INTRODUCTION

Theoretically, quantum cryptography promises unconditional informational security. In 1984, Bennett and Brassard developed a quantum key distribution (QKD) scheme, BB84 [1], with its security guaranteed by the no-cloning theorem and one-time pad encryption. This was the first discrete variable QKD (DV-QKD) protocol, which uses single photons in transmission. This has put forward rigorous conditions for realistic implementations. Some variations within the same category include measurement-device-independent protocols [2,3], decoy-state protocols [4,5], high-dimensional protocols that exploit different carriers [6][7][8][9][10], etc.

Nowadays, as the need for security in communication has vastly grown, realistic implementations of QKD in applicable scenarios have become more important. Thus, thanks to its advantages in experimental implementation, continuous variable QKD (CV-QKD) has become an attractive field, e.g., protocols based on coherent laser light and heterodyne detection [11,12]. However, most research around the security of QKD has assumed that the eavesdropper (Eve) has access to any operation that is allowed by physics law, which is not going to be common in the near future. We have shown the security analysis of realistic secret key distillation with achievable rate calculation in [13,14]. Certain restrictions to Eve's collecting ability were considered, e.g., an exclusion zone around the legitimate receiver or a limited aperture size of Eve's receiver in a wireless channel.

In recent years, the capacity and security of communication between satellites have become relevant with the fast development of satellite-based free-space communications. Since the work on satellite-to-ground quantum key distribution (QKD) in 2017 [15], interests have been rising surrounding free-space secret key distillation for satellites.

In this paper, we examine the typical scenario where Eve's collecting ability is limited by an exclusion zone around the legitimate receiver, as an extension analysis case to our previous work series on this subject [13,16]. In this restriction on Eve, we consider what amounts to letting Eve have all the quantum resources at her disposal but constraining her light-collection capability to a certain fraction of Alice's transmitted photons not collected by Bob through collecting photons only outside of a certain exclusion zone around the legitimate receiver, governed by the practicalities of a line-of-sight free-space channel. In this manuscript, we also explore the secure key rates attainable under such a realistic model with Eve placing her limited-sized aperture in the same plane as Bob. Clearly,

any restriction on eavesdropping will result in the key rate increasing. The important question is to characterize the key rate increment with specific parameters relating to the level of restriction. In Section 2, we introduce and analyze the so-called "exclusion zone" scenario where Eve cannot eavesdrop without alerting the communicating parties in certain regions near Bob. We start with presenting the input power dependency with respect to different reconciliation efficiencies and then optimize the input power to show secure key rate lower and upper bounds as functions of both transmission distance and center frequency. We also incorporate these results with specific continuous-variable and discrete-variable protocols for comparison. Then, in Section 3, we examine the scenario in which Eve's collecting ability is limited by the size of her aperture and study how an exclusion zone can be used to increase security as a defense strategy by presenting lower and upper bounds with respect to different radii of the exclusion zone. We show that an exclusion zone can increase the secure key rate lower bounds and act as a valuable defense strategy, especially when the eavesdropper's aperture size is limited. We demonstrate that, for certain well-chosen parameters, we can achieve significantly higher secure key rate (SKR) lower bounds compared with a traditional unrestricted Eve scenario.

EXCLUSION ZONE SCENARIO

In this section, we evaluate the exclusion zone scenario as is illustrated in Fig. 1, which is called here the "exclusion zone," meaning that Eve cannot collect photons in this zone without being noticed by Alice and Bob. In satellite-based secure communication, setting an "exclusion zone" is one of the most straightforward methods to improve security, as this effectively decreases Eve's collecting ability. Here, we assume that Eve can collect all photons outside of the exclusion zone. In this paper, a multiphoton source governed by the mean photon number without photon-number-resolving detectors is assumed to be used on Alice's side. This will make her limited in knowing whether she is transmitting a multiphoton wave packet. We also only consider thermal noise here since the transmission is in space, and we use the blackbody radiation equation for the noise frequency dependence:

(1) e kT -1 where ne is the mean photon number per mode for the thermal noise, h 6.626 10 -34 m 2 kg/s is the Planck constant, k 1.38064852 10 -23 m 2 kg/(Ks 2 ) is the Boltzmann constant, T 3 K is the space temperature, and f is the center frequency in Hz that we use in transmission.

Generally, the shape of an exclusion zone is defined by the nature of this excluding action such as the communication parties monitoring the surroundings of their receiver apertures, which should be round, as the monitoring action is only limited, roughly, to the distance between the surrounding environment and the aperture. In space, the communication parties could exploit opaque materials to absorb any potential light outside of their aperture, stopping it from further propagation. The size of this opaque structure would also be limited to its material strength, which would roughly reflect as its "radius" to the aperture, whether it is an exact circular shape or not. On the other hand, since we are performing theoretical analysis, using a circular exclusion zone provides a baseline for results on such a model. Even when a noncircular-shaped exclusion zone is applied, we can always use its incircle and excircle to estimate the lower and upper bounds of such an exclusion zone. Thus, in this paper, without loss of generality, we use circular exclusion zones in our analysis.

As illustrated in Fig. 1, we assume the area of the transmitter aperture is Aa (Alice) and the area of the receiver aperture is Ab (Bob), with the area of the so-called "exclusion zone" (dashed line) denoted as Aex. Once more, we define the "exclusion zone" as the area/space in which Eve is not able to collect photons that end in this zone without alerting both communication parties. According to some similar analysis in [17,18], if the frequency used ω is restricted to 0 ω ω0 2πc L/ Aa Ab, then the Alice-to-Bob transmissivity η at frequency ω is denoted as η(ω) (ω/ω0) 2 1. Here, c is the speed of light.

Applying this to our analysis, we will have 2πc The exclusion zone area is denoted as Aex, which is a circular area centered at Bob's aperture.

Here, κ is the restriction factor [13] on Eve, denoting how much power she can collect from the portion not being collected by Bob. Thus, the transmissivity of the Alice-to-Eve channel (ηAE) can be denoted as (1 η(ω))κ(ω) in Eq. ( 6). Equation (4) gives us ω0Ex, which is obtained by replacing Ab with Aex in Eq. ( 2). This allows us to calculate Alice-toexclusion zone transmissivity (ηAEx), assuming there is a virtual receiver covering the entire exclusion zone. Then, assuming Eve has an infinite sized aperture, Alice-to-Eve transmissivity (ηAE) can be calculated by Eq. ( 6).

Recall from [13] that the lower bounds for direct (K →) and reverse (K ←) reconciliation in a quantum thermal noise wiretap channel are given, respectively, by after the measurement of Bob's state. Detailed expressions have been calculated in Eqs. ( 55) and ( 56) in [13].

From Fig. 2, we can see that, although SKR lower bounds for both reconciliation schemes saturate as input power is sufficiently large when no exclusion zone is set (rex rb), we can exceed this saturation threshold by setting an exclusion zone (such as the red curves here with rex rb 5 cm), as this effectively diminishes Eve's collecting ability, so that higher input power would benefit Bob more than Eve. It is also worth noticing in Fig. 2 that an exclusion zone actually benefits a direct reconciliation scheme more than a reverse reconciliation scheme. This is because setting an exclusion zone only diminishes Eve's collecting ability (κ) with no impact on Bob's receiving ability (η); further, from [13], we know that direct reconciliation exceeds reverse reconciliation when κ is small with fixed η.

In Fig. 2 we can also see that the SKR lower bounds are

nondecreasing with increasing input power, meaning that, when the reconciliation efficiency is perfect (β 1), the optimal input power is infinity. In Fig. 3, we show that the optimal input power is finite when the reconciliation effi-

ciency is imperfect (β = 0.95). By setting an exclusion zone 2 2

(rex rb 5 cm in Fig. 3), we are able to achieve higher SKR exceeding this optimal input power. where µ is the mean photon number per input mode from Alice to Bob, and β (0, 1 is the reconciliation efficiency. In Fig. 2, we plot direct (dashed curves) and reverse (solid curves) SKR lower bounds as functions of input power µ for different values of radius rex of Aex. Here, we set the transmission wavelength at 1550 nm. We use blue curves to denote the case with no exclusion zone (rex rb), whereas red curves denote the case with an exclusion zone (rex > rb). The transmission distance is set as 100 km and the reconciliation efficiency is set as β 1.

Here, ν ER (i 1, 2) are the symplectic eigenvalues of the covariance matrix corresponding to the system ER, and ν ER (i = 1, 2) are the symplectic eigenvalues of the covariance matrix corresponding to the post-measurement system ER|y In Fig. 2 we have seen that the optimal input power is infinity when β 1. Further, in Fig. 4, we plot the perfect reconciliation lower bounds as functions of transmission distance assuming equal aperture sizes for Alice and Bob with input power optimized (set to infinity). We also include upper bounds from [13] with dotted curves. The upper-bound derivation can be found in Section IV.B of [13]; we also use similar methods with lower bounds to calculate η, κ, and ne to estimate the upper bound. We can see from Fig. 4 that increasing the exclusion zone area can increase both the direct and reverse reconciliation lower bounds, especially when the transmission distance L is not too large. In Fig. 5, we plot the upper and lower bounds for both reconciliation schemes center frequency (Hz) 10 12 Fig. 6. SKR lower bound versus center frequency. The radius of the exclusion zone and the transmission distance are used as parameters. The reconciliation efficiency is perfect (β 1), and the input power is set to infinity. SKR versus exclusion zone radius with r a =0.2m, r b =0.2m, L=100km 6 5 4 3 2 1 0.2 0.205 0.21 0.215 0.22 0.225 0.23 0.235 0.24 0.245 Exclusion zone radius (r ex (m)) Fig. 5. SKR lower bound versus exclusion zone radius. Wavelength λ is set to 1550 nm. The reconciliation efficiency is perfect (β = 1), and the input power is set to infinity.

as functions of the exclusion zone radius. It is clear that the SKR increment is not linear to the increment of the exclusion zone radius: When the exclusion zone is larger, increasing the exclusion zone can help increase the SKR more. It is also worth noticing that, although the direct reconciliation lower bound is lower than the reverse reconciliation lower bound when no exclusion zone is set, it increases faster as the exclusion zone size increases and eventually the upper bound. In Fig. 6, we plot the SKR lower bound versus the transmission center frequency, wherein the transmission distance L and exclusion zone radius rex are both used as parameters. Here, the red and blue curves represent different transmission distances, the dotted-dashed curves represent a large exclusion zone, and the solid curves represent no exclusion zone. It is worth noting that the lower bounds in Fig. 6 are attained by reverse reconciliation, as the direct reconciliation lower bound is zero for selected parameters.

As we can see in Fig. 6, the SKR lower bounds increase with increasing frequency. Although choosing a higher frequency can always result in higher SKR, this can pose potential challenges to the system design, as we need a higher frequency for longer transmission distance. This downside can be mitigated by imposing a large exclusion zone, as it effectively decreases Eve's receiving ability, relaxing the need for higher frequency as is illustrated in Fig. 6 with dotted-dashed curves.

Next, we analyze specific QKD protocols, the Gaussianmodulated CVQKD protocol (with coherent states, heterodyne detection, and reverse reconciliation) and the decoy state BB84 (DS-BB84) protocol, and compare their performances under the exclusion zone scenario. We assume that Alice uses a weak coherent-state source and transmits signal-state pulses with on average µ photons per pulse at R states/s. For CVQKD SKR, we use the classical-classical-quantum (CCQ) rate (solid curve) as in Eq. ( 69) from [13]; for DS-BB84 (dotted curves), we use Eq. ( 95) from [13] with reconciliation efficiency fL . Here, we also include lower bounds obtained with Eq. ( 8) as the classical-quantum-quantum (CQQ) rate (dashed curves), as it does not assume a specific detection scheme for Bob. Here, CQQ indicates that one of the communication parties, Alice or Bob, has performed a measurement on her/his quantum system, while the other and Eve are yet to measure their respective quantum systems. On the other hand, CCQ indicates that both Alice and Bob have performed measurements [13].

In Fig. 7, we plotted CQQ and CCQ rates for CVQKD and an achievable rate for DS-BB84 versus input power. We can see that, since we assume heterodyne detection for Bob instead of an unspecified detection scheme, the CCQ rate is upper bounded by the CQQ rate. Both rates are largely increased under the exclusion zone (rex = 0.25 m) scenario. 120 140 160 180 200 220 240 transmission distance L (km)

SKR lower bounds versus input power with imperfect radius of the exclusion zone (rex) is used as a parameter. Wavelength λ is set to 1550 nm. R = 1 Gbit/s. information reconciliation. The transmission distance is 100 km. The radius of the exclusion zone (rex) is used as a parameter. Wavelength λ is set to 1550 nm. R = 1 Gbit/s. 10 0 120 140 160 180 200 220 240 transmission distance L (km) information reconciliation. The transmission distance is 100 km. The radius of the exclusion zone (rex) is used as a parameter. Wavelength λ is set to 1550 nm. R = 1 Gbit/s.

In Fig. 7, we can also see that DS-BB84 is upper bounded by the CCQ rate. Although it has an optimal input power when there is no exclusion zone, setting one can increase its rate exceeding this optimal input power, achieving higher rates. In Fig. 8, we include the same comparison case with imperfect information reconciliation (β 0.9, fL 1.2). We can see that imperfect information reconciliation renders finite optimal input power for both the CCQ rate and the DS-BB84 rate with similar comparison results, while an exclusion zone (rex) still largely increases both the CCQ rate and the CQQ rate exceeding this finite optimal input power.

Below, we plot the comparison for these three rates with optimized input power. Figure 9 captures the comparison against transmission distance for different exclusion zone sizes with corresponding optimal input power provided in Fig. 10. Corresponding optimal input power for the CCQ rate and DS-BB84 with or without an exclusion zone in Fig. 9. Fig. 10. We can see that the optimal input power decreases with increasing transmission distance.

LIMITED APERTURE SIZE COMBINED WITH AN EXCLUSION ZONE

In this section, we combine the exclusion zone scenario with the limited-sized aperture scenario where Eve's aperture size is limited [16,19] and investigate how it can improve security as a defense strategy. First, we introduce the problem setup as illustrated in Fig. 11. Similar to Fig. 1, here Aa , Ab, and Aex, respectively, denote the area of the transmitter aperture (Alice), the receiver aperture (Bob), and the exclusion zone. Additionally, we have a limited-sized eavesdropper aperture in the same plane as Bob, denoted as Ae . In this case, Eve's optimal position is shown in Fig. 11 to be tangential to the 10 -1 10 -2 10 -2 10 -1 10 0 10 1 10 2 10 3 10 4 the same plane as Bob's aperture. The exclusion zone area is denoted as Aex, which is a circular area centered at Bob's aperture.

exclusion zone, even though that's difficult to implement in practice but can be used to represent the worst-case scenario.

We assume a Gaussian beam (transverse coordinates x , y ) with beam waist radius W0 ra transmitted with propagation distance L and intensity at the center of the beam waist I0:

Mean photon number per input mode,

IU(ρ,

With Eqs. ( 14)-( 16), the Alice-to-Bob transmissivity (η) and Eve's fraction of collected power (κ) can be expressed as 0 0 follows:

The total power in this Gaussian beam is 2

Also, for noise frequency dependence, we use the blackbody Without loss of generality, we can assume Eve's aperture right above the exclusion zone as in Fig. 11 and derive Bob's received power (PBob) and Eve's received power (PEve) as follows based on Rayleigh-Sommerfeld functions: radiation equation in Eq. ( 1).

With the above expressions, first we plot the input power dependency for different transmission distances. In Fig. 12, we plot the direct and reverse reconciliation achievable rate lower bounds as functions of input power for different exclusion

zone radii. Here, the radii of Alice's, Bob's, and Eve's aper-

tures are all set to 5 cm. The transmission is over 10 km with

reconciliation efficiency β 1. Similarly to Fig. 2, we can see that the optimal input power here is infinity. Not surprisingly, by setting an exclusion zone (rex = 0.5 m), we can increase

the achievable rates for both reconciliation schemes. We can

reverse reconciliation when the transmission distance is small (η is large). Since most power is focused at the center of the transmitted Gaussian beam, which is mostly received by Bob at the line of sight, the corresponding κ is small. This explains why a direct reconciliation scheme achieves a higher rate than a reverse reconciliation scheme in this regime. However, when we increase the transmission distance, as presented in Fig. 13, with only the transmission distance increased to 30 km, we can see that, with no exclusion zone set (rex 5 cm), direct reconciliation is more sensitive to the consequential channel loss increment and performs worse than the reverse reconciliation lower bound. Yet if an exclusion zone is set (rex 0.5 m), compared with the reverse reconciliation, the direct reconciliation benefits more from this strategy specifically designed to decrease κ, and we can see that, in this case, the direct reconciliation lower bound increases to be higher than the reverse reconciliation. This is even more evident if we set the reconciliation efficiency to be imperfect, which is mostly the case in realistic implementation. In Fig. 14, we set reconciliation efficiency β SKR lower bounds versus transmission distance with r Alice =r Bob =r Eve =0.05m 10 0 10 -2

to 0.85. As a result, when no exclusion zone is set (rex = 0.5 m), the optimal input power is shown to decrease from infinity to some finite values. Aside from that, we can see that an imperfect reconciliation process decreases the direct reconciliation

10 -4 10 0 10 1 10 2 10 3 10 4

rate much more than it does to the reverse reconciliation. However, when an exclusion zone is set (rex 0.5 m), still the direct reconciliation rate is higher than the reverse reconciliation rate, exceeding the SKR results corresponding to the original optimal input power. Next, we look into the perfect reconciliation case with optimized input power. In Fig. 15, we plot the direct and reverse reconciliation lower bounds as functions of transmission distance for different exclusion zone radii. Here, the unrestricted Eve case is also included for comparison. We can see that lower bounds exceeding the unrestricted Eve case can be achieved. Here, the direct reconciliation lower bounds exceed the reverse reconciliation lower bounds when the transmission distance is not too large. When the transmission distance increases, the direct reconciliation lower bounds start to decrease in a similar manner to the unrestricted Eve case, while the reverse reconciliation lower bounds converge to a constant rate, as we saw in our previous work [19]. We can see that an exclusion zone mostly helps increase the achievable rate when the transmission distance is not too large. For example, when the exclusion zone radius is large enough (rex 0.5 m), the achievable rate even goes to infinity when the transmission distance is below 10 km, exceeding the constant rate convergence at short transmission distance. We can also see that, with a larger exclusion zone, the achievable rate at a larger transmission distance can be increased. Yet this does not change the constant rate convergence as the transmission distance goes to infinity since an exclusion zone, when its radius is comparable with the communicating parities' aperture radii, does not change the ratio of Eve's collecting ability versus Bob's collecting ability too much at large transmission distance.

In Fig. 16, we plot the lower bounds versus the upper bounds as functions of the transmission distance. Here, the

SUMMARY

In this paper, we have analyzed SKR lower and upper bounds for a realistic scenario in free-space satellite-to-satellite secure communication where an exclusion zone is set around the legitimate receiver and studied the performance with respect to relevant channel parameters. By enlarging the exclusion zone area, we can relax the need for high frequency in long distance transmission and achieve a higher secure key rate. When Eve's aperture size is limited, an exclusion zone can serve as a good defense strategy to increase the secure key rate significantly. Future directions of this work will involve studying the case where Eve can optimize her aperture position, which would be SKR lower and upper bounds versus center frequency with r Alice =r Bob =r Eve =0.05m, transmission distance L=30km 10 1 10 0 10 12 10 13 10 14 10 15 10 16 transmission center frequency (Hz) Fig. 17. SKR lower bounds and upper bounds versus frequency.

lower bound curves are taken as the maximum of the direct and reverse lower bound curves at each given distance value. We also use the same parameters as in Fig. 15. We can see that, when the transmission distance is not too large, the upper bounds closely match the lower bounds, giving the capacity in this region. In Fig. 17, we plot the lower bounds as the maximum of the direct and reverse lower bounds versus upper bounds as functions of the transmission center frequency. We can see that an exclusion zone can help increase the achievable rate, especially when the transmission frequency is high. When the exclusion zone radius increases, similarly to Fig. 6, this helps to achieve a higher achievable rate with low transmission center frequency. It is worth noticing that, when rex = 0.5 m, the achievable rate even goes to infinity instead of converging to a constant rate when the exclusion zone is large. discussed in future works.