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Secret key rate of multi-ring M-APSK continuous variable quantum key distribution

Margarida Almeida, Daniel Pereira, Nelson J. Muga, Margarida Facão, Armando N. Pinto, and Nuno A. Silva

Open Access Open Access

Abstract

Discrete modulation continuous variable quantum key distribution (DM-CV-QKD) is highly considered in real implementations to avoid the complexity of Gaussian modulation (GM), which is optimum in terms of the key rate. DM-CV-QKD systems usually consider M-symbol phase shift keying (M-PSK) constellations. However, this type of constellation cannot reach transmission distances and key rates as high as GM, limiting the practical implementation of CV-QKD systems. Here, by considering M-symbol amplitude and phase shift keying (M-APSK) constellations, we can approximate GM. Indeed, considering finite-size effects, 4 ring 64-APSK can reach 52.0 km, only 7.2 km less than GM and 282% the maximum achievable transmission distance for 8-PSK.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Quantum key distribution (QKD) allows two users, Alice and Bob, to establish an unconditionally secure symmetric secret key [1]. Continuous variable (CV)-QKD systems can be implemented using the phase and amplitude of weak coherent fields, allowing the use of standard coherent detection [2,3]. Discrete modulation (DM)-CV-QKD schemes, with M-symbol phase shift keying (M-PSK) constellations, are usually implemented to avoid the complexity of Gaussian modulation (GM) [4,5]. However, DM-CV-QKD systems cannot reach transmission distances and key rates as high as GM [6]. In order to improve CV-QKD systems other modulation formats can be explored to approach the performance of GM.

CV-QKD systems were first proposed using GM [7,8]. Its practical security bounds were established in [9], and a security analysis of the GM-CV-QKD protocol considering collective and coherent attacks was shown in [10,11]. In [12], authors show that the GM-CV-QKD protocol exhibits optimum theoretical secret key rate. However, GM demonstrates low reconciliation efficiency [13–15] and computationally demanding error-correction resources, and often requires a very high source of randomness on the transmitter’s side [5]. To solve that, DM-CV-QKD protocols were proposed, due to a higher reconciliation efficiency, associated with simpler reconciliation protocols [13,16]. Additionally, DM-CV-QKD has a simpler practical implementation [17,18]. In this context, the majority of the experimental work in CV-QKD has been accomplished assuming DM-CV-QKD schemes [4], usually considering 4-PSK and 8-PSK constellations. In [19], a rigorous secret-key rate lower bound was obtained considering physically reasonable collective attacks in the asymptotic regime for the DM-CV-QKD protocol.

Besides that, in [20], the authors show that, by increasing the constellation size in DM-CV-QKD systems, it is possible to obtain a small increase on the secret key rate using M-PSK. Therefore, eight-state protocols have been considered [21]. Still, even under finite-size effect considerations, the implementation of DM-CV-QKD systems with M-PSK modulation is incapable of reaching transmission distances and key rates as high as GM [6]. Increasing the size of the M-PSK constellation from 8 to 16 points does not significantly improve the performance of the CV-QKD system [6]. In [6], the M-symbol quadrature amplitude modulation (M-QAM) format with a binomial distribution was studied. The results obtained in [6] show that 64-QAM approximates GM for 50 km, without the finite-size effects. Nonetheless, it is known that M-symbol amplitude phase shift keying (M-APSK) can achieve lower error rate and lower bandwidth requirements [22,23]. Comparing M-QAM and M-APSK constellations with the same number M of symbols, the M-APSK contains fewer amplitude levels, associated to a lower peak-to-average power ratio [24].

Here, we applied multi-ring M-APSK modulation to an accurate model for the secret key rate of DM-CV-QKD systems, identifying appropriate limits for the application of M-APSK constellations to DM-CV-QKD. Multi-ring M-APSK is able to approach true GM results, even using more than one photon per symbol. Indeed, for the 4 rings 64-APSK with binomial distribution, we achieve a secret key rate higher than 77% of GM’s result, which is 30 times more than the value achieved with 8-PSK, for a distance as long as 75 km. In the finite-size effect regime, the 4 rings 64-APSK with binomial distribution allows an increase of the maximum achievable transmission distance of 33.5 km with respect to 8-PSK results.

This paper is organized as follows. In Section 2, a generic DM-CV-QKD model is presented to obtain the secret key rate achievable under collective attacks considering the thermal noise at the receiver and the detection efficiency. Section 3 analyses the M-APSK modulation format, considering discrete uniform distribution for the ring’s probability. In Section 4, we optimize the M-APSK modulation using binomial probabilistic shaping. Finally, Section 5 summarizes the most important results.

2. Secret key rate model for DM-CV-QKD constellations

The secret key rate, $K$, obtained between Alice and Bob from a practical implementation of a CV-QKD system is given by [25]:

$$K = \frac{m}{N} \left( \beta I_{BA} - \chi_{BE} - \Delta m\right),$$
where $I_{BA}$ is the mutual information between Bob and Alice, $\chi _{BE}$ represents the Holevo bound between Bob and Eve, and gives the maximum information gained by Eve from Bob’s key when performing a collective attack, $\beta$ is the reconciliation efficiency, $\Delta m$ is a parameter related to the security of privacy amplification (due to the finite size effects), $m$ is the amount of data used for information reconciliation, and $N$ is the total number of states transmitted in the quantum channel. Notice that, the estimation of the channel parameters will not be performed here. However, $m/N$ is set to $1/2$ to account for the data used in parameter estimation. To compute the secret key rate we consider a realistic scenario where Alice and Bob devices are trusted and not manipulated by a potential eavesdropper [26,27].

Alice and Bob only share between them a finite number $N$ of states, since, in a real implementation of the protocol, the quantum channel is used only a finite number $N$ of times, and therefore the finite-size scenario must be accounted for. The finite-size scenario degrades Alice’s and Bob’s knowledge on the channel’s transmission and excess noise [27]. Assuming the raw key to be encoded in bits, the consideration of the finite size effects can be, in a simple manner, introduced by the parameter $\Delta m$, given by,

$$\Delta m = {7} \sqrt{\frac{\log_2(2/\overline{\epsilon})}{m}} + \frac{2}{m} \log_2 (1/\epsilon_{\textrm{PA}}),$$
where $\overline {\epsilon }$ is a smoothing parameter, and $\epsilon _{\textrm {PA}}$ is the failure probability of the privacy amplification procedure [25].

The parameter estimation step requires the estimation of the channel transmission and excess noise, by considering the normal linear model between Alice’s and Bob’s data, $x$ and $y$, respectively. Considering double homodyne detection, Alice’s and Bob’s data follow the normal linear model $y = tx + z$, parametrized by $t = \sqrt {T\eta /2}$ and where $z$ follows a centered normal distribution with variance $\sigma ^2 = \frac {\eta T}{2}\xi + 1 + \xi _{\textrm {thermal}}$, where $T = 10^{-0.02d}$ is the channel transmission, $d$ is the transmission distance, $\xi$ is the excess noise, $\xi _{\textrm {thermal}}$ is the thermal noise of the detection system, and $\eta$ is the photodiodes’ detection efficiency. The thermal noise of the detection system, $\xi _{\textrm {thermal}}$, can be measured during the calibration phase. Since the quality of the estimation depends on the number $N - m$ of states considered in the parameter estimation step, it must be accounted in the finite-size effects, by computing the lower bound of the secret key rate with a probability of at least $1 - \epsilon _{\textrm {PE}}$. The lower bound of the secret key rate is computed by the lower bound of $t$, $t_{\textrm {min}}$, and the upper bound of $\sigma ^2$, $\sigma _{\textrm {max}}^2$, except with a probability of $\epsilon _{\textrm {PE}}/2$, given by [25]

$$t_{\textrm{min}} \approx \sqrt{\frac{T\eta}{2}} - z_{\epsilon_{\textrm{PE}}/2} \sqrt{\frac{\frac{\eta T}{2}\xi + 1 + \xi_{\textrm{thermal}}}{2\left(N-m\right) \left(2 \langle n \rangle\right) }},$$
$$\sigma_{\textrm{max}}^2 \approx \frac{\eta T}{2}\xi + 1 + \xi_{\textrm{thermal}} + z_{\epsilon_{\textrm{PE}}/2} \frac{\left(\frac{\eta T}{2}\xi + 1 + \xi_{\textrm{thermal}}\right)\sqrt{2}}{\sqrt{2\left(N-m\right)}},$$
where $\langle n \rangle$ is the mean number of photons per symbol, and $z_{\epsilon _{\textrm {PE}}/2}$ is the $100(1 - \frac {\epsilon _{\textrm {PE}}}{2})$th percentile of a standard normal distribution, for a confidence interval of $\left (1 - \epsilon _{\textrm {PE}}\right )\cdot 100\%$, and is such that $1 - \textrm {erf}\left (z_{\epsilon _{\textrm {PE}}/2} / \sqrt {2} \right ) = \epsilon _{\textrm {PE}}/2$, where $\textrm {erf} \left (x\right ) = \frac {2}{\pi } \int _0^x \textrm {e}^{-t^2} dt$ is the error function. The factor of $2$ in $2\left (N-m\right )$ accounts for the fact that, for double homodyne detection, both quadratures are measured, and treated independently in the parameter estimation step. Finally, the lower bound of the secret key rate can be computed, except with a probability of $\epsilon _{\textrm {PE}}/2$, considering the minimum value of the channel transmission $T$, $T_{\textrm {min}} = 2 t_{\textrm {min}}^2 / \eta$, and the maximum value of the excess noise $\xi$, $\xi _{\textrm {max}} = \frac {2}{\eta T_{\textrm {min}}}\left ( \sigma _{\textrm {max}}^2 - 1 - \xi _{\textrm {thermal}} \right )$. All the secret key rates presented hereafter that consider the finite-size effects have been computed with $T_{\textrm {min}}$ and $\xi _{\textrm {max}}$.

Considering that Bob uses a double homodyne detection scheme where he measures simultaneously both quadratures sent by Alice, and assuming reverse reconciliation, the mutual information between Bob and Alice, $I_{\textrm {BA}}$, can be computed as [15]

$$I_{\textrm{BA}} = \log_2 \left( 1 + \textrm{SNR} \right) = \log_2 \left( 1 + \frac{2T\eta\langle n \rangle}{2 + T\eta\xi + 2\xi_{\textrm{thermal}}} \right),$$
where $\textrm {SNR}$ is the signal-to-noise ratio. Note that, in Eq. (4) we are assuming that each optical state sent by Alice contains only a few photons per symbol, for which a Gaussian approximation for $I_{\textrm {BA}}$ is valid.

Here, Eve is considered to perform collective attacks, since they are usually optimal in the asymptotic limit [28,29]. In a collective attack, the eavesdropper stores each obtained mixed-state in a quantum memory and measures all of them collectively at once after applying both information reconciliation and privacy amplification steps [15,30]. With this, the Holevo bound between Bob and Eve, $\chi _{\textrm {BE}}$, is given by [20]

$$\chi_\textrm{BE} = \sum_{i = 1}^{2} G\left( \frac{\mu_i - 1}{2} \right) - \sum_{i = 3}^{4} G\left( \frac{\mu_i - 1}{2} \right),$$
where
$$G(x) = (x+1) \log_2 (x+1) - x \log_2 x,$$
$\mu _{1,2}$ are the symplectic eigenvalues of the covariance matrix describing the states shared by Alice and Bob, and $\mu _{3,4}$ are the non-unitary symplectic eigenvalues of the covariance matrix that describes Bob’s projective measurement. Expressions for these parameters can be found in [20]. Both the covariance matrix that describes the states shared between Alice and Bob and the covariance matrix describing Bob’s projective measurement depend on a $Z^{*}$ parameter, which further depends on the type of modulation. For GM it is given by $Z^{*}_{\textrm {G}} = T \sqrt {4\langle n \rangle ^2 + 4\langle n \rangle }$. When other modulation formats are considered, as the M-APSK, the $Z^{*}$ parameter can be computed numerically. To achieve that, we start by defining the density matrix $\tau = \sum _k p_k \vert \alpha _k \rangle \langle \alpha _k \vert$ that describes the state sent by Alice, $\vert \alpha _k \rangle$. From this, we are able to compute the $Z^{*}$ parameter, which is given, for a Gaussian quantum channel, by [6]
$$Z^{*} (T,\xi) = 2\sqrt{T} \textrm{Tr} \left( \tau^{1/2} \hat{a} \tau^{1/2} \hat{a}^{{\dagger}} \right) - \sqrt{2T\xi W},$$
where
$$W = \sum_k p_k \left( \langle \alpha_k \vert \hat{a}_{\tau}^{{\dagger}} \hat{a}_{\tau} \vert \alpha_k \rangle - \left| \langle \alpha_k \vert \hat{a}_{\tau} \vert \alpha_k \rangle \right| ^2 \right),$$
and $\hat {a}_{\tau } = \tau ^{1/2} \hat {a} \tau ^{-1/2}$, with $\hat {a}$ and $\hat {a}^{\dagger }$ representing the annihilation and creation operators on Alice’s system. Note that $\langle n \rangle$ is defined as $\sum _{k} p_k \vert \alpha _k \vert ^2$, that is, it corresponds to the mean value of all the mean numbers of photons per symbol of the states in Alice’s chosen constellation. To the best of our knowledge, analytical expressions for the $Z^{*}$ parameter for M-APSK, in its most general format, both in terms of number of points per ring, and probability associated to each ring, are not available. Thus, in what follows, the $Z^{*}$ parameter for M-APSK was computed numerically using Eq. (7) and Eq. (8). The numerical computation of the $Z^{*}$ parameter can be easily done by writing the coherent state $\vert \alpha _k \rangle$ in the Fock space, up to the order at which the coefficient is negligible.

Figure 1 contains the $Z^{*}$ parameter as a function of the mean number of photons per symbol for 2, 4, 8 and 16-PSK. As expected, the higher the number of points on the constellation, the closer are the curves for M-PSK to the one for GM. Nonetheless, with the increase of the mean number of photons they tend to the lower bound for 2-PSK. Notice however that, by adding one or two rings to the M-PSK constellation, i.e., by considering two or three rings M-APSK modulation, we are able to get even closer to the $Z^{*}$ parameter value for GM, even if considering higher $\langle n \rangle$ values.

 figure: Fig. 1.

Fig. 1. $Z^{*}$ parameter as a function of the mean number of photons per symbol for 2, 4, 8 and 16-PSK. It is also represented the upper bound for GM and the curves considering 2 rings 16-APSK, with 4 states in the inner ring and 12 in the outer one, and 3 rings 32-APSK, with 4 states in the inner ring, 12 in the middle one and 16 in the outer one. The channel parameters are $d = 10$ km, and $\xi = 0.005$ SNU.

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To the best of our knowledge, the information reconciliation problem considering M-APSK constellations is still to be done and is out of the scope of the present work. Thus, since the information reconciliation step is not addressed here, possible options of binning of the complex plane, for each of the constellations, are not studied here. Anyway, as stated in [6], due to the resemblance between large M-APSK and GM, the reconciliation efficiency can be reasonably assumed to be around 95%, as was considered in the results presented in the following sections.

3. Multi-ring M-APSK with discrete uniform distribution

A M-APSK constellation is composed by several concentric M-PSK constellations, named rings, all having a different pulse energy. Thus, the mean number of photons associated with each ring is different. The M-APSK constellation is defined by the alphabet

$$\mathcal{X} = \left\lbrace \beta_p \alpha \textrm{e}^{i k \theta_p}, k = 0, 1,\ldots, M_p - 1, p = 1, 2,\ldots, R \right\rbrace,$$
where $M_p$ is the total number of points in ring $p$, $\theta _p = \frac {2\pi }{M_p}$, $\beta _p$ gives the position of ring $p$, with $\beta _p = 1/R, 2/R,\ldots, 1$, being the position of each ring relative to $\alpha$, and $\alpha \in \mathbb {R}^+$ is the amplitude of the outer ring. Except when said otherwise, all M-APSK constellations have 4, 12, 16, 32, 64, 128, and 256 states in the first to the seventh ring, respectively, with M being the total number of points on the constellation. Therefore, 1 ring 4-APSK has only one ring with 4 states, 16-APSK has two rings, the inner one with 4 states and the outer one with 12 states, and, following the same rule, 512-APSK has 7 rings, with the outer one having 256 states.

Considering that each state of ring $p$ occurs with equal probability, given by $P_k = 1/M_p$, and that each ring occurs with equal probability, given by $P_p = 1/R$, then the probability associated to each state of the constellation is $P_{p,k} = 1 / \left ( R M_p \right )$, with $\alpha$ being chosen such that $\alpha = \sqrt { V_A / \left ( 2 P_{p} \beta _p^2\right ) }$, where $V_A = 2 \langle n \rangle$ is the overall Alice’s modulation variance. Figure 2 presents a 4 rings 64-APSK constellation with discrete uniform distribution for the rings probability.

 figure: Fig. 2.

Fig. 2. Four rings 64-APSK constellation with discrete uniform distribution for the rings probability. The colour gives information on the probability of each coherent state.

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The parameters considered to obtain most the results were $d = 10$ km, $\eta = 0.6$, $\beta = 0.95$, $\xi = 0.005$ shot-noise units (SNU), $\xi _{\textrm {thermal}} = 0.04$ SNU, $\overline {\epsilon } = \epsilon _{\textrm {PA}} = \epsilon _{\textrm {PE}}= 10^{-10}$, $N = 10^8$ points, and $m/N = 1/2$.

In Fig. 3 we present the secret key rate as a function of the mean number of photons per symbol considering both M-APSK modulation and GM with the finite-size effect. Results show that the secret key rate first increases with $\langle n \rangle$, reaching a maximum value, and then decreasing to zero. This maximum point for the secret key rate corresponds to the optimum value of the mean number of photons per symbol. The optimum mean number of photons in 1 ring 4-APSK is 0.25 photons per symbol, resulting in a secret key rate of 0.006 bits/symbol. For 6 rings 256-APSK this value increases to 0.10 bits/symbol, 59.60% of GM’s maximum secret key rate, and this using 5.08 photons per symbol (see Fig. 3).

 figure: Fig. 3.

Fig. 3. Secret key rate as a function of the mean number of photons per symbol having into account both M-APSK modulation and GM with the finite-size effect. Here it was considered $d = 10$ km, $\eta = 0.6$, $\beta = 0.95$, $\xi = 0.005$ SNU, $\xi _{\textrm {thermal}} = 0.04$ SNU, $\overline {\epsilon } = \epsilon _{\textrm {PA}} = \epsilon _{\textrm {PE}}= 10^{-10}$, $N = 10^8$ points, $m/N = 1/2$, and that successive rings of the M-APSK constellations contain 4, 12, 16, 32, 64, 128, and 256 states, from the inner one to the outer one.

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By increasing the number of rings and symbols, the workable region, in terms of the mean number of photons per symbol, also increases. For 1 ring 4-APSK, the maximum value of mean number of photons that can be considered is 0.41 photons per symbol, while for 6 rings 256-APSK it is of 22.68 photons per symbol. Increasing the size of the constellation above 256 symbols results in almost no increase of the secret key rate or workable region, according with the results in Fig. 3. These results imply that we are no longer limited to use sub-one mean number of photons per symbol, as with M-PSK (M-APSK constellations with only one ring), facilitating the practical implementation of DM-CV-QKD. From Fig. 3, it can also be noted that, by increasing the number of rings, the secret key rate is flatter around the optimum point, meaning that using several rings and $\langle n \rangle$ close to the maximum could facilitate the flexibility of the practical implementation without losing considerable performance.

In Fig. 4 we present the secret key rate for M-APSK constellations with 2 and 3 rings with different number of symbols per ring. From Fig. 4 we can see that, in terms of number of symbols per ring, almost no improvement is observed by increasing the number of symbols per ring. In opposition, by increasing the number of rings, for instance, from 1 ring 4-APSK to 2 rings 16-APSK (four points in the inner ring and 12 in the outer one) an increase of 19.1 km is observed, and from 2 rings 16-APSK to 3 rings 32-APSK (four points in the inner ring, 12 in the middle one, and 16 in the outer one) an increase of 7.4 km can be observed. Therefore, by increasing the number of rings, it is possible to obtain higher performance and efficiency for the CV-QKD system.

 figure: Fig. 4.

Fig. 4. Secret key rate as a function of the transmission distance considering a mean number of photons per symbol optimized for each transmission distance for several M-APSK modulation formats and GM with finite-size effect. The M-APSK constellations have 1, 2 and 3 rings with different number of points per ring. The number of points per ring is represented on the label by ($M_1$-$M_2$) for two rings, and ($M_1$-$M_2$-$M_3$) for three rings. Here it was considered $\eta = 0.6$, $\beta = 0.95$, $\xi = 0.005$ SNU, $\xi _{\textrm {thermal}} = 0.04$ SNU, $\overline {\epsilon } = \epsilon _{\textrm {PA}} = \epsilon _{\textrm {PE}}= 10^{-10}$, $N = 10^8$ points, and $m/N = 1/2$.

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In Fig. 5 and Fig. 6 we present the evolution of the secret key rate as a function of the transmission distance, considering (Fig. 6) or not considering (Fig. 5) the finite size effect in Eq. (1). From the results in Fig. 5 and in Fig. 6 we can see that the finite size effect clearly limits the performance of the DM-CV-QKD system in terms of transmission distance. Moreover, from Fig. 5 and Fig. 6 we see that constellation sizes beyond 4 rings 64-APSK only provide low gains in terms of secret key rate.

 figure: Fig. 5.

Fig. 5. Secret key rate as a function of the transmission distance considering a mean number of photons per symbol optimized for each transmission distance for several M-APSK modulation formats and GM without the finite-size effect. Here it was considered $\eta = 0.6$, $\beta = 0.95$, $\xi = 0.005$ SNU, $\xi _{\textrm {thermal}} = 0.04$ SNU, and that successive rings of the M-APSK constellations contain 4, 12, 16, 32, 64, 128, and 256 states, from the inner one to the outer one.

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 figure: Fig. 6.

Fig. 6. Secret key rate as a function of the transmission distance with the mean number of photons per symbol optimized for each transmission distance for several M-APSK modulation formats and GM with the finite-size effect. Here it was considered $\eta = 0.6$, $\beta = 0.95$, $\xi = 0.005$ SNU, $\xi _{\textrm {thermal}} = 0.04$ SNU, $\overline {\epsilon } = \epsilon _{\textrm {PA}} = \epsilon _{\textrm {PE}}= 10^{-10}$, $N = 10^8$ points, $m/N = 1/2$, and that successive rings of the M-APSK constellations contain 4, 12, 16, 32, 64, 128, and 256 states, from the inner one to the outer one.

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Finally, in Fig. 7, we present results for the secret key rate given by Eq. (1) as a function of the excess of noise. From results in Fig. 7 we can see that a considerable gap is still present between the performance of M-APSK with discrete uniform distribution and GM, for large values of excess noise.

 figure: Fig. 7.

Fig. 7. Secret key rate as a function of the excess noise with the mean number of photons per symbol optimized for each transmission distance for several M-APSK modulation formats and GM with the finite-size effect. Here it was considered $d = 10$ km, $\eta = 0.6$, $\beta = 0.95$, $\xi _{\textrm {thermal}} = 0.04$ SNU, $\overline {\epsilon } = \epsilon _{\textrm {PA}} = \epsilon _{\textrm {PE}}= 10^{-10}$, $N = 10^8$ points, $m/N = 1/2$, and that successive rings of the M-APSK constellations contain 4, 12, 16, 32, 64, 128, and 256 states, from the inner one to the outer one.

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4. Multi-ring M-APSK with non-uniform distribution

To increase M-APSK performance, non-uniform probabilistic shaping can be applied. One of the choices is to use half of the binomial distribution. Thus, the probability of each ring, $P_p$, is given by

$$P_{p} = \frac{2}{2^{\left(2R-1\right)}} {2R-1 \choose p-1},$$
with $P_{p,k} = {P_p / M_p}$, and $\alpha$ being chosen such that $\alpha = \sqrt { V_A / \left ( 2 P_{p} \beta _p^2\right ) }$, where $V_A = 2 \langle n \rangle$ is Alice’s modulation variance. Another choice is Gaussian distribution, for which $P_{p}$ is given by $P_{p} = \exp \left (-\nu \beta _p^2 \right )$, with $\nu$ such that $\sum _{p} P_{p} = 1$. Both distributions were tested, with binomial distribution showing better performance than Gaussian distribution. Therefore, we shall consider the binomial distribution type of probabilistic shaping in the results that follow.

In Fig. 8 we show results for the secret key rate as a function of $\langle n \rangle$, considering a binomial distribution for the M-APSK constellation. From Fig. 8, we can see that the maximum achievable secret key rate for 6 rings 256-APSK, with binomial distribution, is 0.11 bits/symbol, for 2.13 photons per symbol, which corresponds to 66.29% of GM’s result. This represents an increase of 11.12% when comparing with a discrete uniform distribution, using 41.88% the number of photons per symbol (Fig. 3). Moreover, comparing the full set of results of the secret key rate as a function of the mean number of photons per symbol for discrete uniform distribution (Fig. 3) and binomial distribution (Fig. 8) we may observe the trade-off between the ability of discrete uniform distribution to use a higher mean number of photons per symbol having smaller values of secret key rate associated, and the ability of binomial distribution to obtain higher secret key rate values for lower mean number of photons per symbol. Note that the maximum mean number of photons to be used for 6 rings 256-APSK is of 6.52 photons per symbol, 0.29 times the value of considering a discrete uniform distribution.

 figure: Fig. 8.

Fig. 8. Secret key rate as a function of the mean number of photons per symbol having into account both M-APSK modulation considering binomial distribution and GM with the finite-size effect. Here it was considered $d = 10$ km, $\eta = 0.6$, $\beta = 0.95$, $\xi = 0.005$ SNU, $\xi _{\textrm {thermal}} = 0.04$ SNU, $\overline {\epsilon } = \epsilon _{\textrm {PA}} = \epsilon _{\textrm {PE}}= 10^{-10}$, $N = 10^8$ points, $m/N = 1/2$, and that successive rings of the M-APSK constellations contain 4, 12, 16, 32, 64, 128, and 256 states, from the inner one to the outer one.

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In Fig. 9 we present results for the secret key rate given by Eq. (1) as a function of the transmission distance considering finite size effects and binomial distribution for the M-APSK constellation. Table 1 compares transmission distances and secret key rate values, considering specific cases, namely, 8-PSK, 64-PSK, 128-PSK, and 256-PSK, with binomial distribution, without and with finite-size effects. Secret key rates are closer to GM’s rate than in the case of uniformly distributed M-APSK in both cases, considering and not considering the finite size effects (see Fig. 5 and Fig. 6). Considering finite-size effects, the maximum achievable transmission distance for 4 rings 64-APSK, with binomial distribution, is equivalent to 87.80% of the value for GM, being 2.8 times the value for 8-PSK, and 1.6 times the value for the regular 8-APSK with 2 rings, each with 4 points. The regular 8-APSK with 2 rings, each with 4 points constellation was also considered for a more complete analysis. Improvements can also be noted in terms of the secret key rate. For 15 km, the secret key rate of the 4 rings 64-APSK, considering binomial distribution, is 3.5 times higher than for regular 8-APSK with 2 rings, each with 4 points, and represents 0.62 of the value for GM. Small improvements in terms of achievable transmission distance can be seen by increasing the number of rings in the constellation. Nonetheless, the improvements become smaller with the rings addition.

 figure: Fig. 9.

Fig. 9. Secret key rate as a function of the transmission distance with the mean number of photons per symbol optimized for each transmission distance considered having into account both M-APSK modulation considering binomial distribution and GM with the finite-size effect. The blue and red stars signal the secret key rate for 15 km for 8-PSK and 8-APSK regular with 2 rings, each with 4 points, respectively, while the blue and red arrows signal the maximum achievable transmission distance for 8-PSK and 8-APSK regular with 2 rings, each with 4 points, respectively. Here it was considered $\eta = 0.6$, $\beta = 0.95$, $\xi = 0.005$ SNU, $\xi _{\textrm {thermal}} = 0.04$ SNU, $\overline {\epsilon } = \epsilon _{\textrm {PA}} = \epsilon _{\textrm {PE}}= 10^{-10}$, $N = 10^8$ points, $m/N = 1/2$, and that successive rings of the M-APSK constellations contain 4, 12, 16, 32, 64, 128, and 256 states, from the inner one to the outer one.

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Tables Icon

Table 1. Transmission distance for $10^{-6}$ bits/symbol without and with finite size effects and secret key rate for 75 km and for 15 km without and with finite-size effects, respectively, for 8-PSK, M-APSK, with M between 64 and 256, considering binomial distribution, and GM. The percentage values were computed in relation to GM considering 6 significant digits for transmission distance and 5 significant digits for secret key rate values.

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In Fig. 10 we show the results obtained considering the optimum $\langle n \rangle$ value that maximizes the secret key rate as a function of the transmission distance. From the results in Fig. 10 we can see clearly that the M-APSK constellation with binomial distribution allows for a higher $\langle n \rangle$ value when compared with the typical 8-PSK constellation [21] used in most of experimental work and with regular 8-APSK with 2 rings, each with 4 points, with higher secret key rates and achievable transmission distances (Fig. 9).

 figure: Fig. 10.

Fig. 10. Optimum mean number of photons per symbol as a function of the transmission distance for M-APSK with binomial distribution and finite size effects (Left axis for M-PSK modulation and right axis for GM). The blue and red stars signal the optimum mean number of photons per symbol for 15 km for 8-PSK and 8-APSK regular with 2 rings, each with 4 points, respectively, related to left axis. Here it was considered $\eta = 0.6$, $\beta = 0.95$, $\xi = 0.005$ SNU, $\xi _{\textrm {thermal}} = 0.04$ SNU, $\overline {\epsilon } = \epsilon _{\textrm {PA}} = \epsilon _{\textrm {PE}}= 10^{-10}$, $N = 10^8$ points, $m/N = 1/2$, and that successive rings of the M-APSK constellations contain 4, 12, 16, 32, 64, 128, and 256 states, from the inner one to the outer one.

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The increase on the maximum compatible excess noise is also remarkable. In Fig. 11, we present results for the secret key rate given by Eq. (1) as a function of the CV-QKD system’s excess of noise for the M-APSK constellation with binomial distribution. From the results we can see that the 4 rings 64-APSK is able to accept 0.12 SNU, 8.9 times the value for 8-PSK, and 3.4 times the value for the regular 8-APSK with 2 rings, each with 4 points. Increasing the number of rings and states in the M-APSK constellation brings the possibility to increase this value even further. A gap is still observed between 5 rings 128-APSK considering binomial distribution and GM. Nonetheless, this gap is less than half the one considering discrete uniform distribution (see Fig. 7).

 figure: Fig. 11.

Fig. 11. Secret key rate as a function of the excess noise with the mean number of photons per symbol optimized for each excess noise value considered having into account both M-APSK modulation considering binomial distribution and GM with the finite-size effect. The blue and red arrows signal the maximum accepted excess noise for 8-PSK and 8-APSK regular with 2 rings, each with 4 points, respectively. Here it was considered $d = 10$ km, $\eta = 0.6$, $\beta = 0.95$, $\xi _{\textrm {thermal}} = 0.04$ SNU, $\overline {\epsilon } = \epsilon _{\textrm {PA}} = \epsilon _{\textrm {PE}}= 10^{-10}$, $N = 10^8$ points, $m/N = 1/2$, and that successive rings of the M-APSK constellations contain 4, 12, 16, 32, 64, 128, and 256 states, from the inner one to the outer one.

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Figure 12 presents the secret key rate given by Eq. (1) as a function of the transmission distance for 4 rings 64-APSK modulation, considering binomial distribution and GM with the finite-size effect for different values of excess noise. As expected, the increase of the excess noise results in a decrease of the maximum achievable transmission distance. The decrease of the maximum achievable transmission distance is higher for the 4 rings 64-APSK than for the GM. Even so, considering $\xi = 0.025$ SNU, the 4 rings 64-APSK is still able to reach 45.4 km, 83.4% of the value for GM, while 4-APSK (equivalent to 4-PSK) and 8-PSK cannot extract keys for transmission distances above 10 km (Fig. 11).

 figure: Fig. 12.

Fig. 12. Secret key rate as a function of the transmission distance with the mean number of photons per symbol optimized for each transmission distance value considered having into account both 4 rings 64-APSK modulation considering binomial distribution and GM with the finite-size effect for different values of excess noise ($\xi = \left \lbrace 0.005, 0.01, 0.025 \right \rbrace$ SNU). Here it was considered $\eta = 0.6$, $\beta = 0.95$, $\xi _{\textrm {thermal}} = 0.04$ SNU, $\overline {\epsilon } = \epsilon _{\textrm {PA}} = \epsilon _{\textrm {PE}}= 10^{-10}$, $N = 10^8$ points, $m/N = 1/2$, and that successive rings of the 64-APSK constellation contain 4, 12, 16, and 32 states, from the inner one to the outer one.

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5. Conclusion

In conclusion, DM-CV-QKD with M-APSK constellation can approximate the results observed for GM-CV-QKD with greater performance than with the usually considered M-PSK constellation. For the conditions here considered, while 8-PSK, considering finite-size effects with our choice of parameters, can only reach 18.5 km (increasing the number of states in the M-PSK constellation brings no advantage), 4 rings 64-APSK, with binomial probability, has a maximum achievable transmission distance of 52.0 km, i.e., only 7.2 km less than for GM. In this work, we showed that binomial distribution is the one that better approximates GM. Going from 4 rings 64-APSK to 5 rings 128-APSK increases the performance of the system in terms of transmission distance. Results also show that an increase to even higher orders brings barely no advantage. Using 4 rings 64-APSK, or above, also turns on the ability to work with more than one photon per symbol, which facilitates the practical implementation of the system. In what accounts for the excess noise, it cannot be left unnoticed the capacity of 4 rings 64-APSK to accept more 0.11 SNU of excess noise than 8-PSK, i.e., less 0.08 SNU than GM. Further increase of $M$ in M-APSK increases the ability to accept more noise in the system. Little to no improvement is expected from considering M-QAM constellations instead of M-APSK constellations. Nonetheless, a broader comparison is left as future work.

Funding

European Regional Development Fund (Q.DOT Ref. POCI-01-0247-FEDER-039728); FCT/MCTES (UIDB/50008/2020-UIDP/50008/2020); Fundação para a Ciência e a Tecnologia (PhD Grant SFRH/BD/139867/2018).

Acknowledgments

This work is supported in part by the Fundação para a Ciência e a Tecnologia (FCT) through national funds, by the European Regional Development Fund (FEDER), through the Competitiveness and Internationalization Operational Programme (COMPETE 2020) of the Portugal 2020 framework under the project Q.DOT Ref. POCI-01-0247-FEDER-039728, by FCT/MCTES through national funds and when applicable co-funded EU funds under the project UIDB/50008/2020-UIDP/50008/2020 (actions DigCORE, QUESTS, and QuRUNNER), and by FCT through the PhD Grant SFRH/BD/139867/2018.

Disclosures

The authors declare no conflicts of interest.

Data availability

No data were generated or analyzed in the presented research.

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Data availability

No data were generated or analyzed in the presented research.

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Figures (12)
Fig. 1.
Fig. 1. $Z^{*}$ parameter as a function of the mean number of photons per symbol for 2, 4, 8 and 16-PSK. It is also represented the upper bound for GM and the curves considering 2 rings 16-APSK, with 4 states in the inner ring and 12 in the outer one, and 3 rings 32-APSK, with 4 states in the inner ring, 12 in the middle one and 16 in the outer one. The channel parameters are $d = 10$ km, and $\xi = 0.005$ SNU.

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Fig. 2.
Fig. 2. Four rings 64-APSK constellation with discrete uniform distribution for the rings probability. The colour gives information on the probability of each coherent state.

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Fig. 3.
Fig. 3. Secret key rate as a function of the mean number of photons per symbol having into account both M-APSK modulation and GM with the finite-size effect. Here it was considered $d = 10$ km, $\eta = 0.6$, $\beta = 0.95$, $\xi = 0.005$ SNU, $\xi _{\textrm {thermal}} = 0.04$ SNU, $\overline {\epsilon } = \epsilon _{\textrm {PA}} = \epsilon _{\textrm {PE}}= 10^{-10}$, $N = 10^8$ points, $m/N = 1/2$, and that successive rings of the M-APSK constellations contain 4, 12, 16, 32, 64, 128, and 256 states, from the inner one to the outer one.

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Fig. 4.
Fig. 4. Secret key rate as a function of the transmission distance considering a mean number of photons per symbol optimized for each transmission distance for several M-APSK modulation formats and GM with finite-size effect. The M-APSK constellations have 1, 2 and 3 rings with different number of points per ring. The number of points per ring is represented on the label by ($M_1$-$M_2$) for two rings, and ($M_1$-$M_2$-$M_3$) for three rings. Here it was considered $\eta = 0.6$, $\beta = 0.95$, $\xi = 0.005$ SNU, $\xi _{\textrm {thermal}} = 0.04$ SNU, $\overline {\epsilon } = \epsilon _{\textrm {PA}} = \epsilon _{\textrm {PE}}= 10^{-10}$, $N = 10^8$ points, and $m/N = 1/2$.

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Fig. 5.
Fig. 5. Secret key rate as a function of the transmission distance considering a mean number of photons per symbol optimized for each transmission distance for several M-APSK modulation formats and GM without the finite-size effect. Here it was considered $\eta = 0.6$, $\beta = 0.95$, $\xi = 0.005$ SNU, $\xi _{\textrm {thermal}} = 0.04$ SNU, and that successive rings of the M-APSK constellations contain 4, 12, 16, 32, 64, 128, and 256 states, from the inner one to the outer one.

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Fig. 6.
Fig. 6. Secret key rate as a function of the transmission distance with the mean number of photons per symbol optimized for each transmission distance for several M-APSK modulation formats and GM with the finite-size effect. Here it was considered $\eta = 0.6$, $\beta = 0.95$, $\xi = 0.005$ SNU, $\xi _{\textrm {thermal}} = 0.04$ SNU, $\overline {\epsilon } = \epsilon _{\textrm {PA}} = \epsilon _{\textrm {PE}}= 10^{-10}$, $N = 10^8$ points, $m/N = 1/2$, and that successive rings of the M-APSK constellations contain 4, 12, 16, 32, 64, 128, and 256 states, from the inner one to the outer one.

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Fig. 7.
Fig. 7. Secret key rate as a function of the excess noise with the mean number of photons per symbol optimized for each transmission distance for several M-APSK modulation formats and GM with the finite-size effect. Here it was considered $d = 10$ km, $\eta = 0.6$, $\beta = 0.95$, $\xi _{\textrm {thermal}} = 0.04$ SNU, $\overline {\epsilon } = \epsilon _{\textrm {PA}} = \epsilon _{\textrm {PE}}= 10^{-10}$, $N = 10^8$ points, $m/N = 1/2$, and that successive rings of the M-APSK constellations contain 4, 12, 16, 32, 64, 128, and 256 states, from the inner one to the outer one.

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Fig. 8.
Fig. 8. Secret key rate as a function of the mean number of photons per symbol having into account both M-APSK modulation considering binomial distribution and GM with the finite-size effect. Here it was considered $d = 10$ km, $\eta = 0.6$, $\beta = 0.95$, $\xi = 0.005$ SNU, $\xi _{\textrm {thermal}} = 0.04$ SNU, $\overline {\epsilon } = \epsilon _{\textrm {PA}} = \epsilon _{\textrm {PE}}= 10^{-10}$, $N = 10^8$ points, $m/N = 1/2$, and that successive rings of the M-APSK constellations contain 4, 12, 16, 32, 64, 128, and 256 states, from the inner one to the outer one.

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Fig. 9.
Fig. 9. Secret key rate as a function of the transmission distance with the mean number of photons per symbol optimized for each transmission distance considered having into account both M-APSK modulation considering binomial distribution and GM with the finite-size effect. The blue and red stars signal the secret key rate for 15 km for 8-PSK and 8-APSK regular with 2 rings, each with 4 points, respectively, while the blue and red arrows signal the maximum achievable transmission distance for 8-PSK and 8-APSK regular with 2 rings, each with 4 points, respectively. Here it was considered $\eta = 0.6$, $\beta = 0.95$, $\xi = 0.005$ SNU, $\xi _{\textrm {thermal}} = 0.04$ SNU, $\overline {\epsilon } = \epsilon _{\textrm {PA}} = \epsilon _{\textrm {PE}}= 10^{-10}$, $N = 10^8$ points, $m/N = 1/2$, and that successive rings of the M-APSK constellations contain 4, 12, 16, 32, 64, 128, and 256 states, from the inner one to the outer one.

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Fig. 10.
Fig. 10. Optimum mean number of photons per symbol as a function of the transmission distance for M-APSK with binomial distribution and finite size effects (Left axis for M-PSK modulation and right axis for GM). The blue and red stars signal the optimum mean number of photons per symbol for 15 km for 8-PSK and 8-APSK regular with 2 rings, each with 4 points, respectively, related to left axis. Here it was considered $\eta = 0.6$, $\beta = 0.95$, $\xi = 0.005$ SNU, $\xi _{\textrm {thermal}} = 0.04$ SNU, $\overline {\epsilon } = \epsilon _{\textrm {PA}} = \epsilon _{\textrm {PE}}= 10^{-10}$, $N = 10^8$ points, $m/N = 1/2$, and that successive rings of the M-APSK constellations contain 4, 12, 16, 32, 64, 128, and 256 states, from the inner one to the outer one.

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Fig. 11.
Fig. 11. Secret key rate as a function of the excess noise with the mean number of photons per symbol optimized for each excess noise value considered having into account both M-APSK modulation considering binomial distribution and GM with the finite-size effect. The blue and red arrows signal the maximum accepted excess noise for 8-PSK and 8-APSK regular with 2 rings, each with 4 points, respectively. Here it was considered $d = 10$ km, $\eta = 0.6$, $\beta = 0.95$, $\xi _{\textrm {thermal}} = 0.04$ SNU, $\overline {\epsilon } = \epsilon _{\textrm {PA}} = \epsilon _{\textrm {PE}}= 10^{-10}$, $N = 10^8$ points, $m/N = 1/2$, and that successive rings of the M-APSK constellations contain 4, 12, 16, 32, 64, 128, and 256 states, from the inner one to the outer one.

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Fig. 12.
Fig. 12. Secret key rate as a function of the transmission distance with the mean number of photons per symbol optimized for each transmission distance value considered having into account both 4 rings 64-APSK modulation considering binomial distribution and GM with the finite-size effect for different values of excess noise ($\xi = \left \lbrace 0.005, 0.01, 0.025 \right \rbrace$ SNU). Here it was considered $\eta = 0.6$, $\beta = 0.95$, $\xi _{\textrm {thermal}} = 0.04$ SNU, $\overline {\epsilon } = \epsilon _{\textrm {PA}} = \epsilon _{\textrm {PE}}= 10^{-10}$, $N = 10^8$ points, $m/N = 1/2$, and that successive rings of the 64-APSK constellation contain 4, 12, 16, and 32 states, from the inner one to the outer one.

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Tables (1)

Tables Icon

Table 1. Transmission distance for 10 − 6 bits/symbol without and with finite size effects and secret key rate for 75 km and for 15 km without and with finite-size effects, respectively, for 8-PSK, M-APSK, with M between 64 and 256, considering binomial distribution, and GM. The percentage values were computed in relation to GM considering 6 significant digits for transmission distance and 5 significant digits for secret key rate values.

View Table

Equations (11)

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K = m N ( β I B A − χ B E − Δ m ) ,
Δ m = 7 log 2 ⁡ ( 2 / ϵ ¯ ) m + 2 m log 2 ⁡ ( 1 / ϵ PA ) ,
t min ≈ T η 2 − z ϵ PE / 2 η T 2 ξ + 1 + ξ thermal 2 ( N − m ) ( 2 ⟨ n ⟩ ) ,
σ max 2 ≈ η T 2 ξ + 1 + ξ thermal + z ϵ PE / 2 ( η T 2 ξ + 1 + ξ thermal ) 2 2 ( N − m ) ,
I BA = log 2 ⁡ ( 1 + SNR ) = log 2 ⁡ ( 1 + 2 T η ⟨ n ⟩ 2 + T η ξ + 2 ξ thermal ) ,
χ BE = ∑ i = 1 2 G ( μ i − 1 2 ) − ∑ i = 3 4 G ( μ i − 1 2 ) ,
G ( x ) = ( x + 1 ) log 2 ⁡ ( x + 1 ) − x log 2 ⁡ x ,
Z ∗ ( T , ξ ) = 2 T Tr ( τ 1 / 2 a ^ τ 1 / 2 a ^ † ) − 2 T ξ W ,
W = ∑ k p k ( ⟨ α k | a ^ τ † a ^ τ | α k ⟩ − | ⟨ α k | a ^ τ | α k ⟩ | 2 ) ,
X = { β p α e i k θ p , k = 0 , 1 , … , M p − 1 , p = 1 , 2 , … , R } ,
P p = 2 2 ( 2 R − 1 ) ( 2 R − 1 p − 1 ) ,
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