Phase compensation for free-space continuous-variable quantum key distribution

Shiyu Wang; Peng Huang; Miaomiao Liu; Tao Wang; Ping Wang; Guihua Zeng

doi:10.1364/OE.387402

1. Introduction

Quantum key distribution (QKD) [1–5] enables the generation of a secret key shared by two legitimate parties conventionally called Alice and Bob. By transmitting quantum states carrying the key information through free-space channels, free-space QKD shows the possibility of establishing global quantum networks. Low-loss transmission in outer space allows satellite-based QKD (for reviews see [6,7]) which has been realized and may fulfill the demand of connecting distant nodes. To implement terrestrial free-space QKD (see reviews [8,9]) on Earth, inevitably, atmospheric turbulence effects and attenuation must be tackled; this appears to be a daunting task, but in practice the difficulties can still be surmounted and the communication range has reached 144 km [10,11]. In practical terms, continuous-variable QKD (CVQKD) deserves attention as it has an inherent characteristic, that is, the resistance to background noise [12–15]. Various techniques are investigated and introduced to improve the performance of free-space CVQKD, e.g., high-dimensional multiplexing [16,17], transmittance fluctuation stabilization [18], and many others [19–22]. Phase encoding is a crucial part of free-space optical communications (e.g., see [23–25]); and phase modulation is also applied in CVQKD where information is encoded on field quadratures. However, free-space channels usually experience fading which makes the way of phase compensation for free-space CVQKD remain unclear.

In the present paper, we address the implementation of phase compensation for CVQKD over free-space fading channels. We adopt an approach similar to a method proposed in our earlier work [26] which is aimed at in-fiber transmission. Effective compensation is demonstrated by considering experimentally acquired transmittance of a real fading channel. The results of further performance analysis suggests a wide range of applications in free-space CVQKD experiments and real-world systems.

The paper is organized as follows. In section 2, we theoretically study the correlation between the data sent by Alice and those measured by Bob after free-space transmission; and a phase compensation scheme which utilizes the aforementioned correlation is described. In section 3, a real free-space channel is characterized by the distribution and the spectrum of experimentally measured transmittance; a demonstration of the compensation scheme is then given via applying the transmittance. In section 4, a simulation is performed to numerically analyze the performance of the compensation method. Finally, we come to the conclusion in section 5.

2. Phase compensation for fading channels

We consider the scheme that signal and local oscillator (LO) pulses are co-transmitted [26,27] through a fading channel, of which the transmittance $T$ is a random variable. The transmission and detection of field quadratures with encoded information can be described by

(1)$$X_B = \sqrt{N_0}\left\lbrace \sqrt{\eta} \left[ \sqrt{T}\left(X_A + \delta X_A \right) + \sqrt{1-T} X_{v1} + X_E \right] + \sqrt{1-\eta} X_{v2} \right\rbrace + X_{ele},$$

where $X_A$ represents the Gaussian modulation of the signal with a variance of $V_A$, and $X_B$ is the measured result of homodyne detection (here we take the X-quadrature as an example; it is basically the same for the P-quadrature). $\delta X_A$, $X_{v1}$, and $X_{v2}$ are vacuum states with unity variance; $X_E$ and $X_{ele}$ are the excess noise and electronic noise with variance $\xi$ and $V_{ele}$, respectively. The shot noise unit (SNU) can be given by $N_0 = \eta T |\alpha _\textrm {LO}|^2$ [28], where $|\alpha _\textrm {LO}|^2$ is the mean photon number of the LO sent by Alice, and $\eta$ is the detection efficiency. Of course, strictly speaking, there should be a conversion factor $g$ (the SNU becomes $g N_0$), which is usually determined by the amplifier in a homodyne detector and can be regarded as a constant; without loss of generality, we take $g=1$ for simplicity. Equation (1) can be normalized to SNU and rearranged as

(2)$$\begin{aligned} X_{Bs} &= \sqrt{\eta T} X_A + \left[ \sqrt{\eta} \left( \sqrt{T}\delta X_A + \sqrt{1-T} X_{v1} \right) + \sqrt{1-\eta} X_{v2}\right] + \sqrt{\eta} X_E + X_{el} \\ &=t X_A + z, \end{aligned}$$

where $X_{el} = X_{ele} / \sqrt {N_0}$, and $t = \sqrt {\eta T}$; $z$ is zero-mean and has a variance of $\sigma _{z}^2 = 1 + \eta \xi + v_{el}$ with $v_{el} = V_{ele} \left \langle 1/N_0 \right \rangle$; $X_B = \sqrt {N_0} X_{Bs} = \sqrt {N_0} \times t X_A + z'$, where $z'$ is still zero-mean with variance $\sigma _{z'}^2 = \left \langle N_0 \right \rangle \left ( 1 + \eta \xi + V_{ele}/\left \langle N_0 \right \rangle \right )$. To model phase drift occurs during the whole process from Alice to Bob, the above equations should be modified:

(3)$$ X_B' = \sqrt{N_0} \times t X_A' + z', $$

(4)$$ X_{Bs}' = t X_A' + z, $$

where $X_A' = r \cos \left (\theta + \Delta \theta \right )$, $r$ and $\theta$ are the amplitude and phase of the modulation $X_A = r \cos \theta$ respectively, and $\Delta \theta$ is the phase drift.

To probe $\Delta \theta$, the two parties announce a small portion of data transmitted during the time of a stable frame, within which $\Delta \theta$ varies so small that it can be assumed constant; then they can estimate $\Delta \theta$ by simply examining the correlation between their disclosed data. This approach has been shown to work well for fiber channels [26]; here we extend it to fading channels (the implementation is basically the same). Before computing the correlation, Alice rotates her data by $\Delta \varphi$; so $X_A$ becomes

(5)$$X_A = r \cos \left( \theta + \Delta\varphi \right).$$

If Bob’s data in SNU is used, the correlation is given by

(6)$$\left\langle X_{Bs}' X_A \right\rangle = \sqrt{\eta} \left\langle \sqrt{T} \right\rangle V_A \cos\left( \Delta\theta - \Delta\varphi \right).$$

The fluctuating characteristic of $T$ vanishes in this equation; thus Alice can just scan $\Delta \varphi$ from 0 to $2\pi$ and obtain the estimated value of $\Delta \theta$ when $\left \langle X_{Bs}' X_A'' \right \rangle$ reaches its maximum value; we denote the estimated phase as $\widetilde {\Delta \theta }$. However, to do so, there will be a precondition, i.e., the SNU is precisely monitored or measured in real time, as the LO power received by Bob fluctuates with the channel transmittance; this is not an easy task and will increase the hardware and software complexity of systems. Therefore, it is necessary to check whether the correlation between $X_B'$ and $X_A''$ also has the good property; and we find that

(7)$$\left\langle X_B' {X_A^{\prime\prime}} \right\rangle = \eta |\alpha_\textrm{LO}| \left\langle T \right\rangle V_A \cos\left( \Delta\theta - \Delta\varphi \right).$$

This means there is no need to normalize measured data to SNU, thus exempting the SNU monitoring or measurement. But we should point out that $\Delta \theta$ can be estimated by both Eq. (6) and Eq. (7), so one can choose which way to use according to specific systems and working conditions (e.g., the choice may be subject to data availability). Now, Alice can retrieve the linear relationship indicated in Eq. (1) and Eq. (2) by shifting the phase of her data to $\theta + \widetilde {\Delta \theta }$. Meanwhile, the disclosed data should be discarded.

3. Demonstration of phase compensation

To verify the phase compensation scheme, real fading channel transmittance is obtained by performing direct detection of received light, which passes through a 150-m free-space channel located on the Minhang Campus of Shanghai Jiao Tong University in an urban environment, as shown in Fig. 1. The sender Alice is on the fourth floor of a School of Electronic Information and Electrical Engineering (SEIEE) building; and the receiver Bob is on the second level of another SEIEE’s building. At Alice’s side, the light emitted from a 1550-nm continuous-wave laser (RIO Orion) is attenuated by a variable optical attenuator, and then collimated to the free-space channel by a collimator with focal length 100 mm. At Bob’s side, the arriving beam is captured by a 10X expander with an aperture of 35 mm; and subsequently, the reduced beam is carefully aligned via a mirror and coupled into a single-mode fiber (SMF). A balanced photodetector (THORLABS PDB435C) with one input port blocked (denoted as PD in Fig. 1) is used to measure the intensity of the coupled light with an oscilloscope. The transmitted light intensity is measured by an optical power meter and converted to the electric output of the PD via a response curve measured beforehand (see Fig. 2). The transmittance is now obtained by computing the relative intensity value.

Fig. 1. Experimental set-up and aerial view of the free-space channel within the city of Shanghai. VOA: variable optical attenuator, PD: photodetector, Osc: oscilloscope.

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Fig. 2. The response curve of the PD. The last three measurements with high input power are excluded when performing curve fitting, as they may be within the saturation region. The fitted curve is used for the conversion.

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A transmittance distribution measured on a clear night is shown in Fig. 3(a). The data is recorded for 10 seconds at a sample rate of 200 kHz. The spectrum of these transmittance data is obtained by computing their fast Fourier transform (FFT) as shown in Fig. 3(b). Since the frequency resolution is 0.1 Hz, there is no need to zero pad the data. To suppress spectral leakage, a Hanning window is applied. The spectrum of electronic noise (scaled by the transmitted intensity) of the PD is displayed for comparison. It is obvious that low-frequency components dominate the transmittance spectrum; and the transmittance fluctuation is basically below 0.2 kHz.

The real transmittance data are used to perform simulation according to Eq. (3) and Eq. (4). To demonstrate the process of phase compensation, we consider that Alice sends phase scan signal (the amplitude is still Rayleigh distributed) over the recorded fading channel. As can be seen in Fig. 4, the phase of compensated signals are well matched with that of received signals whether normalized to SNU or not. Note the signals used for phase estimation and collected transmittance data have equal length in the simulation, but one should not expect such a long stable frame in practical circumstances; here it is just for a clear demonstration. The choice of signal length is never a easy question and will be discussed in the following section.

4. Performance analysis

The uncertainty of $\Delta \theta$, i.e., $\Delta \phi = \widetilde {\Delta \theta } - \Delta \theta$ depends on the signal-to-noise ratio (SNR) and the number of data $N_{est}$ used in a estimation. According to Section 2, the SNR of $X_B'$ and $X_{Bs}'$ for a fading channel can be given by

(8)$$\textrm{SNR}_B = \frac{\eta \left\langle T \right\rangle V_A}{1 + \eta\xi + V_{ele}\left\langle 1/N_0 \right\rangle }$$

and

(9)$$\textrm{SNR}_{Bs} = \frac{\eta \left\langle T^2 \right\rangle V_A}{\left\langle T \right\rangle \left( 1 + \eta\xi + V_{ele}/\left\langle N_0 \right\rangle \right) },$$

respectively. Naturally, for a fixed channel, above equations reduce to $\textrm {SNR}_B = \textrm {SNR}_{Bs} = \eta T V_A / \left ( 1 + \eta \xi + v_{el} \right )$. Small $\Delta \phi$ can be achieved with high SNR and large $N_{est}$, which however is usually not the case in practical situations; the quantum signal becomes quite weak after traveling over a long distance or being affected by atmospheric effects, leading to a low SNR level; the price of improving estimation accuracy by using large $N_{est}$ is the sacrifice of raw keys, thus bringing a trade-off. Besides, a large $N_{est}$ is available when turbulence-induced phase variation is relatively slow and the feedback bandwidth is sufficient, whereas a smaller one is suitable for fast changes and narrow bandwidths. Thereby, efforts should be devoted to improve SNR and select an appropriate value of $N_{est}$. Note the uncertainty can further induce excess noise

(10)$$\xi_{\Delta\phi} = 2 \left\langle T \right\rangle V_A \sigma_{\Delta\phi}^2,$$

which is obtained by adapting the result given in [29]; $\sigma _{\Delta \phi }^2$ is the variance of $\Delta \phi$. Thus secret key rates will be affected.

To evaluate $\Delta \phi$ and key rates (see Appendix) at various SNR levels and $N_{est}$, another set of transmittance data is acquired on the same night. These data are sampled at 1 MHz as typical system repetition rates are of the order of MHz [30]. The distribution and the spectrum are shown in Fig. 5; they are similar to those in Fig. 3 but the spectrum has a few components from 0.2 to 0.4 kHz, which may due to a deep fade, i.e., a drastic drop in channel transmittance and thus SNR, occurs at around the 16th segment as shown in Fig. 6. The phrase “deep fade” [31] comes from wireless communications where various techniques such as interleaving, space-time coding are developed to mitigate it [31]. Despite the casues—multipath propagation of radio waves and failure of light coupling (in our experiment)—are not the same, the consequences are similar; so we still follow this phrase here.

Fig. 3. The transmittance measured on the night of November 14, 2019. The bin width of (a) the distribution is 0.01. (b) The amplitude spectrum of transmittance (blue) and electronic noise (orange).

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Fig. 4. Demonstration of phase compensation. (a) Normalized to SNU; (b) Not normalized to SNU. Without loss of generality, $|\alpha _\textrm {LO}|^2$ is assumed unity. Further parameters: $V_A = 50$, $\xi = 0.03$, $\eta = 0.35$, $v_{el} = 0.6$.

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Fig. 5. The transmittance measured at 1 MHz sample rate on the night of November 14, 2019. It is recorded for 2 seconds which result in frequency resolution of 0.5 Hz.

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Fig. 6. SNR (filled areas) and the standard deviation of $\Delta \phi$ (solid lines) at different transmittance segments for $N_{est}=2000$, 4000, 8000, and 10000 (blue, orange, green, and red). The SNR values are basically the same for different $N_{est}$ so the filled areas overlap each other. Further parameters: $V_A=5$, $\xi =0.01$, $\eta =0.35$, $V_{ele}=0.0735$, reconciliation efficiency $\beta =0.96$.

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We divide the whole recorded channel into 1000, 500, 250, and 200 segments; each segment contains 2000, 4000, 8000, and 10000 samples, respectively. 100 uniformly-spaced segments (from the beginning to the end) of each division are selected to conduct Monte Carlo simulation. Over each transmittance segment signal generation, detection (according to Eq. (3)), and phase estimation are repeated for 10000 times. Figure 6 shows the standard deviation of $\Delta \phi$ varying with SNR for various $N_{est}$ (as a reference, one may find “compensation accuracy of $0.1^\circ$ per frame” is achieved in [26]); it rises as SNR decreases and vice versa; and larger $N_{est}$ leads to overall reduction in the standard deviation. However, there is no need to use a very large $N_{est}$ in a practical system as only slight improvements can be brought about via increasing the same amount of $N_{est}$ when it becomes really large; besides, more raw keys can be obtained with smaller $N_{est}$. Here we define the deterioration in secret key rates $\Delta K = K - K_{\Delta \phi }$, where $K_{\Delta \phi }$ and $K$ are the key rate with and without the excess noise $\xi _{\Delta \phi }$, respectively. $\xi _{\Delta \phi }$ and $\Delta K$ can also be reduced by increasing $N_{est}$ as shown in Fig. 7 ($K$ is also plotted in Fig. 8 for reference); and still a certain number of $N_{est}$ is sufficient. In Fig. 7(a), except for the deep fade, the $\xi _{\Delta \phi }$ for each $N_{est}$ does not have strong fluctuations, therefore SNR dominates the key rates and the variations of $\Delta K$ are similar to those of SNR. The deep fade causes a sharp rise in $\xi _{\Delta \phi }$ but not in $\Delta K$; on the contrary, $\Delta K$ is “mitigated” by the SNR fall. This is because $K$ itself becomes quite small at this time and does not have much to lose.

Fig. 7. (a) $\xi _{\Delta \phi }$ and (b) $\Delta K$ for $N_{est}=2000$, 4000, 8000, and 10000 (blue, orange, green and red). Other parameters are specified in the caption of Fig. 6.

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Fig. 8. The $K$ that corresponds to the $\Delta K$ of Fig. 7(b) at a repetition rate of 1 MHz. The lines overlap because the SNR is basically the same as mentioned in the caption of Fig. 6. The key rates would usually become lower for a longer free-space link due to the increasing scaling of the amplitude and phase noise.

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5. Conclusion

We examined the correlation between data held by Alice and Bob for free-space CVQKD. A key feature of fading channels, i.e., the fluctuation of transmittance disappears in the correlation, thus enabling phase compensation for signals over such fluctuant channels. We demonstrated the phase compensation scheme with experimentally measured real channel transmittance; precise phase matches are shown to be achievable even though transmittance is fluctuating. The compensation accuracy is highly related to SNR and the number of data used. Thus we analyzed the performance of the compensation method. A high SNR level and a large number of data are preferred, but the larger the number of data is the less further improvements can be expected. Therefore, the number of data should be determined according to the conditions of optimization of secret key rates or other parameters required. The phase compensation approach is promising and could be applied to practical implementations of free-space CVQKD.

Appendix: secret key rate calculation

The key rate $K$ is given by

(11)$$K = \beta I_\textrm{AB} - \chi_\textrm{BE},$$

where $\beta$ is the reconciliation efficiency. $I_\textrm {AB}$ is the Shannon mutual information and can take the form

(12)$$I_\textrm{AB} = \frac{1}{2}\log_2 \frac{1}{1-\frac{\left\langle\sqrt{T} \right\rangle^2 \left(V-1 \right) }{\left\langle T \right\rangle \left( V-1 \right) + \xi + (1+v_{el})/\eta}}.$$

The Holevo quantity is given by [32]

(13)$$\chi_\textrm{BE} = \sum_{i=1}^{2}G\left(\frac{\lambda_i-1}{2}\right) - \sum_{i=3}^{5}G\left(\frac{\lambda_i-1}{2}\right),$$

where $G(x)=(x+1)\log _2(x+1)-x\log _2x$; the symplectic eigenvalues can be expressed as

(14)$$\begin{aligned}\lambda_{1,2}^2 &= \frac{1}{2} \left[ A \pm \sqrt{A^2 -4B} \right], \\ A &= a^2 + b^2 -2c^2, \\ B &= \left( ab-c^2 \right)^2, \end{aligned}$$

and

(15)$$\begin{aligned}\lambda_{3,4}^2 &= \frac{1}{2} \left[ C \pm \sqrt{C^2 -4D} \right], \\ C &= \frac{A\chi + a\sqrt{B} + b}{b+\chi}, \\ D &= \sqrt{B}\frac{a+\sqrt{B}\chi}{b+\chi}, \\ \chi &= \frac{1}{\eta} \left( 1-\eta + v_{el} \right), \end{aligned}$$

with $\lambda _5 = 1$, where

(16)$$\begin{aligned}a &= V, \\ b &= \left\langle T \right\rangle \left( V - 1 \right) + 1 + \xi, \\ c &= \left\langle \sqrt{T} \right\rangle \sqrt{V^2 - 1}. \end{aligned}$$

Funding

National Key Research and Development Program of China (2016YFA0302600); National Natural Science Foundation of China (61631014, 61671287, 61971276).

Disclosures

The authors declare no conflicts of interest.

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Phase compensation for free-space continuous-variable quantum key distribution

Abstract

1. Introduction

2. Phase compensation for fading channels

3. Demonstration of phase compensation

4. Performance analysis

5. Conclusion

Appendix: secret key rate calculation

Funding

Disclosures

References

Cited By

Figures (8)

Equations (16)

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