Performance improvement of free-space continuous-variable quantum key distribution with an adaptive optics unit

Abstract

We propose a method to enhance the performance of the free-space continuous-variable quantum key distribution (FSCVQKD) by utilizing an adaptive optics (AO) unit at the receiver’s side to suppress wavefront distortions caused by high-order wavefront aberrations for fault-tolerant detection. Benefiting from the AO unit, the high-order wavefront aberrations can be corrected and thus effectively improve the mixing efficiency of homodyne detection, which enhances the performance of FSCVQKD. Considering the fact that the closed-loop control bandwidth of AO unit and the atmospheric coherence length have an important effect in performance improvement for FSCVQKD, in this paper, we analyze the performance of our protocol with AO unit under different closed-loop control bandwidths and atmospheric coherence lengths, respectively. The analysis is performed by considering the three specific scenarios of turbulence which are frequent challenges in FSCVQKD protocol, namely beam wandering, randomly blocked and log-normal distribution. In addition, we also study the impact of AO-added noise on our protocol. Simulation results show that all in the beam wandering case, the randomly blocked case and the log-normal distribution, the use of AO unit can enhance the performance of FSCVQKD protocol by adjusting the closed-loop control bandwidth of AO unit and the atmospheric coherence length in appropriate ranges with the case of fixed Greenwood frequency in the presence of AO-added noise.

Appendices

Appendix A: Compute of Holevo bound

Here, we compute the Holevo bound $\chi _\mathrm{E}$ based on the reverse reconciliation $(\chi _\mathrm{E}=\chi _\mathrm{BE})$. Considering the fact that state mentioned previously can be purified by Eve, hence we can calculate the Holevo bound $\chi _\mathrm{BE}$ as

$$\begin{aligned} \chi _\mathrm{BE}=S(\rho _\mathrm{AB_{1}})-S(\rho ^{m_\mathrm{B}}_\mathrm{AFG}). \end{aligned}$$

(16)

According to Ref. [54], the expression of $\chi _\mathrm{BE}$ can be further simplified as

$$\begin{aligned} \chi _\mathrm{BE}=\sum ^{2}_{i=1}G\left( \frac{\nu _{i}-1}{2}\right) -\sum ^{5}_{i=3}G\left( \frac{\nu _{i}-1}{2}\right) , \end{aligned}$$

(17)

where $G(x)=(x+1)\mathrm{log}(x+1)-x\mathrm{log}x$ and the symplectic eigenvalues $\nu _{1,2}$ are given by

$$\begin{aligned} \nu ^{2}_{1,2}=\frac{1}{2}\left[ A\pm \sqrt{A^{2}-4B}\right] , \end{aligned}$$

(18)

with the denotations

$$\begin{aligned} A= & {} V^{2}(1-2T_\mathrm{nf})+2T_\mathrm{nf}+T^{2}_\mathrm{nf}(V+\chi _\mathrm{line})^{2}, \nonumber \\ B= & {} T^{2}_\mathrm{nf}(V\chi _\mathrm{line}+1)^{2}. \end{aligned}$$

(19)

Furthermore, the second part of Eq. (17) can be calculated by utilizing the symplectic eigenvalues of the covariance matrix $\rho ^{m_\mathrm{B}}_\mathrm{AFG}$, which can be expressed by

$$\begin{aligned} \rho ^{m_\mathrm{B}}_\mathrm{AFG}=\rho _\mathrm{AFG}-\sigma ^{T}_\mathrm{AFGB_{3}}(X\rho _\mathrm{B_{3}}X)^{MP}\sigma _\mathrm{AFGB_{3}}, \end{aligned}$$

(20)

where $X=\mathrm{diag}(1,0)$ and MP represents the inverse on the range. The decomposed covariance matrix $\rho _\mathrm{AFGB_{3}}$ is expressed by

$$\begin{aligned} \rho _\mathrm{AFGB_{3}}=\left( \begin{array}{cc} \rho _\mathrm{AFG} &{} \sigma ^{T}_\mathrm{AFGB_{3}} \\ \sigma _\mathrm{AFGB_{3}}&{} \rho _\mathrm{B_{3}}\\ \end{array} \right) . \end{aligned}$$

(21)

It is obvious that the sub-matrices $\rho _\mathrm{AFG}$, $\rho _\mathrm{B_{3}}$ and $\sigma _\mathrm{AFGB_{3}}$ can be derived from this matrix. Note that the matrix $\rho _\mathrm{AFGB_{3}}$ can be achieved by utilizing appropriate rearrangement of lines and columns from the matrix $\rho _{\mathrm{AB}_{3}FG}$, which can be written as

$$\begin{aligned} \rho _\mathrm{AB_{3}FG}=(\varXi ^\mathrm{BS})^{T}[\rho _\mathrm{AB_{2}}\bigoplus \rho _\mathrm{F_{0}G}]\varXi ^\mathrm{BS}, \end{aligned}$$

(22)

where $\rho _\mathrm{F_{0}G}$ is the matrix which describes the EPR state of variance $\upsilon$ used to model the electronic noise of the detector. It can be given by

$$\begin{aligned} \rho _\mathrm{F_{0}G}=\left( \begin{array}{cc} \upsilon I &{} \sqrt{\upsilon ^{2}-1}\sigma _{z} \\ \sqrt{\upsilon ^{2}-1}\sigma _{z}&{} \upsilon I \\ \end{array} \right) . \end{aligned}$$

(23)

For the homodyne detection in our system, the $\upsilon$ can be allowed to take the appropriate value. Ultimately, the matrix $\varXi ^\mathrm{BS}$ stands for the operation of the beam splitter, which models the inefficiency of Bob’s detector. It can be expressed by

$$\begin{aligned} \varXi ^\mathrm{BS}=I\bigoplus \varXi ^\mathrm{BS}_\mathrm{B_{2}F}\bigoplus I, \end{aligned}$$

(24)

with

$$\begin{aligned} \varXi ^\mathrm{BS}_\mathrm{B_{2}F}=\left( \begin{array}{cc} \sqrt{\gamma _\mathrm{h}}I &{} \sqrt{1-\gamma _\mathrm{h}}I \\ -\sqrt{1-\gamma _\mathrm{h}}I&{} \sqrt{\gamma _\mathrm{h}}I\\ \end{array} \right) . \end{aligned}$$

(25)

We now can calculate the symplectic eigenvalues $\nu _{3,4,5}$ of the covariance matrix $\rho ^{m_\mathrm{B}}_\mathrm{AFG}$ based on the above elements. Namely, for the homodyne case, the eigenvalues $\nu _{3,4}$ are given by

$$\begin{aligned} \nu ^{2}_{3,4}=\frac{1}{2}[C\pm \sqrt{C^{2}-4D}], \end{aligned}$$

(26)

with the denotations

$$\begin{aligned} C= & {} \frac{A(\chi _\mathrm{w}+\chi _\mathrm{h})+V\sqrt{B}+T_\mathrm{nf}(V+\chi _\mathrm{line})}{T_\mathrm{nf}(V+\chi _\mathrm{tot})},\nonumber \\ D= & {} \frac{\sqrt{B}V+B(\chi _\mathrm{w}+\chi _\mathrm{h})}{T_\mathrm{nf}(V+\chi _\mathrm{tot})}, \end{aligned}$$

(27)

where A, B have been given in Eq. (19). The last symplectic eigenvalue is $\nu _{5}=1$. Based on above equations, the Holevo quantity $\chi _\mathrm{BE}$ can be calculated.

Appendix B: Analysis of beam wandering

In realistic free-space channels, the main causes of fluctuations are complex. Fortunately, we can use a distribution of values T to character such fluctuating-loss channels with a probability density distribution P(T). According to other recent works [19], we will focus on the important phenomenon of beam wandering, which is caused by turbulence and unstable adjustment of the radiation source. Considering that the fluctuating-loss channels are mainly characterized by the probability distribution of the transmission coefficient (PDTC), thus for the convenience of analysis, the approximate analytical expression of the transmission efficiency is used here to evaluate the PDTC, which is given by

$$\begin{aligned} T^{2}=T^{2}_{0}\exp \left[ -\left( \frac{r}{Q}\right) \right] ^{\varOmega }, \end{aligned}$$

(28)

where r stands for the beam-deflection distance and $\varOmega$ and Q stand for the shape and scale parameter, respectively. The parameter $T_{0}$ represents the maximal transmission coefficient for the given beam-spot radius W, which can be expressed by

$$\begin{aligned} T^{2}_{0}=1-\exp [-2\varPhi ], \end{aligned}$$

(29)

where $\varPhi =\frac{R^{2}}{W^{2}}$. As shown in Fig. 10a, the transmittance decreases with increasing beam-deflection distance r under different beam-spot radius. Based on the work in Ref. [19], the beam-deflection distance r follows the Rice distribution with parameters d and $\tau ^{2}$. Under the assumption that the beam spatially fluctuates around the aperture center $(d=0)$, P(T) can be described by the log-negative Weibull distribution, which is given by

$$\begin{aligned} P(T)=\frac{2Q^{2}}{\tau ^{2}\varOmega T}\left( 2\ln \frac{T_{0}}{T}\right) ^{\frac{2}{\varOmega }-1}\exp \left[ - \frac{Q^{2}}{2\tau ^{2}}\left( 2\ln \frac{T_{0}}{T}\right) ^{\frac{2}{\varOmega }}\right] , \end{aligned}$$

(30)

for $T\in [0,T_{0}]$, with $P(T)=0$ otherwise. The parameters $\varOmega$ and Q can be expressed as

$$\begin{aligned} \varOmega= & {} 8\varPhi \frac{\exp (-4\varPhi )I_{1}[4\varPhi ]}{1-\exp (-4\varPhi )I_{0}[4 \varPhi ]}\left[ \ln \left( \frac{2T^{2}_{0}}{1-\exp (-4\varPhi )I_{0}[4\varPhi ]}\right. \right] ^{-1}, \end{aligned}$$

(31)

$$\begin{aligned} Q= & {} R\left[ \ln \left( \frac{2T^{2}_{0}}{1-\exp (-4\varPhi )I_{0}[4\varPhi ]}\right) \right] ^{-\frac{1}{\varOmega }}, \end{aligned}$$

(32)

where $I_{0}[.]$ and $I_{1}[.]$ are the modified Bessel functions. Then we can obtain $\langle T\rangle =\int ^{T_{0}}_{0}TP(T)\text {d}T$ and $\langle \sqrt{T}\rangle =\int ^{T_{0}}_{0}\sqrt{T}P(T)\text {d}T$. When the distance $d=0$, the relationship between the factors (the parameters $T_\mathrm{nf}$ and $\xi _\mathrm{nf}$) of the fluctuating channel and the variance $\tau ^{2}$ is given in Fig. 10b. It is clear that the parameter $T_\mathrm{nf}$ decreases with $\tau ^{2}$ and finally tends to a constant, while $\xi _\mathrm{nf}$ increases and ultimately reaches a maximum.

Fig. 10

(Color online) Simulations based on the fluctuations in the transmissivity caused by beam wandering. a The relationship between transmittance squared and the beam-deflection distance r with different values of the beam-spot radius W. Here, for simplicity the aperture radius R is assumed $R=1$. b Parameters $T_\mathrm{nf}$ and $\xi _\mathrm{nf}$ as functions of beam-deflection variance $\tau ^{2}$

Appendix C: Analysis of randomly blocked quantum channel

The randomly blocked quantum channel means that the transmittance of the fluctuating channel is blocked randomly, as shown in Fig. 11. That is to say, the entangled state can be transmitted perfectly $(T=1)$ with probability $p_{1}$, while it is entirely effaced ($T=0$) with probability $1-p_{1}$.

Fig. 11

The randomly blocked channel which shows the transmittance is fading randomly with $T=1$ or $T=0$

Fig. 12

The parameters $T_\mathrm{nf}$ and $\xi _\mathrm{nf}$ as functions of perfect transmitted probability $p_{1}$

Different from beam wandering case, in the randomly blocked case the parameters $T_\mathrm{nf}=p^{2}_{1}$ and $\xi _\mathrm{nf}=(p_{1}-p^{2}_{1})(V-1+\xi )$. In Fig. 12, we show $T_\mathrm{nf}$ and $\xi _\mathrm{nf}$ as functions of probability $p_{1}$. It is clear that $T_\mathrm{nf}$ increases with the increase of probability $p_{1}$ and can reach perfect transmittance when $p_{1}=1$, while $\xi _\mathrm{nf}$ reaches a maximum of 0.5 when $p_{1}=0.5$.

Appendix D: Analysis of log-normal model

The transmittance of the log-normal probability distribution is given by

$$\begin{aligned} P_{N}=\frac{1}{T\tau _{n}\sqrt{2\pi }}\exp \left[ -\frac{(-\ln {T}-\mu )^{2}}{2\tau ^{2}_{n}}\right] , \end{aligned}$$

(33)

where $\mu =-\ln (\frac{\langle T\rangle ^{2}}{\sqrt{\langle T^{2}\rangle }})$ and $\tau ^{2}_{n}=\ln (\frac{\langle T^{2}\rangle }{\langle T\rangle ^{2}})$ represent parameters of the log-normal distribution. Here, the first and second moments of transmittance is given by

$$\begin{aligned} \langle T\rangle= & {} \int _{\partial } d^{2}\varUpsilon \varGamma _{2}(\varUpsilon ), \end{aligned}$$

(34)

$$\begin{aligned} \langle T^{2}\rangle= & {} \int _{\partial } d^{2}\varUpsilon _{1}d^{2}\varUpsilon _{2}\varGamma _{4}(\varUpsilon _{1}, \varUpsilon _{2}), \end{aligned}$$

(35)

where $\varUpsilon _{i}=(x_{i}, y_{i})'$ (i=1,2) stands for the vector of transverse coordinates, $\varGamma _{2}$ and $\varGamma _{4}$ are the field coherence functions (detailed expressions of $\varGamma _{2}$ and $\varGamma _{4}$ are shown in Ref. [23]), and $\partial$ represents the circular aperture opening area. Note that $\mu$ and $\tau ^{2}_{n}$ are related to $\langle T\rangle$ and $\langle T^{2}\rangle$.

The $\langle T\rangle$ shown in Eq. (34) can be definitely estimated as [23]

$$\begin{aligned} \langle T\rangle =1-\exp \left[ -\frac{2R^{2}}{\langle W^{2}\rangle }\right] , \end{aligned}$$

(36)

where R represents the aperture radius and

$$\begin{aligned} \langle W^{2}\rangle =\langle S_{xx}\rangle +4\langle x^{2}_{0}\rangle \end{aligned}$$

(37)

is called “long-term” beam mean-square radius [24]. The $\langle S_{xx}\rangle$ and $\langle x^{2}_{0}\rangle$ are expressed as

$$\begin{aligned} \langle S_{xx}\rangle =4[\int _{{\mathbb {R}}^{2}}d^{2}\varUpsilon x^{2}\varGamma _{2}(\varUpsilon ;L)-\int _{{\mathbb {R}}^{4}}d^{2}\varUpsilon _{1}d^{2}\varUpsilon _{2}x_{1}x_{2}\varGamma _{4}(\varUpsilon _{1},\varUpsilon _{2};L)] \end{aligned}$$

(38)

and

$$\begin{aligned} \langle x^{2}_{0}\rangle =\int _{{\mathbb {R}}^{4}}d^{2}\varUpsilon _{1}d^{2}\varUpsilon _{2}x_{1}x_{2}\varGamma _{4}(\varUpsilon _{1},\varUpsilon _{2};L), \end{aligned}$$

(39)

where L stands for the distance at which a Gaussian beam propagates along the z axis onto the aperture plane.