Practical aspects of terahertz wireless quantum key distribution in indoor environments

Abstract

We discuss the feasibility of continuous-variable wireless quantum key distribution (WQKD) with thermal Gaussian states in terahertz (THz) band. We rigorously analyze the secret key rate of practical WQKD system using direct reconciliation and homodyne detection against collective Gaussian attacks. The results show that in the case of low-gain antenna, the free space loss is the dominant limiting factor of secure transmission distance of THz WQKD system. When the antenna diameter increases from 1 to 5 cm, we can get the maximum security distance and its corresponding optimal frequency in the range of 0.1–1 THz. We also obtain that a security distance of 1.95 m can be achieved when using a 5-cm diameter antenna and controlling the excess noise below 0.25 simultaneously. Specifically, we study the finite size effects and show that the transmission distance can reach 78 cm with a 5-cm diameter antenna at the frequency of 300 GHz. This work takes an important step toward short distance wireless QKD system.

Appendices

Appendix A: Calculation of information entropy

The information entropy is described by the second-order statistical properties, including variance, covariance, and conditional variance. Firstly, we define the quadrature row vector $\hat{\mathbf {Y}}=(\hat{Q}_1,\hat{P}_1,\ldots ,\hat{Q}_k,\hat{P}_k)$, which describes the bosonic system of k-mode and its elements satisfy the commutator relation [50]. The correlation matrix (CM) of the Gaussian bosonic state $\rho$ can be calculated by using the following formulas. The off-diagonal terms of CM are defined by

$$\begin{aligned} V_{ij}=\frac{1}{2}(\langle \hat{Y}_{i}\hat{Y}_{j}+\hat{Y}_{j}\hat{Y}_{i}\rangle )-\langle \hat{Y}_{i}\rangle \langle \hat{Y}_{j}\rangle , \end{aligned}$$

(A.1)

and the diagonal terms of CM are

$$\begin{aligned} V_{ii}=\langle \hat{Y}_{i}^{2}\rangle -\langle \hat{Y}_{i}\rangle ^{2}. \end{aligned}$$

(A.2)

The conditional variance is

$$\begin{aligned} V(\hat{X}|Y)=V(\hat{X})-\frac{|\langle \hat{X}Y\rangle |^{2}}{V(Y)}. \end{aligned}$$

(A.3)

The Shannon entropy is calculated as

$$\begin{aligned} H(\hat{X}_{A})=\frac{1}{2}\log V(\hat{X}_{A}). \end{aligned}$$

(A.4)

The von Neumann entropy of a Gaussian state $\rho$ containing k-mode can be written as [5]

$$\begin{aligned} S(\rho )=\sum _{i=1}^kg(\nu _{i}), \end{aligned}$$

(A.5)

where

$$\begin{aligned} g(x)=\frac{(x+1)}{2}\log \frac{(x+1)}{2}-\frac{(x-1)}{2}\log \frac{(x-1)}{2}, \end{aligned}$$

(A.6)

where $\nu _{i}$ corresponds to the symplectic eigenvalues of the ith of the k-mode Gaussian bosonic system and is given by

$$\begin{aligned} \nu =|i\varvec{\Omega \Gamma }|, \end{aligned}$$

(A.7)

where $\nu \ge 1$, $\varvec{\Gamma }$ corresponds to the CM of the Gaussian bosonic state $\rho$ and

$$\begin{aligned} {\varvec{\Omega }}:=\bigoplus _{i=1}^k \begin{bmatrix} 0&\quad 1\\ -1&\quad 0 \end{bmatrix}, \end{aligned}$$

(A.8)

where $\bigoplus$ is direct sum.

We consider the case of two-mode bosonic Gaussian system to calculate the secure key rate, i.e., $k=2$, $\hat{\mathbf {Y}}=(\hat{Q}_{A},\hat{P}_{A},\hat{Q}_{B},\hat{P}_{B})$. Based on Eqs. (5), (A.1), and (A.2), the covariance matrix ${{\varvec{\Gamma }}_{AB}}$ of the state $\rho _{AB}$ shared by Alice and Bob can be obtained.

$$\begin{aligned} \begin{aligned}\Gamma _{11}&=\Gamma _{22}=V(\hat{Q}_{A})=V,\\\Gamma _{33}&=\Gamma _{44}=V(\hat{Q}_{B})=V(\sqrt{T}\hat{Q}_{B_{0}}+\sqrt{1-T}\hat{E})\\ {}&=TV+(1-T)W, \\ \Gamma _{13}&=\Gamma _{31}=\frac{1}{2}\langle \hat{Q}_{A}\hat{Q}_{B_{0}}+\hat{Q}_{B_{0}}\hat{Q}_{A}\rangle -\langle \hat{Q}_{A}\rangle \langle \hat{Q}_{B_{0}}\rangle \\&=\sqrt{T(V^2-1)}. \end{aligned} \end{aligned}$$

(A.9)

Similarly, other elements of ${{\varvec{\Gamma }}_{AB}}$ can be calculated. ${{\varvec{\Gamma }_{AB}}}$ is given by

$$\begin{aligned} \varvec{\Gamma }_{AB}=\begin{bmatrix}V\mathbf {I}&\quad \sqrt{(T(V^2-1)}\varvec{\sigma }_{z}\\ \sqrt{T(V^2-1)}\varvec{\sigma }_{z}&\quad (TV+(1-T)W)\mathbf {I}\end{bmatrix}, \end{aligned}$$

(A.10)

where $\mathbf {I}$ is the $2\times 2$ identity matrix and $\varvec{\sigma }_{z}={\hbox {diag}}(1,-1)$. To represent concisely, we denote

$$\begin{aligned} \begin{aligned}a&=V,\\b&=TV+(1-T)W,\\c&=\sqrt{T(V^2-1)},\end{aligned} \end{aligned}$$

(A.11)

therefore

$$\begin{aligned} \varvec{\Gamma }_{AB}=\begin{bmatrix}a\mathbf {I}&\quad c\varvec{\sigma }_{z}\\ c\varvec{\sigma }_{z}&\quad b\mathbf {I} \end{bmatrix}. \end{aligned}$$

(A.12)

We first calculate the covariance matrix $\varvec{\Gamma }_{A_{0}C_{0}B}$

$$\begin{aligned} \varvec{\Gamma }_{A_{0}C_{0}B}=\begin{bmatrix}a\mathbf {I}&\quad 0&\quad c\varvec{\sigma }_{z}\\ 0&\quad \mathbf {I}&\quad 0\\c\varvec{\sigma }_{z}&\quad 0&\quad b\mathbf {I}\end{bmatrix}. \end{aligned}$$

(A.13)

As shown in Fig. 1, $\hat{A}_{0}$ and $\hat{C}_{0}$ are coupled through the beam splitter with $T_{A}=0.5$. $\varvec{\Gamma }_{ACB}$ can be calculated as

$$\begin{aligned} \varvec{\Gamma }_{ACB}=\begin{bmatrix}\frac{1}{\sqrt{2}}\mathbf {I}&\quad \frac{1}{\sqrt{2}}\mathbf {I}&\quad 0\\ -\frac{1}{\sqrt{2}}\mathbf {I}&\quad \frac{1}{\sqrt{2}}\mathbf {I}&\quad 0\\0&\quad 0&\quad \mathbf {I}\end{bmatrix}^\mathrm{T}\varvec{\Gamma }_{A_{0}C_{0}B} \begin{bmatrix}\frac{1}{\sqrt{2}}\mathbf {I}&\quad \frac{1}{\sqrt{2}}\mathbf {I}&\quad 0\\ -\frac{1}{\sqrt{2}}\mathbf {I}&\quad \frac{1}{\sqrt{2}}\mathbf {I}&\quad 0\\0&\quad 0&\quad \mathbf {I}\end{bmatrix}. \end{aligned}$$

(A.14)

After Alice performs homodyne detection according to Eq. (7), we can get

$$\begin{aligned} \varvec{\Gamma }_{BC|x}=\begin{bmatrix}b-\frac{c^{2}}{a}&\quad 0&\quad \frac{\sqrt{2}c}{a+1}&\quad 0\\0&\quad b&\quad 0&v\frac{-c}{\sqrt{2}}\\ \frac{\sqrt{2}c}{a+1}&\quad 0&\quad \frac{2a}{a+1}&\quad 0 \\ 0&\quad \frac{-c}{\sqrt{2}}&\quad 0&\quad \frac{a+1}{2}\end{bmatrix}, \end{aligned}$$

(A.15)

$S(\rho _{AB} )$ is a function of the symplectic eigenvalues $\nu _{1,2}$ of $\varvec{\Gamma }_{AB}$ and $S(\rho _{BC}|x)$ is a function of the symplectic eigenvalues $\nu _{3,4}$ of $\varvec{\Gamma }_{BC|x}$. Combined with Eqs. (A.5)–(A.8), (A.11), and (A.15), we can get

$$\begin{aligned} \begin{aligned}\nu _{1,2}^{2}&=\frac{1}{2}\left[ \varDelta _{1}\pm \sqrt{\varDelta _{1}^{2}-4D_{1}}\right] ,\\ \nu _{3,4}^{2}&=\frac{1}{2}\left[ \varDelta _{2}\pm \sqrt{\varDelta _{2}^{2}-4D_{2}}\right] , \end{aligned} \end{aligned}$$

(A.16)

where

$$\begin{aligned} \begin{aligned}\varDelta _{1}&=a^2+b^2-2c^2,\\D_{1}&=(ab-c^2)^2,\\ \varDelta _{2}&=\frac{a+a^2+b^2-2c^2+ab^2-bc^2}{a+1},\\D_{2}&=\frac{(ab-c^2)^2+ab^2-bc^2}{a+1},\end{aligned} \end{aligned}$$

(A.17)

Finally, by Eqs. (11), (12), (A.5), and (A.16), we can get the security key rate R.

Appendix B: The estimation of CM

We consider here a normal model for x and y as

$$\begin{aligned} y=tx+z, \end{aligned}$$

(B.1)

where t is $\sqrt{T}$ and z is a normal variable with unknown variance $\sigma ^{2}$. Therefore, $S_{\varepsilon _\mathrm{PE}}(x:E)$ is a function of t and $\sigma ^2$. At this point, it is worth considering the dependence of $S_{\varepsilon _\mathrm{PE}}(x:E)$ on the variables t and $\sigma ^{2}$:

$$\begin{aligned} \begin{aligned}\frac{\partial S(x:E)}{\partial \sigma ^{2}}|_{t}&>0,\\\frac{\partial S(x:E)}{\partial t}|_{\sigma ^{2}}&<0.\end{aligned} \end{aligned}$$

(B.2)

For maximizing the Holevo information between Eve and Alice’s classical data, we need to estimate $t_\mathrm{min}$ and $\sigma ^2_\mathrm{max}$ through the sampling of m couples of correlated variables $(x_{i}, y_{i})_{i=1, \ldots , m}$. We can obtain the estimated value of t and $\sigma ^{2}$ by the maximum likelihood estimation:

$$\begin{aligned} \begin{aligned} \hat{t}&=\frac{\sum _{i=1}^{m}x_{i}y_{i}}{\sum _{i=1}^{m}x_{i}^2},\\ \hat{\sigma ^{2}}&=\frac{1}{m}\sum _{i=1}^{m}(y_{i}-tx_{i})^{2}, \end{aligned} \end{aligned}$$

(B.3)

where $\hat{t}$ and $\hat{\sigma }^{2}$ are independent estimators. By the large number theorem, we can get the following distributions:

$$\begin{aligned} \begin{aligned} \hat{t}&\sim N \left( t,\frac{\sigma ^{2}}{\sum _{i=1}^{m}x_{i}^2}\right) , \\ \frac{m\hat{\sigma }^{2}}{\sigma ^{2}}&\sim \chi ^{2}(m-1), \end{aligned} \end{aligned}$$

(B.4)

where t and $\sigma ^2$ are the values of the parameters. We expect their estimated values to be as close as possible to the true values, i.e., $E(\hat{t} )=\sqrt{T}$ and $E(\hat{\sigma }^2 )=(1-T)W+T$. As we mentioned before, the failure probability of parameters estimation is $\varepsilon _\mathrm{PE}$ and the error function is defined as

$$\begin{aligned} erf(x)=\frac{2}{\sqrt{\pi }}\int _{0}^xe^{-t^2}{\hbox {d}}t, \end{aligned}$$

(B.5)

Therefore, when the confidence probability $\delta$ is $\varepsilon _\mathrm{PE}/2$, we can compute the confidence interval for $\hat{t}$, $\hat{\sigma }^2$ as

$$\begin{aligned} \begin{aligned}t&\in \left[ \hat{t}-z_{\delta }\sqrt{\frac{\hat{\sigma }^2}{mV_{A}}}, \hat{t}+z_{\delta }\sqrt{\frac{\hat{\sigma }^2}{mV_{A}}}\right] ,\\ \sigma ^2&\in \left[ \hat{\sigma }^2-z_{\delta }\frac{\sqrt{2}\hat{\sigma }^2}{\sqrt{m}} ,\hat{\sigma }^2+z_{\delta }\frac{\sqrt{2}\hat{\sigma }^2}{\sqrt{m}}\right] , \end{aligned} \end{aligned}$$

(B.6)

where $z_{\delta }$ satisfies $1-erf(\frac{z_{\delta }}{\sqrt{2}})/2=\delta$.