⇦Chinese Physics Letters, 2019, Vol. 36, No. 10, Article code 100302 Four-State Modulation in Middle of a Quantum Channel for Continuous-Variable Quantum Key Distribution Protocol with Noiseless Linear Amplifier * Yu Mao (毛宇)1, Qi Liu (刘淇)2, Ying Guo (郭迎)3, Hang Zhang (张航)1, Jian Zhou (周健)3** Affiliations 1School of Automation, Central South University, Changsha 410083 2College of Science, Central South University of Forestry and Technology, Changsha 410004 3School of Computer Science and Engineering, Central South University, Changsha 410083 Received 20 April 2019, online 21 September 2019 *Supported by the National Natural Science Foundation of China under Grant No 61572529.
**Corresponding author. Email: 13142153489@163.com Citation Text: Mao Y, Liu Q, Guo Y, Zhang H and Zhou J et al 2019 Chin. Phys. Lett. 36 100302    Abstract We characterize a modified continuous-variable quantum key distribution (CV-QKD) protocol with four states in the middle of a quantum channel. In this protocol, two noiseless linear amplifiers (NLAs) are inserted before each detector of the two parts, Alice and Bob, with the purpose of increasing the secret key rate and the maximum transmission distance. We present the performance analysis of the new four-state CV-QKD protocol over a Gaussian lossy and noisy channel. The simulation results show that the NLAs with a reasonable gain $g$ can effectively enhance the secret key rate as well as the maximum transmission distance, which is generally satisfied in practice. DOI:10.1088/0256-307X/36/10/100302 PACS:03.67.Dd, 03.67.Hk, 42.50.Ex © 2019 Chinese Physics Society Article Text Quantum key distribution (QKD) protocols provide the interaction of two parties, Alice and Bob, to share a secret key over an authenticated classical channel and a quantum channel that is controlled by an eavesdropper Eve.[1] Generally speaking, there are two main types of QKD protocols, includes discrete-variable (DV) QKD and continuous-variable (CV) QKD.[2,3] DV-QKD encodes information in properties of single photon pulses, which has been increased experimental efforts recently, such as the high-speed Faraday–Sagnac–Michelson QKD system[4] and the twin-field quantum key distribution (TFQKD) protocol.[5,6] However, CV-QKD encodes key bits in the quadratures ($\hat{x}$ and $\hat{p}$) of the optical field, which offers the prospect of high detection efficiency and high key rate. Most of the CV-QKD protocols[7–11] belong to Gaussian modulated CV-QKD, which is the most extensively applied in CV-QKD. Recently, it is further sufficiently proved to be secure against Gaussian collective attacks via a Gaussian de Finetti reduction to obtain security against general attacks,[12] which rigorously confirms the belief that Gaussian collective attacks are indeed optimal against Gaussian modulated CV-QKD. However, the CV-QKD protocol implemented with Gaussian modulation is restricted only to short transmission distance. It is hard to reconcile correlated Gaussian variables, especially at low SNRs, which are inherent in long-distance experiments.[11] There is another approach to solve the problem; that is, using discrete modulation such as the four-state CV-QKD protocol, proposed by Leverrier and Grangier,[13,14] which generates four nonorthogonal coherent states and exploits the sign of the measured quadrature of each state to encode information rather than using the quadrature $\hat{x}$ or $\hat{p}$ itself. This is the reason why the sign of the measured quadrature is already the discrete value to which the most excellent error-correcting codes are suitable even at very low SNR.[15] However, they do not yield a general security proof for discrete modulated CV-QKD.[13,14] Recently, a lower bound on the asymptotic secret key rate of CV-QKD with a discrete modulation has been established, which is valid against collective attacks and is obtained by formulating the problem as a semidefinite program.[16] An idea placing the untrusted source in the middle between Alice and Bob for quantum teleportation was put forward to keep the high secret key rate and as a defence against attack.[17] In this way, a secure key can still be generated between Alice and Bob via the previous analytical techniques. As mentioned earlier, for Gaussian modulation strategy, the obvious contradiction is that working with high SNR requires a good reconciliation, yet high reconciliation efficiency could not reach an ideal value at a low SNR.[18] Hence, based on the passive restriction both on the maximum transmission distance and Gaussian modulation,[15] we prepare a four-state protocol as the non-Gaussian source in the middle of the CV-QKD protocol. It is interesting to introduce a device in the CV-QKD system called noiseless linear amplifier (NLA), which has been proven to be available on experimental operation.[19,20] Using an NLA can probabilistically applied in the CV-QKD protocol to increase the maximum transmission distance. In this study, we describe an improved scheme based on the original CV-QKD protocol. A two-mode entangled state is initially prepared as non-Gaussian source in the middle between Alice and Bob by unsecure third party, Charlie. Using the NLAs before Alice and Bobs' detections in the presence of lossy and noisy Gaussian channels. For the four-state modulation in the middle of the CV-QKD protocol with NLAs, the gain $g$ of the NLAs can lead to improvement of the maximum transmission distance and the secret key rate. To show the effect of the quantum network with four-state non-Gaussian source in the middle, we first describe the prepare-and-measure (PM) and entanglement-based (EB) versions of the four-state CV-QKD protocol. In the PM scheme of the four-state CV-QKD protocol,[21] Alice randomly sends one of the four coherent states: $\rho=\frac{1}{4}\sum\limits_{k=0}^3|\alpha_{k}\rangle\langle\alpha_{k}|$ with $\alpha_{k}=\alpha e^{i\pi(2k+1)/4}$ to Bob through a quantum channel, see Fig. 1(a), where $\alpha$ is chosen to be a real positive number. The probability of each state sent is 1/4. For a certain channel, that is, given the transmittance $T$ and excess noise $\epsilon$, we can set the value of the $\alpha$ to maximize the secret key rate.[14] After receiving the coherent state, Bob measures randomly one of the quadratures in homodyne detection and decodes the information by the sign of his measurement result. Afterwards, Bob sends the absolute value results to Alice through classical channel. In this approach, Alice and Bob share correlated strings of bits. Then through the operations of error reconciliation and privacy amplification, the secret key can be obtained. The PM version is relatively easy to implement, but its security is hard to be proved directly. Hence, the security that we describe is established by considering the equivalent entanglement-based version of the protocol.
cpl-36-10-100302-fig1.png
Fig. 1. (a) PM scheme of the four-state CV-QKD protocol: Alice obtained the random number $k$ from a random number generator (RNG), and then modulated the original state sent from the source by modulator. (b) The equivalent EB version of the protocol: Alice prepared a two-mode entangled state, one mode is measured to obtain the result $k$, and the other state of the mode is sent to Bob.
We next describe the four-state CV-QKD protocol performed in the EB version,[22] see Fig. 1(b). Alice prepares the entangled state $|{\it \Phi}_{\rm AB}\rangle$ and performs the projective measurement $\{|\psi_{0}\rangle\langle\psi_{0}|$, $|\psi_{1}\rangle\langle\psi_{1}|$, $|\psi_{2}\rangle\langle\psi_{2}|$, $|\psi_{3}\rangle\langle\psi_{3}|\}$ on her half. When her measurement gives the result $k$, the coherent state $|\alpha_{k}\rangle$ is sent through the quantum channel to Bob who measures either one of the quadratures with a homodyne detection. The EB version provides a valid security proof against collective attacks through the covariance matrix $\gamma_{\rm AB}$ of the state before their respective measurements.[23] The covariance matrix $\gamma_{A_{0}B_{0}}$ of the original four state $|{\it \Phi}_{\rm AB}(\alpha)\rangle$ is $$ \gamma_{A_{0}B_{0}}=\begin{bmatrix} VI_{2}& Z\sigma _{z}\\ Z\sigma _{z} & VI_{2} \end{bmatrix},~~ \tag {1} $$ where $I_{2}$ is the $2\times2$ identity matrix, and $\sigma_{z}={\rm diag} (1,-1)$, $V=2\alpha^{2}+1=V_{\rm A}+1$ is the variance of quadratures for modes A and B, and $$ Z=V_{\rm A}\sum\limits_{k=0}^3\frac{\lambda_{k-1}^{3/2}}{\lambda_{k}^{1/2}}~~ \tag {2} $$ reflects the relationship between mode A and mode B. The detailed formula derivation of $Z$ can be seen in Ref. [14]. We begin by introducing the scheme that four coherent states placed in the middle as non-Gaussian source between Alice and Bob. We also place two noiseless linear amplifiers (NLAs), one at Alice's side and the other at Bob's side. We then analyze the security under the Gaussian collective attacks, and calculate the secret key rate of the modified protocol.
cpl-36-10-100302-fig2.png
Fig. 2. (a) Modified four-state CV-QKD with two NLAs. (b) A virtually equivalent protocol without NLAs. A state $|{\it \Phi}_{\rm AB}(\alpha)\rangle$ sent in the middle through two Gaussian channels with transmittance $T_{1}$ and excess noise $\epsilon_{1}$ to Alice and $T_{2}$, $\epsilon_{2}$ to Bob, respectively, with a successful amplification followed. The protocol has the same final covariance matrix with a state $\zeta$ sent through two Gaussian channels of transmittance $\eta_{1}$, $\eta_{2}$ and excess noise $\epsilon_{1}^{g}$, $\epsilon_{2}^{g}$, without two NLAs.
As shown in Fig. 2, an entangled four-state with variance $V$ is in the middle between two participants, Alice and Bob. The source is employed by Alice and Bob to generate a secure key for encryption.[24] The entangled source we select is based on the consideration that the four-state scheme could reach a long transmission distance at very low SNR under the practical lossy and noisy Gaussian channel. There are two entangling cloner attacks by Eve on each side of the source. The loss is simulated by two separate beam splitters with transmittances $T_{1}$ and $T_{2}$, where $T_{i} \in[0,1]$. The scheme of the four-state CV-QKD protocol with an untrusted source in the middle can be described with two steps.[25] First, the untrusted third party, Charlie encodes information in the quadratures of two-mode coherent states, and then he sends one of the four-state to Alice and Bob respectively with equal probability through unsecure channels 1 and 2 which could completely controlled by the eavesdropper, and Eve may perform her attack. Then Alice and Bob perform either homodyne (Hom) or heterodyne (Het) detection on the received modes $A_{2}$ and $B_{2}$. Once Alice and Bob have collected a sufficiently large set of correlated data, they proceed with classical data post-processing, privacy amplification and error reconciliation, which can be performed as either direct reconciliation (DR) or reverse reconciliation (RR).[15] The security of discrete modulated CV-QKD has recently been proven to be secure against Gaussian collective attacks,[16] which is obtained by solving a semidefinite program that computes the covariance matrix of the state shared by Alice and Bob in the entanglement-based version of the protocol. However, it has not been completely proven to be secure for discrete modulated CV-QKD with general attacks. Based on these considerations, the secure analysis in this work mainly aims at collective attacks. Before the security analysis, we assume that for each of the two quantum channels, Eve performs Gaussian collective attacks, and Eve interacts her independent ancilla modes with the resultant modes of Alice and Bob to generate a memoryless Gaussian channel. For calculation, we assume that reverse reconciliation is set to the default.[18] Eve is generally confined to the Holevo bound $S_{\rm BE}$. The secret key rate of the entangled in the middle CV-QKD protocol for reverse reconciliation is defined as $$ K(\alpha,T,\epsilon)\geq\beta I_{\rm AB}(\alpha,T,\epsilon)-S_{\rm BE}(\alpha,T,\epsilon),~~ \tag {3} $$ where $I_{\rm AB}$ is the mutual information of Alice and Bob, $S_{\rm BE}$ is the Holevo bound of the mutual information between Eve and Bob, and $\beta$ represents reconciliation efficiency. In our calculation, $\beta=95\%$, which could be achieved using an irregular LDPC technique; i.e., MET-LDPC code[26] and multi-edge quasi-cyclic LDPC codes.[27] The mutual information of Alice and Bob could be expressed as $$\begin{alignat}{1} \!\!\!\!\!\!\!\!I_{\rm AB}(\alpha,T,\epsilon)=\frac{1}{2}\log_{2}\Big(\frac{V\!+\!\chi}{1\!+\!\chi}\Big) =\frac{1}{2}\log_{2}(1\!+\!SNR),~~ \tag {4} \end{alignat} $$ where $\chi=\frac{1}{T}+\epsilon-1$ is the equivalent total noise at input. It can be noted that $S_{\rm BE}$ could obtain the maximum value under the condition that the state $\rho$ is of Gaussian states.[13] Hence, we could replace the state $\rho$ with $\rho^{G}$ to simplify the calculation of $S_{\rm BE} ^G$. Clearly, one has $$ S_{\rm BE}\leq S_{\rm BE}^G=G\Big(\frac{\upsilon_{1}-1}{2}\Big)+G\Big(\frac{\upsilon_{2}-1}{2}\Big) -G\Big(\frac{\upsilon_{3}-1}{2}\Big),~~ \tag {5} $$ where $G(x)=(x+1)\log_{2}(x+1)-x\log_{2}x$, and $\upsilon_{1,2}$ represent the symplectic eigenvalues of the covariance matrix $\gamma_{\rm AB}$. Furthermore, after passing through the Gaussian channel 1 with transmittance $T_{1}$ and excess noise $\epsilon_{1}$ and the Gaussian channel 2 with transmittance $T_{2}$ and excess noise $\epsilon_{2}$, the state $|{\it \Phi}_{\rm AB}(\alpha)\rangle$ becomes $\rho_{\rm AB}$ with covariance matrix $$\begin{alignat}{1} \gamma_{\rm AB}(\alpha,T,\epsilon)=\,&\begin{bmatrix} aI_{2}& c\sigma_{z}\\ c\sigma_{z}& bI_{2} \end{bmatrix}\\ =\,&\begin{bmatrix} T_{1}(V+\chi)I_{2}&\sqrt{T_{1}T_{2}}Z\sigma_{z}\\ \sqrt{T_{1}T_{2}}Z\sigma_{z}& T_{2}(V+\chi)I_{2}\end{bmatrix}.~~ \tag {6} \end{alignat} $$ We assume that the states sent to Alice and Bob in the quantum channels have the unconditional same circumstance; i.e., the transmittance $T_{1}$ equals $T_{2}$ and excess noise $\epsilon_{1}$ equals $\epsilon_{2}$ in transmission.[13] Based on the covariance matrix $\gamma_{\rm AB}(\alpha,T,\epsilon)$, the symplectic eigenvalues $\upsilon_{1,2}$ could be given by $$ \upsilon_{1,2}=\sqrt{\frac{1}{2}({\it \Delta}\pm\sqrt{{\it \Delta}^{2}-4D^{2}})},~~ \tag {7} $$ where $$ \begin{cases} {\it \Delta}=a^{2}+b^{2}-2c^{2}=2T^{2}(V+\chi)^{2}-2T^{4}Z^{2},\\ D=ab-c^{2}=T^{2}(V+\chi)^{2}-T^{2}Z^{2}. \end{cases}~~ \tag {8} $$ Ultimately, we could obtain the lower bound of the secret key rate as follows: $$ \underline{K}(\alpha,T,\epsilon,\beta)=\beta I_{\rm AB}(\alpha,T,\epsilon)-S_{\rm BE}^G(\alpha,T,\epsilon).~~ \tag {9} $$ For the proposed protocol, when the final secret key rate satisfies $\underline{K}(\alpha,T,\epsilon,\beta)>0$, it could be concluded that the communication scheme is unconditional secure against any attacks.[13,14] Based on this, we consider to add NLAs before the detectors of Alice and Bob for four-state modulation in the middle scheme, which is assumed to be perfect for simplicity of analysis, see Fig. 2(a). In this part, Eve implements two entangling cloner attacks on each side of the source, which is a common example of collective Gaussian attacks. The quantum network with NLAs can be reformulated in a simplified system using equivalent parameters. The covariance matrix $\gamma_{\rm AB}(\lambda,T_{1},\epsilon_{1},g_{1},T_{2},\epsilon_{2},g_{2})$ of the amplified state with four-state parameter $\lambda$ is equal to the covariance matrix $\gamma_{\rm AB}^{'}(\zeta,\eta_{1}, \epsilon_{1}^{g},g_{1}=1,\eta_{2}, \epsilon_{2}^{g},g_{2}=1)$ without using the NLAs, as shown in Fig. 2(b). The mapping relations of these parameters are given by[28] $$ \begin{cases} \zeta=\lambda\sqrt{\frac{[(g_{1}^{2}-1)(\epsilon-2)T-2] \cdot[(g_{2}^{2}-1)(\epsilon-2)T-2]}{[(g_{1}^{2}-1)\epsilon T-2]\cdot[(g_{2}^{2}-1)\epsilon T-2]}},\\ \eta_{1}=\frac{4Tg_{1}^{2}}{T(g_{1}^{2}-1)\cdot[(g_{1}^{2}-1)(\epsilon-2)\epsilon T-4(\epsilon-1)]+4},\\ \epsilon_{1}^{g}=\epsilon-\frac{1}{2}(g_{1}^{2}-1)(\epsilon-2)\epsilon T\\ \eta_{2}=\frac{4Tg_{2}^{2}}{T(g_{2}^{2}-1)\cdot[(g_{2}^{2}-1)(\epsilon-2)\epsilon T-4(\epsilon-1)]+4},\\ \epsilon_{2}^{g}=\epsilon-\frac{1}{2}(g_{2}^{2}-1)(\epsilon-2)\epsilon T \end{cases},~~ \tag {10} $$ where we assume that the quantum channels $1$ and $2$ are identical, thus the states go through their channels with the same parameters $(T=T_{1}=T_{2}, \epsilon=\epsilon_{1}=\epsilon_{2})$. Those parameters can be interpreted as physical parameters in an equivalent system if they are able to meet the conditions, as follows: $$ \begin{cases} 0\leq\zeta < 1,\\ 0\leq\eta_{1}\leq1,~\epsilon_{1}^{g}\geq0,\\ 0\leq\eta_{2}\leq1,~\epsilon_{2}^{g}\geq0. \end{cases}~~ \tag {11} $$ Since $\lambda$ is the only factor that affects parameter $\zeta$, $\eta_{1}$, $\epsilon_{1}^{g}$, $\eta_{2}$ and $\epsilon_{2}^{g}$ do not depend on $\lambda$. Thus the first condition is always satisfied on the condition that $\lambda$ conforms to the inequation $$\begin{alignat}{1} \!\!\!\!\!\!\!&0\leq\lambda\\ \!\!\!\!\!\!\! < \,&\sqrt{\frac{[(g_{1}^{2}\!-\!1)\epsilon T\!-\!2]\cdot[(g_{2}^{2}\!-\!1)\epsilon T\!-\!2]}{[(g_{1}^{2}\!-\!1)(\epsilon-2)T\!-\!2]\cdot[(g_{2}^{2}\!-\!1)(\epsilon-2)T\!-\!2]}}.~~ \tag {12} \end{alignat} $$
cpl-36-10-100302-fig3.png
Fig. 3. Maximum value of the gain $g_{\rm max}$ as functions of the loss and the excess noise. Maximum value $g_{\rm max}$ rises with the increase of the loss while the excess noise has a negative effect on $g_{\rm max}$.
Only when $\epsilon$ is smaller than 2 and the gains of the two NLAs are smaller than the maximum value can we obtain the maximum value of the gain $g_{\rm max}$ given by $$\begin{alignat}{1} \!\!\!\!\!g_{1}^{\rm max}=\,&g_{2}^{\rm max}\\ \!\!\!\!\!=\,&\sqrt{\frac{\epsilon[T(\epsilon-2)+2] -2\sqrt{\epsilon[T(\epsilon-2)+2]}}{T\epsilon(\epsilon-2)}}.~~ \tag {13} \end{alignat} $$ The function of $g_{\rm max}$ has been plotted, as shown in Fig. 3, which is valuable for us to find the optimal parameter $g$ according to the values of loss and excess noise.[29] It is clear that under the same channel lossy, decreasing the channel noise would improve the maximum amplification gain $g_{\rm max}$. What is more, when the gain $g$ is too high, the excess noise $\epsilon$ and transmittance $T$ would be increased accordingly, which will bring about the negative effect of the secret key rate and the maximum transmission distance. We can obtain the final secret key rate with NLA $K_{\rm NLA}$ via multiplying the secret key rate for successful amplifications $K(\zeta,\eta,\epsilon^{g})$ by the probability of success $P_{\rm ss}$.[22] The theoretical bound of the probability of success $P_{\rm ss}$ denoted as $1/g^{2}$ has been discussed.[30] Afterwards, the secret key rate $K_{\rm NLA}$ of the four-state in the middle QKD protocol with NLAs can be derived, and $K_{\rm NLA}$ can be obtained, $$ K_{\rm NLA}=P_{\rm ss}K(\zeta,\eta,\epsilon^{g}),~~ \tag {14} $$ where the success probability $P_{\rm ss}$ for NLAs with gain $g$ is bounded by $1/g^{2}$. The precise value $P_{\rm ss}$ will depend on practical implementations, where we set its value as $1/g^{2}$ for simplify.[30] It can be notice that with the increase of the gain $g$, the probability of successful implementation of amplifier would be decreased. Figure 4 illustrates the effects of Gaussian channel excess noise $\epsilon$ on the transmission performance. It can be noticed that the excess noise has negative impacts on both transmission distance and the secret key rate. The numeral results show that with the decrease of the excess noises, the secret key rates could be enhanced, which means that the performance of the scheme has better security, and the maximal transmission could be enhanced at the same time. Meanwhile, based on the structure of NLAs, other noise would not be produced since the noise has been discussed with $P_{\rm ss}$.
cpl-36-10-100302-fig4.png
Fig. 4. The secret key rate for the four-state in the middle CV-QKD protocol using NLAs against the transmission distance with different excess noises $\epsilon=0.2$, 0.02, 0.002, and 0.0002. In the simulations, $V=2.25$, $\beta=0.95$, $g=2$ and $P_{\rm ss}=1/g^{2}$.
The transmittance $T$ is also an unavoidable factor during the transmission over a lossy and noise channel. The transmittance $T=10^{-ad/10}$, where $a=0.2$ dB/km is the loss coefficient of the optical fibres, and $d$ is the quantum channel transmission distance.[28] In this study, we have taken into account all of two noises that may appear in our scheme. In the following we discuss the performance about the gain $g$ of NLAs.
cpl-36-10-100302-fig5.png
Fig. 5. The secret key rate for the four-state source in the middle CV-QKD protocol with ($g=1.5$, 2 and 2.5) and without ($g=1$) the NLA against the transmission distance. In the simulations, $V=2.25$, $\beta=0.95$, $\epsilon=0.02$ and $P_{\rm ss}=1/g^{2}$.
Based on this analysis, we illustrate the secret key rates (bit/pulse) as a function of transmission distance (km) see Fig. 5. The various parameters are chosen from theoretical simulation values $V=2.25$, $\beta=0.95$, and $\epsilon=0.02$. The simulation results indicate that the blue line represents the protocol with no NLAs ($g=1$), which could reach the maximum transmission distance about 120 km. With the increase of the gain $g$, the maximum secure distances improve obviously. NLAs with $g=2.5$ inserted in four-state protocol enhances almost 80 km of the communication distance longer than the original four-state protocol. Using the NLAs, the maximal tolerable excess noise can be increased. It is noted that the secret key rate decreases slowly before the transmission distance to 180 km, where we can obtain the optimizing of performance between the secret key rate and the secure distance. The modified CV-QKD protocol has a desirable improvement on the limitation of short transmission distance, whose maximum secure distance can reach over 190 km. Fig. 5 shows that the maximum transmission distance can be increased by adding the values of the gain $g$ in a specified range. This happens because of the competition between decreasing probability $P_{\rm ss}$ of success $1/g^{2}$ and the increase of the secret key rate for the successfully amplified states, see Eq. (14). We need to notice that with the increase of the gain $g$, the probability of success $P_{\rm ss}=1/g^{2}$ would decrease as a power function of 2, which is the key reason that the secret key rate would decline simultaneously. In summary, we have proposed a CV-QKD protocol with four-state source in the middle of channel by placing two NLAs before each receiving terminal, and have analyzed the performances of the functional relationship between secret key rate $K$ and transmission distance $d$. The simulation results show that the inserted NLAs in the four-state protocol can increase the communication distance and maximal transmission tolerable losses by almost 16 dB compared with the original four-state protocol. With increasing the value of gain $g$, the secure distance would improve simultaneously, while with the increase of the gain, the secret key rate would decline slightly. Thus, it is remarkable that setting a suitable parameter $g$ is necessary in practical operation. Furthermore, the excess noise from the quantum channel has negative impacts on both transmission distance and the secret key rate. It is therefore clear that using practical devices such as optical parametric amplifiers to overcome the distance limitation of CV-QKD systems is indeed possible. It is worth noting that up to date the security analysis of the discrete modulated CV-QKD against general attacks has not been completely proven to be secure, thus in this work we restrict Eve to perform Gaussian collective attacks. The study of general attacks on discrete modulated CV-QKD still needs further investigations. 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