and $f=50$ THz.
To achieve the maximum value of the SKR, we further analyze the influence of the modulation variance on protocol performance. As the plot in Fig. 8 and Fig. 9, the optimal areas of $V_A$ can be obtained in different scenarios. In Fig. 8, we plot the relationship between the composable finite-size SKR and $V_A$ with different block sizes. It is observed that as the block size increases, the optimal areas of $V_A$ increase. It is remarkable that for a fixed distance, we can find a public optimal value of $V_A$. Here we take the heterodyne protocol [Fig. 8(a)] as an example to study the optimal $V_A$, and the homodyne protocol [Fig. 8(b)] can be analyzed in the same way. For the heterodyne protocol, when $d=500$ km, the public optimal value of $V_A$ in the finite-size-UN case can be set to 6, and when $d=1000$ km, the optimal $V_A$ is 7.8 in both the finite-size-TN and TLTN cases.
Fig. 8. The relationship between the composable finite-size SKR and the modulation variance with different block size. (a) LLO for the heterodyne protocol. (b) LLO for the homodyne protocol. Simulation parameters are $\eta =0.7$, $\beta = 0.98$ and $f=50$ THz.
Fig. 9. The relationship between the composable finite-size SKR and the modulation variance with different detector efficiency. (a) LLO for the heterodyne protocol. (b) LLO for the homodyne protocol. Simulation parameters are $\beta = 0.98$, $N=10^8$ and $f=50$ THz.
Due to the detector efficiency playing an important role in the protocol performance, we focus on the optimal areas of $V_A$ with different detector efficiency in Fig. 9. It is clear that as the detector efficiency decreases, the optimal areas of $V_A$ are constricted gradually. Fortunately, for a fixed distance, we can still achieve a public optimal $V_A$; from Fig. 9(a), when $d=900$ km, the optimal $V_A$ can be set to 8 for the case of finite-size-UN and when $d=350$ km, the optimal $V_A$ is 6.8 in both the finite-size-TN and TLTN cases. For the homodyne protocol, we can make a similar analysis from Fig. 9(b).
In the following, we plot the relationship between the SKR and reconciliation efficiency with different transmission distances as depicted in Fig. 10, Note that we take the case of UN-detector here as an example to show the usable range of reconciliation efficiency in the finite-size scenario, as other cases of trust levels can be evaluated in the same way. We can observe that the required reconciliation efficiency increases as the transmission distance increases. Fortunately, our protocol (with transmission distance $d=400$ km) can still achieve a reasonable finite-size SKR with low reconciliation efficiency ($\beta =0.92$) even in the case of UN-detector. However, when $d \geq 600$ km, the reconciliation efficiency needs to exceed 0.97. Thus, it is necessary to take advantage of an efficient reconciliation efficiency for long-distance inter-satellite communication.
Fig. 10. The relationship between the composable finite-size SKR and the reconciliation efficiency for the heterodyne protocol with LLO and UN-detector. Simulation parameters are $V_A=10$, $\eta =0.7$, $N=10^9$ and $f=50$ THz.
6. Conclusions
We have analyzed the composable finite-size SKR that are achievable assuming three different trust levels for Bob’s receiver setup, from the case where the loss and noise of the detector are considered to be trusted to the worst case of the fully-untrusted detector. We have also shown how these levels of trust for realistic assumptions on the receiver can nontrivially increase the composable SKRs and the transmission distance. Interestingly, we have demonstrated the feasibility of the long-distance inter-satellite THz-CVQKD even in the worst case of the fully-untrusted detector. In particular, the results show that the secure transmission distance can be almost 600 km for the heterodyne protocol and 800 km for the homodyne protocol in the UN-detector case. Besides, we have also analyzed various parameters that may optimize the SKR. However, in the process of practical application, the optimal performance needs to combine various technologies, e.g., accurate parameter estimation algorithm, high gain antenna, and noise suppression, etc. These are what we need to overcome in the future.
Funding
National Natural Science Foundation of China (62001351, 61372076, 61971348); Foundation of Shaanxi Key Laboratory of Information Communication Network and Security (ICNS201802); Natural Science Basic Research Program of Shaanxi Province (2021JM-142); Key Research and Development Program of Shaanxi Province (2019ZDLGY09-02).
Acknowledgments
The authors would like to thank Xu Hou for helpful discussions.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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