Long-distance continuous-variable quantum key distribution using non-Gaussian state-discrimination detection

In general, the four-state CVQKD protocol is derived from discretely modulated CVQKD, which can be generalized to the one with N coherent states $| {\alpha }_{k}^{N}\rangle =| \alpha {{\rm{e}}}^{{\rm{i}}2k\pi /N}\rangle$, where k ∈ {0, 1, ... , N} [10]. For the four-state CVQKD protocol, we have $| {\alpha }_{k}^{4}\rangle =| \alpha {{\rm{e}}}^{{\rm{i}}(2k+1)\pi /4}\rangle$, where k ∈ {0, 1, 2, 3}, α is a positive number related to the modulation variance of coherent state as V_M = 2α².

Let us consider the PM version of the four-state CVQKD protocol first. Alice randomly chooses one of the coherent states $| {\alpha }_{k}^{4}\rangle$ and sends it to the remote Bob through a lossy and noisy quantum channel, which is characterized by a transmission efficiency η and an excess noise ε. When Bob receives the modulated coherent states, he can apply either homodyne or heterodyne detector with detection efficiency τ and electronics noise v_el to measure arbitrary one of the two quadratures $\hat{x}$ or $\hat{p}$ (or both quadratures). The mixture state that Bob received can be expressed with the following form

$$\begin{eqnarray}&&{\rho }_{4}=\displaystyle \frac{1}{4}\displaystyle \sum _{k=0}^{3}| {\alpha }_{k}^{4}\rangle \langle {\alpha }_{k}^{4}| .\end{eqnarray} \tag{ 1 }$$

After measurement, Bob then reveals the absolute values of measurement results through a classical authenticated channel and keeps their signs. Alice and Bob exploit the signs to generate the raw key. After conducting post-processing procedure, they can finally establish a correlated sequence of random secure key.

The PM version of the protocol is equivalent to the EB version, which is more convenient for security analysis. In EB version, Alice prepares a pure two-mode entangled state

$$\begin{eqnarray}| {{\rm{\Psi }}}_{4}\rangle & = & \displaystyle \sum _{k=0}^{3}\sqrt{{\lambda }_{k}}| {\phi }_{k}\rangle | {\phi }_{k}\rangle \\ & = & \displaystyle \frac{1}{2}\displaystyle \sum _{k=0}^{3}| {\psi }_{k}\rangle | {\alpha }_{k}^{4}\rangle ,\end{eqnarray} \tag{ 2 }$$

where the states

$$\begin{eqnarray}&&| {\psi }_{k}\rangle =\displaystyle \frac{1}{2}\displaystyle \sum _{m=0}^{3}{{\rm{e}}}^{{\rm{i}}(1+2k)m\pi /4}| {\phi }_{m}\rangle \end{eqnarray} \tag{ 3 }$$

are the non-Gaussian states, and the state $| {\phi }_{m}\rangle$ is given by

$$\begin{eqnarray}&&| {\phi }_{k}\rangle =\displaystyle \frac{{{\rm{e}}}^{-{\alpha }^{2}/2}}{\sqrt{{\lambda }_{k}}}\displaystyle \sum _{n=0}^{\infty }{(-1)}^{n}\displaystyle \frac{{\alpha }^{4n+k}}{\sqrt{(4n+k)!}}| 4n+k\rangle ,\end{eqnarray} \tag{ 4 }$$

with

$$\begin{eqnarray}&&{\lambda }_{\mathrm{0,2}}=\displaystyle \frac{1}{2}{{\rm{e}}}^{-{\alpha }^{2}}[\cosh ({\alpha }^{2})\pm \cos ({\alpha }^{2})],\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{\lambda }_{\mathrm{1,3}}=\displaystyle \frac{1}{2}{{\rm{e}}}^{-{\alpha }^{2}}[\sinh ({\alpha }^{2})\pm \sin ({\alpha }^{2})].\end{eqnarray} \tag{ 6 }$$

Consequently, the mixture state ρ₄ can be expressed by

$$\begin{eqnarray}{\rho }_{4} & = & \mathrm{Tr}(| {{\rm{\Psi }}}_{4}\rangle \langle {{\rm{\Psi }}}_{4}| )\\ & = & \displaystyle \sum _{k=0}^{3}{\lambda }_{k}| {\phi }_{k}\rangle \langle {\phi }_{k}| .\end{eqnarray} \tag{ 7 }$$

Let A and B, respectively, denote the two output modes of the bipartite two-mode entangled state $| {{\rm{\Psi }}}_{4}\rangle ,\hat{a}$ and $\hat{b}$ denote the annihilation operators applying to mode A and B, respectively. We have the covariance matrix Γ_AB of the bipartite state $| {{\rm{\Psi }}}_{4}\rangle$ with the following form

$$\begin{eqnarray}{{\rm{\Gamma }}}_{{AB}}=\left(\begin{array}{cc}X{\mathbb{I}} & {Z}_{4}{\sigma }_{z}\\ {Z}_{4}{\sigma }_{z} & Y{\mathbb{I}}\end{array}\right),\end{eqnarray} \tag{ 8 }$$

where ${\mathbb{I}}$ and σ_z represent diag(1,1) and diag(1, − 1), respectively, and

$$\begin{eqnarray}X & = & \langle {{\rm{\Psi }}}_{4}| 1+2{a}^{\dagger }a| {{\rm{\Psi }}}_{4}\rangle =1+2{\alpha }^{2},\\ Y & = & \langle {{\rm{\Psi }}}_{4}| 1+2{b}^{\dagger }b| {{\rm{\Psi }}}_{4}\rangle =1+2{\alpha }^{2},\\ {Z}_{4} & = & \langle {{\rm{\Psi }}}_{4}| {ab}+{a}^{\dagger }{b}^{\dagger }| {{\rm{\Psi }}}_{4}\rangle =2{\alpha }^{2}\displaystyle \sum _{k=0}^{3}{\lambda }_{k-1}^{3/2}{\lambda }_{k}^{-1/2}.\end{eqnarray} \tag{ 9 }$$

Note that the addition arithmetic should be operated with modulo 4. The detailed derivation of the four-state CVQKD protocol can be found in [12].

After preparing the two-mode entangled state $| {{\rm{\Psi }}}_{4}\rangle$ with variance V = 1 + V_M, Alice performs projective measurements $| {\psi }_{k}\rangle \langle {\psi }_{k}| \ (k=0,1,2,3)$ on mode A, which projects another mode B onto a coherent state $| {\alpha }_{k}^{4}\rangle$. Alice subsequently sends mode B to Bob through the quantum channel. Bob then applies homodyne (or heterodyne) detection to measure the incoming mode B. Finally, the two trusted parties Alice and Bob extract a string of secret key by using error correction and privacy amplification.

In what follows, we elaborate the long-distance discretely modulated CVQKD scheme. This novel scheme is based on the four-state CVQKD protocol so that its transmission distance could be extended more largely comparing with the continuous modulation counterparts. We focus on the principle of the whole long-distance CVQKD scheme first, leaving the detailed techniques description to the next section.

As shown in figure 1, a source of the two-mode entangled state (Einstein–Podolsky–Rosen (EPR) state) is used for creating a secure key [30]. After Alice prepares the entangled state $| {{\rm{\Psi }}}_{4}\rangle$, she performs heterodyne detection on one half (mode A) of the state and sends another half (mode B) to the photon-subtraction operation (the module within the green box). This non-Gaussian operation is modeled by a beam splitter (BS) with transmittance μ and a vacuum state $| 0\rangle$ imports the unused port of the BS. As a result, the incoming signal (mode B) is then divided into two parts by the photon-subtraction operation. There are two advantages for applying photon-subtraction operation. Firstly, putting a proper non-Gaussian operation at Alice's side has been proven to be beneficial for lengthening the maximal transmission distance of the traditional CVQKD [25], because this operation can be deemed the preparation trusted noise controlled by Alice, which can well prevent the eavesdropper from acquiring communication information [23, 31]. Secondly, the photon-subtraction operation tactfully provides a method to divide the incoming signal into two parts, which are the mode B₁ containing most photons for homodyne (or heterodyne) detector and the mode C containing a few subtracted j photons (or even one) for state-discrimination detector, respectively. The two parts of signal are subsequently recombined by using the polarization-multiplexing technique with a polarizing BS (PBS). The recombinational mode B₂ is then sent to the lossy and insecure quantum channel.

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In the EB CVQKD scheme, the quantum channel is replaced by an eavesdropper (say Eve) who performs the collective Gaussian attack strategy. This attacks is proved to be an optimal attack strategy in direct and reverse reconciliation (RR) protocols. Very recently, Leverrier [32] shows that it is sufficient to prove the security of CVQKD against collective Gaussian attacks in order to obtain security against general attacks, therefore confirming rigorously the belief that collective Gaussian attacks are indeed optimal against CVQKD. In this kind of attacks, Eve usually prepares her ancillary system in a product state and each ancilla interacts individually with a single pulse sent by Alice, being later stored in a quantum memory [33]. The tripartite state then reads,

$$\begin{eqnarray}&&{\rho }_{{\rm{ABE}}}={\left[\displaystyle \sum _{a}P(a)| a\rangle \langle a{| }_{a}\otimes {\psi }_{{\rm{BE}}}^{a}\right]}^{\otimes n}.\end{eqnarray} \tag{ 10 }$$

After eavesdropping the communication revealed by Alice and Bob in the data post-processing, Eve applies the optimal collective measurement on the ensemble of stored ancilla to steal the secret information. In particular, Eve can launch the so called entangling cloner [17, 21, 34] attack which is a kind of collective Gaussian attack. Specifically, Eve replaces the channel with transmittance η and excess noise referred to the input χ by preparing the ancilla $| E\rangle$ with variance W and a BS with transmittance η. The value W can be tuned to match the noise of the real channel χ_line = (1 − η)/η + ε. After that, Eve keeps one mode E₁ of $| E\rangle$ and injects the mode E₂ into the unused port of the BS and thus acquires the output mode E₃. After repeating this process for each pulse, Eve stores her ancilla modes, E₁ and E₃, in quantum memories. Finally, Eve measures the exact quadrature on E₁ and E₃ after Alice and Bob reveal the classical communication information. The measurement of E₁ allows her to decrease the noise added by E₃.

After passing the untrusted quantum channel, Bob applies another PBS with dynamic polarization controller to demultiplex the incoming signal. One of the demultiplexed modes B₄ is then sent to Bob's homodyne or heterodyne detector which is modeled by a BS with transmittance τ and its electronic noise is modeled by an EPR state with variance v_el. The mode D is synchronously sent to the state-discrimination detector to improve the system's performance.

This long-distance CVQKD scheme subtly combines the merits of the four-state CVQKD protocol and photon-subtraction operation in terms of lengthening maximal transmission distance, surpassing the SQL via the state-discrimination detector.

As shown in figure 1, we suggest the EB CVQKD with photon-subtraction operation (the green box) applied at Alice's station, where other modules are temporarily ignored. Alice uses a BS with transmittance μ to split the incoming mode B and the vacuum state C₀ into modes B₁ and C. The yielded tripartite state ${\rho }_{{{ACB}}_{1}}$ can be expressed by

$$\begin{eqnarray}&&{\rho }_{{{ACB}}_{1}}={U}_{{\rm{BS}}}[| {\rm{\Psi }}{\rangle }_{4}\langle {\rm{\Psi }}{| }_{4}\otimes | 0\rangle \langle 0| ]{U}_{{\rm{BS}}}^{\dagger }.\end{eqnarray} \tag{ 11 }$$

Subsequently a photon-number-resolving detector (PNRD, black dotted box at Alice's side) is adopted to measure mode C by applying positive operator-valued measurement (POVM) $\{{\hat{{\rm{\Pi }}}}_{0},{\hat{{\rm{\Pi }}}}_{1}\}$ [35]. The photon number of subtraction j depends on ${\hat{{\rm{\Pi }}}}_{1}=| j\rangle \langle j|$. Only when the POVM element ${\hat{{\rm{\Pi }}}}_{1}$ clicks can Alice and Bob keep A and B₁. The photon-subtracted state ${\rho }_{{{AB}}_{1}}^{{\hat{{\rm{\Pi }}}}_{1}}$ is given by

$$\begin{eqnarray}&&{\rho }_{{{AB}}_{1}}^{{\hat{{\rm{\Pi }}}}_{1}}=\displaystyle \frac{{\mathrm{tr}}_{C}({\hat{{\rm{\Pi }}}}_{1}{\rho }_{{{ACB}}_{1}})}{{\mathrm{tr}}_{{{ACB}}_{1}}({\hat{{\rm{\Pi }}}}_{1}{\rho }_{{{ACB}}_{1}})},\end{eqnarray} \tag{ 12 }$$

where tr_X(·) is the partial trace of the multi-mode quantum state and ${\mathrm{tr}}_{{{ACB}}_{1}}({\hat{{\rm{\Pi }}}}_{1}{\rho }_{{{ACB}}_{1}})$ is the success probability of subtracting j photons, which can be calculated as

$$\begin{eqnarray}{P}_{(j)}^{{\hat{{\rm{\Pi }}}}_{1}} & = & {\mathrm{tr}}_{{{ACB}}_{1}}({\hat{{\rm{\Pi }}}}_{1}{\rho }_{{{ACB}}_{1}})\\ & = & (1-{\xi }^{2})\displaystyle \sum _{n=j}^{\infty }{C}_{n}^{j}{\xi }^{2n}{(1-\mu )}^{j}{\mu }^{n-j}\\ & = & \displaystyle \frac{(1-{\xi }^{2}){(1-\mu )}^{j}{\xi }^{2j}}{{(1-\mu {\xi }^{2})}^{j+1}},\end{eqnarray} \tag{ 13 }$$

where C_n^j is combinatorial number and $\xi =\tfrac{\alpha }{\sqrt{1+{\alpha }^{2}}}$.

After passing the BS, it is worth noticing that the subtracted state ${\rho }_{{{AB}}_{1}}^{{\hat{{\rm{\Pi }}}}_{1}}$ is not Gaussian anymore, while its entanglement degree increases with the introduction of the photon-subtraction operation [23, 25].

Due to the fact that heterodyne detection on one half of the EPR state will project the other half onto a coherent state, which is convenient to implement in experimentation, we take into account a situation where Alice performs heterodyne detection and Bob executes homodyne detection. Suppose ${{\rm{\Gamma }}}_{{{AB}}_{1}}^{(j)}$ represents the covariance matrix of ${\rho }_{{{AB}}_{1}}^{{\hat{{\rm{\Pi }}}}_{1}}$, and it can be given by

$$\begin{eqnarray}{{\rm{\Gamma }}}_{{{AB}}_{1}}^{(j)}=\left(\begin{array}{cc}X^{\prime} {\mathbb{I}} & {Z}_{4}^{{\prime} }{\sigma }_{z}\\ {Z}_{4}^{{\prime} }{\sigma }_{z} & Y^{\prime} {\mathbb{I}}\end{array}\right),\end{eqnarray} \tag{ 14 }$$

where

$$\begin{eqnarray}{Z}_{4}^{{\prime} } & = & \displaystyle \frac{\sqrt{\mu }\xi (j+1)}{1-\mu {\xi }^{2}},\\ X^{\prime} & = & \displaystyle \frac{\mu {\xi }^{2}+2j+1}{1-\mu {\xi }^{2}},\\ Y^{\prime} & = & \displaystyle \frac{\mu {\xi }^{2}(2j+1)+1}{1-\mu {\xi }^{2}}.\end{eqnarray} \tag{ 15 }$$

See [24] for the detailed calculations.

Note that for the proposed long-distance CVQKD scheme, the PNRD which is placed at Alice's side is removed, whereas the subtracted mode C which is supposed to enter the PNRD is recombined with mode B₁ in a PBS by using polarization-multiplexing technique. The task of resolving subtracted photon number is therefore handed over to the state-discrimination detector at Bob's side.

We design a state-discrimination detector to increase the performance of the CVQKD coupled with photon-subtraction operation. This quantum detector can unconditionally discriminate four nonorthogonal coherent states in QPSK modulation with the error probabilities lower than the SQL.

As shown in figure 2, we depict the structure of the improved state-discrimination detector using photon number resolving and adaptive measurements [36–38] in the form of fast feedback. This state-discrimination detector contains M times adaptive measurements in the field of $| \alpha \rangle$. For each measurement i(i ∈ {0,1, ⋯, M}), the strategy first prepares a predicted state $| {\beta }_{i}\rangle$ which has the highest probability based on the current data in classical memory. Subsequently, a displacement $\hat{D}({\beta }_{{i}})$ is adopted to displace $| \alpha \rangle$ to $| \alpha -{\beta }_{{i}}\rangle$ and a PNRD is used to detect the number of photons of the displaced field. If the predicted state is correct, i.e., $| {\beta }_{{i}}\rangle =| \alpha \rangle ,{{\rm{\Pi }}}_{0}$ will click, because the input field is displaced to vacuum so that the PNRD cannot detect any photon [26]. Note that different from the photon-subtraction operation where Π₀ clicks represents the failure of subtracting photon, Π₀ clicks here denotes that the improved state-discrimination strategy has correctly predicted the input state. This successful prediction is marked as l_i = 0, otherwise l_i = 1. After the ith adaptive measurement, the strategy calculates the posterior probabilities of all possible states ($| {\alpha }_{i0}\rangle ,| {\alpha }_{i1}\rangle$,$| {\alpha }_{i2}\rangle$ and $| {\alpha }_{i3}\rangle$) using Bayesian inference according to the present label history L_Hist and predicted history ${\hat{D}}_{{\rm{Hist}}}$ (Note that for now β_i has already been added to the ${\hat{D}}_{{\rm{Hist}}}$ with previous data to collectively calculate these probabilities), and designates the most probable state as $| {\beta }_{i+1}\rangle$, which is deemed as an input for next feedback. In each feedback period, the probabilities of all possible states are updated dynamically and the posterior probabilities of period i become prior probabilities in period i + 1. The rule of Bayesian inference can be expressed as

$$\begin{eqnarray}&&{{\boldsymbol{P}}}_{\mathrm{po}}(\{| \alpha \rangle \}| {\beta }_{i},{l}_{i})=A\,{\mathbb{P}}({l}_{i}| {\beta }_{i},\{| \alpha \rangle \}){{\boldsymbol{P}}}_{\mathrm{pr}}(\{| \alpha \rangle \}),\end{eqnarray} \tag{ 16 }$$

where ${{\boldsymbol{P}}}_{\mathrm{po}}(\{| \alpha \rangle \}| {\beta }_{i},{l}_{i})$ and ${{\boldsymbol{P}}}_{\mathrm{pr}}(\{| \alpha \rangle \})$ are the posterior and prior probabilities, respectively, ${\mathbb{P}}({l}_{i}| {\beta }_{i},\{| \alpha \rangle \})$ is conditional Poissonian probability of observing the detection result l_i for $| \alpha \rangle$ displaced by field β_i, and A is the normalization factor calculated by summing equation (16) over all possible states. Bayesian inference is a method of statistical inference in which Bayes theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Therefore, the final decision $| {\beta }_{M+1}\rangle$ of the input state $| \alpha \rangle$ can be predicted in the last adaptive measurement M using iterative Bayesian inference [28].

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This kind of strategies could surpass the SQL and approach the Helstrom bound with the help of high bandwidth and high detection efficiency. Mathematically, the SQL for discriminating the four nonorthogonal coherent state in QPSK modulation can be expressed by

$$\begin{eqnarray}&&{P}_{{\rm{SQL}}}=1-{\left[1-\displaystyle \frac{1}{2}\mathrm{erfc}\left(\sqrt{\displaystyle \frac{| \alpha {| }^{2}}{2}}\right)\right]}^{2},\end{eqnarray} \tag{ 17 }$$

where

$$\begin{eqnarray}&&\mathrm{erfc}(x)=\displaystyle \frac{2}{\sqrt{\pi }}{\int }_{x}^{\infty }{{\rm{e}}}^{-{t}^{2}}{\rm{d}}t,\end{eqnarray} \tag{ 18 }$$

and the Helstrom bound for the QPSK signals can be approximated by using the square-root measure [10], which can be calculated by

$$\begin{eqnarray}&&{P}_{{\rm{Hel}}}=1-\displaystyle \frac{1}{16}{\left(\displaystyle \sum _{k=1}^{4}\sqrt{{\omega }_{k}}\right)}^{2},\end{eqnarray} \tag{ 19 }$$

where ${\omega }_{k}={{\rm{e}}}^{-{\alpha }^{2}}{\sum }_{n=1}^{4}{\rm{\exp }}\left[(1-k)\tfrac{2\pi {\rm{i}}{n}}{4}+{\alpha }^{2}{\rm{\exp }}\left(\tfrac{2\pi {\rm{i}}{n}}{4}\right)\right]$ are eigenvalues of Gram matrix for QPSK signals. As the improved state-discrimination detector is parallel with homodyne or heterodyne detector at Bob's side, the ultimate detection of the states is codetermined by the coherent detector and the state-discrimination detector. From the perspective of information-theoretical sense, we can define an improvement ratio ζ to depict how much performance could the state-discrimination detector enhance the CVQKD system, namely

$$\begin{eqnarray}&&\zeta =\displaystyle \frac{1-{P}_{{\rm{rec}}}^{(M)}}{1-{P}_{{\rm{SQL}}}},\end{eqnarray} \tag{ 20 }$$

where ${P}_{{\rm{rec}}}^{(M)}$ represents the error probability of state-discrimination detector with M adaptive measurements. Theoretically, the detector could reach the Helstrom bound when M is large enough. Therefore, the optimal improvement ratio ζ_opt can be calculated by considering the minimum error probability allowed by quantum mechanics. Thus we have

$$\begin{eqnarray}&&{\zeta }_{{\rm{opt}}}=\displaystyle \frac{1-{P}_{{\rm{Hel}}}}{1-{P}_{{\rm{SQL}}}}.\end{eqnarray} \tag{ 21 }$$

In figure 3, we illustrate the error probabilities of discriminating the four nonorthogonal coherent states and the improvement ratio as functions of mean photon number $\langle n\rangle$. The blue dashed line shows the SQL to which the conventional and ideal coherent detector can achieve, while the blue solid line shows that the state-discrimination detector with 10 adaptive measurements is below the SQL and approaches the Helstrom bound (the blue dotted line). The red dashed line with squares denotes the optimal improvement ratio which descends quickly with the increased mean photon number but still above 1. Therefore, the proposed detection strategy that consists of state-discrimination detector and coherent detector could improve the performance of CVQKD system, satisfying the requirement of long-distance transmission.

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We first demonstrate the optimal values of simulated parameters before giving the performance of secret key rate. It is known that the optimal photon-subtraction operation in Gaussian-modulated CVQKD can be achieved when only one photon is subtracted [23, 24], which means that subtracting one photon is the preferred operation to improve the transmission distance. For the proposed long-distance discretely-modulated CVQKD scheme, we show the success probability of subtracting j(j = 1, 2, 3, 4, 5) photons as a function of transmittance μ in figure 4. Similar to its Gaussian-modulated counterpart, the success probability of subtracting one photon (j = 1, blue line) outperforms other numbers of photon subtraction and the success probability decreases with the increase of the number of subtracted photons. Meanwhile, as shown in figure 3, the red dashed line with squares depicts the improvement ratio of the improved state-discrimination detector. It is obvious that the highest value of the improvement can be obtained with mean photon number $\langle \bar{n}\rangle =1$. This coincidence ($j=\langle \bar{n}\rangle =1$) allows us to obtain the optimal performance by tactfully combining photon-subtraction operation and state-discrimination detector together. More specifically, the one photon subtracted by photon-subtraction operation at Alice' side is detected by state-discrimination detector at Bob' side, and both modules perform optimally as one photon meets the optimal requirements. Therefore, we consider the optimal one-photon subtraction operation in subsequent simulations to show the best performance of the proposed scheme.

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Because channel loss and excess noise are two of the most important factors that would have an effect on the performance of CVQKD system [39], the performance of these parameters with different modulation variance V_M needs to be illustrated. In figures 5 and 6, solid lines denote the performance of the proposed long-distance CVQKD scheme with the optimal one-photon subtraction operation, while dashed lines represent the four-state CVQKD protocol as a comparison, and their secret key rates change as V_M changes. The global simulation parameters are as follows: reconciliation efficiency is β = 95%, quantum efficiency of Bob's detection is τ = 0.6 and electronic noise is v_el = 0.05. In figure 5, excess noise ε and other parameters are fixed to legitimate values, the numerical areas of V_M are compressed for the four-state CVQKD protocol when channel loss increases, and the secret key rate decreases rapidly with the increase of channel loss. While for the proposed long-distance CVQKD scheme, V_M can be set to a large range of values and its secret key rate increases with the increased V_M even though the secret key rate also decreases as channel loss increases, which means the performance of the proposed long-distance CVQKD scheme would be consecutively improved theoretically when the modulation variance is set large enough. However, this cannot be realized in practice, thus the modulation variance V_M must be set to a reasonable value in simulations. In figure 6, transmission distance, which is proportional to the channel loss (0.2 dB km⁻¹), and other parameters are fixed. For the four-state CVQKD protocol, its optimal regions of V_M are also compressed with the increased excess noise ε. Fortunately, there is only slight impact on the proposed long-distance CVQKD scheme with one-photon subtraction when excess noise ε changes. It shows the proposed scheme greatly outperforms the four-state CVQKD protocol in terms of tolerable channel excess noise. The reasons may be given as follows. Firstly, excess noise can be deemed channel imperfections which deteriorate the correlation between the two output modes, while photon-subtraction operation can well enhance the correlation which is positively related to entanglement degree of EPR state and thus improves the performance of CVQKD system [23, 25]. Rendering the CVQKD system that applied this non-Gaussian operation tolerates more higher excess noise. Secondly, the proposed long-distance CVQKD scheme is not very sensitive to the noise with the help of state-discrimination detector, which means Bob can obtain more correct results without performing very precise measurement on quadrature $\hat{x}$ or/and $\hat{p}$. The reason is that the raw key in Gaussian modulated CVQKD protocol is tremendously affected by channel excess noise (and imperfect coherent detector) since its information is directly encoded in quadratures. In the four-state CVQKD protocol, the information is encoded in QPSK modulation which can be unconditionally discriminated by the state-discrimination detector [26]. Therefore, the detection strategy could predict incoming state using probability-based method, i.e. Bayesian inference, thus alleviating the impact of excess noise.

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Up to now, we have derived the parameters that may largely affect the CVQKD system. In what follows, we consider the secret key rate of the proposed CVQKD scheme. In general, the asymptotic secret key rate can be calculated with the form

$$\begin{eqnarray}&&{K}_{{\rm{asym}}}=\beta I(A\,:B)-S(E\,:B),\end{eqnarray} \tag{ 22 }$$

where β is the efficiency for RR, I(A : B) is the Shannon mutual information between Alice and Bob, and S(E : B) is the Holevo bound [40] of the mutual information between Eve and Bob. For the proposed CVQKD protocol, the asymptotic secret key rate in equation (22) can be rewritten by

$$\begin{eqnarray}&&{K}_{{\rm{asym}}}={P}_{(j)}^{{\hat{{\rm{\Pi }}}}_{1}}[\beta {\zeta }_{{\rm{opt}}}I(A\,:B)-S(E\,:B)].\end{eqnarray} \tag{ 23 }$$

As previously mentioned, ${P}_{(j)}^{{\hat{{\rm{\Pi }}}}_{1}}$ represents the probability of successful subtracting j photons and ζ_opt depicts the improvement ratio of the introduced state-discrimination detector. Detailed calculation of the asymptotic secret key rate can be found in appendix A.

In figure 7, we depict the asymptotic secret key rate as a function of transmission distance for the CVQKD protocol. Red line shows the original four-state protocol proposed in [13], yellow line denotes the optimal one-photon subtraction scheme for Gaussian modulated coherent state proposed in [24], blue line represents the scheme of four-state protocol with one-photon subtraction, and the green line denotes the proposed long-distance CVQKD scheme using non-Gaussian state-discrimination detector. The modulation variance V_M of above protocols is optimized except for the proposed long-distance CVQKD scheme since its secret key rate is monotonic increasing in a large range of the modulation variance V_M, which means that the performance of the proposed scheme can be further improved when V_M is set to larger value. However, from the perspective of fair comparison and practical significance, the modulation variance V_M of the proposed long-distance CVQKD scheme is reasonably set as same as its fundamental communication protocol, i.e., the four-state CVQKD protocol. As shown in figure 7, the proposed long-distance CVQKD scheme outperforms all other CVQKD protocols in terms of maximum transmission distance up to 330 km. Therefore, the proposed long-distance CVQKD scheme using non-Gaussian state-discrimination detector could be more suitable for long-distance transmission. Note that this distance record is limited by the secret key rate more than 10⁻⁶ bits per pulse, and it can be further extended when one considers the secret key rate below this bound.

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In addition, finite-size effect [41] needs to be taken into consideration, since the length of secret key is impossibly unlimited in practice. Moreover, one can make the assumption in the asymptotic case that the quantum channel is perfectly known before the transmission is performed, while in finite-size scenario, one actually does not know the characteristics of the quantum channel in advance. Because a part of exchanged signals has to be used for parameter estimation rather than generates the secret key. As shown in figure 8, the performance of the proposed CVQKD scheme in finite-size regime is outperformed by that obtained in asymptotic limit. The maximum transmission distance significantly decreases when the number of total exchanged signals N decreases. However, it still has a large improvement when comparing with original four-state CVQKD protocol and its Gaussian-modulated protocol counterpart which also take finite-size effect into account. Notice that the performance in the finite-size regime will converge to the asymtotic case if N is large enough. The detailed calculation of secret key rate in the finite-size regime can be found in appendix B.

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Finally, we demonstrate the performance of the proposed long-distance CVQKD scheme in composable security framework. The composable security is the enhancement of security based on uncertainty of the finite-size effect [18] so that one can obtain the tightest secure bound of the protocol by carefully considering every detailed step in CVQKD system [20]. In figure 9, we show the secret key rate of the proposed long-distance CVQKD scheme with one-photon subtraction operation in the case of composable security, as a function of total exchanged signals N. The performance is more pessimistic than that obtained in the finite-size regime, let alone in the asymptotic limit. For example, assuming that N = 10¹⁴ and the minimal secret key rate is limited to above 10⁻⁶ bis per pulse, the maximal transmission distance in finite-size regime is approximate 320 km (purple line in figure 8), while the maximal transmission distance is reduced to approximate 260 km (light blue line in figure 9) when one considers the proposed scheme in composable security framework. Therefore, the composable security, which takes the failure probabilities of every step into account, is the strictest theoretic security analysis of CVQKD system so that one can obtain more practical secure bound. In addition, the composable secret key rate also approaches the asymptotic value for very large N (dashed lines). The detailed calculation of the secret key rate for composable security is shown in appendix C.

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The asymptotic limit, the finite-size scenario and the composable security framework are the efficient approaches to evaluate the performance of CVQKD system. Although the results vary with the different approach, the trends of the performance are similar. Therefore, the proposed long-distance CVQKD using non-Gaussian state-discrimination detector can beat other existing CVQKD protocols in terms of maximal transmission distance and thus meet the requirement of long-distance transmission.