Secret key rates of free-space optical continuous-variable quantum key distribution

In this letter, we derive the maximal achievable secret key rates for continuous-variable quantum key distribution (CVQKD) over free-space optical (FSO) quantum channels. We provide a channel decomposition for FSO-CVQKD quantum channels and study the SNR (signal-to-noise ratio) characteristics. The analytical derivations focus particularly on the low-SNR scenarios. The results are convenient for wireless quantum key distribution and for the quantum Internet.

devices for experimental implementations. [4][5][6]11,[17][18][19][20][21][22][23][24][25][26][27][43][44][45][46][47][48][49] The multicarrier CVQKD has been recently introduced through the adaptive quadrature division modulation (AMQD) scheme. 44 The multicarrier CVQKD injects several additional degrees of freedom into the transmission, which are not available for a standard, single-carrier CVQKD setting. [45][46][47][48]50 The achievable secret key rates in a multicarrier CVQKD setting have been proven in Gyongyosi and Imre. 47 The secret key rates confirm the multimode bounds determined in Pirandola et al. 1 The FSO systems bring several new attributes to both the theoretical and experimental side of CVQKD. An FSO quantum link's special characteristics require a specific mathematical description. The channel characteristics of the FSO quantum links are approachable via the mathematical framework of the GG (gamma-gamma) distribution. [6][7][8][9][10] The secret key rates for CVQKD schemes over FSO links and the performance of free-space quantum links in diverse environmental conditions raise several questions and call for further examination. Another interesting problem is the private classical capacity 4 of a GG link. Without loss of generality, the private classical capacity measures the amount of classical information that can be privately transmitted from a sender (Alice) to a receiver (Bob) in the presence of an eavesdropper (Eve). For further information on the rate-loss scaling in quantum optical communications, we suggest the derivations in Pirandola et al. 1 Our results on the private classical capacity also confirm the bounds of 2 on private quantum communications in a CVQKD setting.
The private classical capacity imposes a theoretical upper bound on the achievable secret key rates in QKD implementations. Since practical QKD implementations operate in the low-SNR regime, 11,[17][18][19][20][21][22][23][24][25][26][27]44 we will analyze the behavior of private classical capacity in the low-SNR domain for CVQKD over FSO (referred to as FSO-CVQKD). By theory, the DVQKD and CVQKD implementations require different channel models. For DVQKD, the resulting channel noise distribution is analogous to the binary-symmetric channel (BSC), 23 while for the DVQKD setting, the resulting noise is Gaussian. 11,[17][18][19][20][21][22][23][24][25][26][27] Another important difference from traditional crypto systems is that the correlation measure functions are non-traditional. In the theoretical analysis, it is assumed that the legal parties and the eavesdropper have quantum memories and can perform joint measurements, 11,17-27 therefore the Holevo information 4 is the appropriate correlation measure function for deriving the private classical capacity.
The novel contributions of our paper are as follows: • We derive an upper bound for the secret key rates of wireless CVQKD over FSO channels.
• We provide a decomposition model for the FSO quantum channel in a CVQKD setting.
• We investigate the SNR attributes of the GG-channel and the complementary channel for the information leakage.
This paper is organized as follows: Section 2 provides the channel model for FSO-CVQKD. Section 3 focuses on the private classical capacity of an FSO link in a CVQKD setting. Finally, Section 4 concludes the results.

SYSTEM MODEL
In this section, the general formulas and equations are briefly summarized.
The  (  ) private classical capacity identifies the maximum rate at which classical information can be transmitted privately (ie, an eavesdropper has no knowledge about the original message) over a quantum channel  . To derive the  (  ) private classical capacity of  in the FSO setting, the physical quantum link  is divided into logical channels  AB ,  AE , and  BE . Logical channel  AB denotes information transmission through the GG quantum channel between Alice (A) and Bob (B). Logical channels  AE and  BE are complementary channels that model the information leakage from Alice to eavesdropper Eve (E), and the information leakage from Bob to Eve, respectively. In our setting,  AE is relevant for the DV case, while  BE is important in the reverse reconciliation of CVQKD (Bob starts the reconciliation to minimize information leakage). Let identify a jth input coherent state (Gaussian state) in the phase space , with i.i.d. Gaussian random position and momentum quadratures x ∈  ( 0, 2 , where 2 0 is the modulation variance. The coherent state | | ⟩ in the phase space  can be modeled as a zero-mean, circular symmetric complex Gaussian random variable with variance and with i.i.d. real and imaginary zero-mean Gaussian random components, Re . The transmission of this complex variable over the Gaussian quantum channel  can be characterized by the T is the transmission of the momentum quadrature. During the evaluation of the private classical capacity for an FSO setting, we assume that the CVQKD protocol operates in the low-SNR regime, SNR → 0, 8 which is precisely the situation for practical CVQKD scenarios.
Utilizing an FSO channel for the transmission of a given input z j , logical channel  AB is defined as where j is a zero-mean, circular symmetric complex Gaussian random variable that identifies the Gaussian noise added Eve , and with i.i.d. real and imaginary zero-mean Gaussian random components, Re where is the effective photocurrent conversion ratio of the receiver, while I is the normalized irradiance with the gamma-gamma (GG) probability density 7-10 where K v (·) is the modified Bessel function of the second kind and of order v, 8 Γ (·) is the Gamma function, while a ≥ 0, and b ≥ 0 are the distribution-shaping parameters expressed as is the Rytov variance, C 2 is the altitude-dependent turbulence strength, l is the length of the link, and |k| is the optical wave number, while d = √ |k| D 2 ∕4l, where D is the receiver's aperture diameter. 8 The complementary channel  BE for a reverse reconciliation in the CVQKD case is defined as where 2  BE is the noise variance for  BE . Assuming that the parties have quantum memories and can perform joint measurement on their quantum registers in the QKD protocol run, the appropriate correlate measure functions for the logical channels  AB ,  AE , and  BE are the Holevo quantities 4 AB , AE , and BE . Without loss of generality, for the S (  ) secret key rate over a quantum channel  , the following relation holds: where P (  ) is the private classical capacity of  . Assuming reverse reconciliation in CVQKD with GG channel  AB and Eve's Gaussian channel where f is the reconciliation efficiency, AB is the Holevo information between Alice and Bob and BE is the Holevo information between Bob and Eve are the Holevo quantities between Alice and Bob and Bob and Eve; S ( ) = −Tr ( log ( ) ) is the von Neumann entropy, while AB = ∑ i p i i and BE = ∑ i p i i . The quantity BE is the Holevo information between Bob and Eve, which plays a role in a reverse reconciliation CVQKD. We also use this approach to derive  (  ) for the CVQKD case, since reverse reconciliation is proved to minimize the eavesdropper's Holevo information compared with the direct-reconciliation case.
Thus, P (  ) at a reverse reconciliation with reconciliation efficiency f is evaluated as where  ( i ) represents the ith output density matrix. Specifically, the D ( ·‖ ·) quantum relative entropy function between density matrices and is The Holevo quantity can be expressed by the quantum relative entropy function as = D ( k ‖ ), where k denotes an optimal channel output state (for which the Holevo quantity will be maximal) and = ∑ p k k . The Holevo information can be derived in terms of D ( ·‖ ·) as Therefore, AB is rewritten as The quantity BE measures the Holevo information leaked to Eve from Bob during a reverse reconciliation and is written as Using (19) and (20), can be expressed as where AB−BE k is the final optimal density matrix, while AB−BE refers to the final output average density matrix. Figure 1 depicts the model of FSO-CVQKD channel  used for the derivation of  (  ) .

FIGURE 1 Decomposition of a physical
quantum link  into logical channels  AB and  BE in an FSO-CVQKD setting at a reverse reconciliation. The input system of the link is A, Bob's system is B, and Eve's system is E. The logical channel  AB refers to the FSO link between Alice and Bob with the GG characteristics and Gaussian noise. The complementary channel  BE is a Gaussian channel between Bob and Eve

Theorem 1. (Scalability of the secret key rate). At a direct or reverse reconciliation with channels  AE and  BE between Alice and Eve and Bob and Eve, the S
secret key rate over an FSO link  in any CVQKD protocol is scalable by the SNR AB of  AB .
Proof. Let  AB be the quantum channel between Alice and Bob. For the transmission of a given z i , let the SNR of  AB be evaluated as where 2 0 is the modulation variance of a quadrature component and 2 Eve is the variance of the Gaussian noise. To evaluate the private classical capacity, we consider three scenarios 8 for the values of coefficients a and b. First, assume that 8 a + b = 13 2 , which yields while coefficient is as and is a Lagrange multiplier. [8][9][10] Then, let SNR be a Lagrange multiplier at an average power constraint c, where 2̃2 0 refers to the average input power 8-10 associated to a jth input z j and = 2 I 2 . [8][9][10] The resulting modulation variance 2 2 0 for  AB with SNR AB is then which allows us to evaluate  (  ) at a + b = 13 2 via (13), where AB is the Holevo information between Alice and Bob over  AB , while BE is the Holevo information derived via the Gaussian channel  BE between Bob and Eve, where , while tot is the overall noise tot = line + M , with an SNR BE of the Gaussian channel  BE , as where 2 2′ is the modulation variance of Bob's noisy z ′ i . Second, let's assume that 8-10 a + b > 13 2 , thus after some calculations, SNR is as where m < 0, and SNR AB of  AB is Thus, the resulting modulation variance 2 2 0 is as 13 2 is yielded from the Holevo information AB of  AB , written as where  (·) is the product logarithm function,  −1 refers to the lower branch of the function , 7-10 and BE is given by (29). Finally, at a + b < 13 2 , after some calculations, SNR AB is yielded as where m > 0; thus the resulting modulation variance 2 2 0 is from which  (  ) at a + b < 13 2 is approachable from the Holevo information AB of  AB , where AB is as follows: where  0 is the principal branch of , 8 while BE , as given by (29). The proof is concluded here.
The next theorem focuses on the reverse reconciliation case.
and the SNR AB Lagrange multiplier is evaluated as The Holevo information BE of Eve is as given by (29), the proof is therefore concluded here.

CONCLUSIONS
This letter studied the performance of CVQKD over free-space optical quantum channels. The analysis provided the private classical capacity of an FSO link for the CVQKD protocols at a reverse reconciliation. Our derivations were focused on the low-SNR setting. The results prove to be convenient for wireless quantum communications, wireless quantum key distribution and quantum Internet scenarios.