Impact of non-orthogonal measurement in Bell detection on continuous-variable measurement-device-independent quantum key distribution

Abstract

Continuous-variable measurement-device-independent quantum key distribution (CV-MDI QKD), whose core is Bell detection, can effectively resist attacks on the detection system. However, in CV-MDI QKD, Charlie is untrusted; the external environment will cause non-ideal Bell detection that will open security loopholes. We give a complete security analysis under non-ideal Bell detection caused by on-orthogonal measurement. The simulation results indicate that compared with no angle error, only 1$^\circ$ angle error will cause the transmission distance to drop by 7 km. Therefore, our research further improves and perfects the security of CV-MDI QKD.

Appendices

Appendix A Detailed expression of matrix ${{r}_{{\text {A}_{2}}\text {B}_{2}^{'}}}$

In order to calculate the secret key rate, we need to know the specific values of matrix ${{r}_{{\text {A}_{2}}\text {B}_{2}^{'}}}$. ${{r}_{{\text {A}_{2}}\text {B}_{2}^{'}}}$ can be divided into three parts: the covariance matrix of Alice’s quantum state, the covariance matrix of Bob’s quantum state, and the covariance matrix between Alice and Bob. Firstly, the covariance matrix of Alice’s quantum state is

$$\begin{aligned} {{r}_{{\text {A}_{2}}}}=\left( \begin{matrix} {{V}_\text {A}} &{} 0 \\ 0 &{} {{V}_\text {A}} \\ \end{matrix} \right) . \end{aligned}$$

(A1)

Then, the covariance matrix of Bob’s quantum state is

$$\begin{aligned} {{r}_{\text {B}_{2}^{'}}}=\left( \begin{matrix} \langle {{x}_{\text {B}_{2}^{'}}}{{x}_{\text {B}_{2}^{'}}}\rangle &{} \langle {{x}_{\text {B}_{2}^{'}}}{{p}_{\text {B}_{2}^{'}}}\rangle \\ \langle {{p}_{\text {B}_{2}^{'}}}{{x}_{\text {B}_{2}^{'}}}\rangle &{} \langle {{p}_{\text {B}_{2}^{'}}}{{p}_{\text {B}_{2}^{'}}}\rangle \\ \end{matrix} \right) . \end{aligned}$$

(A2)

In ${{r}_{\text {B}_{2}^{'}}}$, the expression of each component is

$$\begin{aligned} \langle x{{(p)}_{\text {B}_{2}^{'}}}x{{(p)}_{\text {B}_{2}^{'}}}\rangle =&{{T}_\text {x(p)}}({{V}_\text {A}}-1)+1+{{T}_\text {x(p)}}{{\varepsilon }_\text {x(p)}}, \end{aligned}$$

(A3)

$$\begin{aligned} \langle {{x}_{\text {B}_{2}^{'}}}{{p}_{\text {B}_{2}^{'}}}\rangle =&\langle {{p}_{\text {B}_{2}^{'}}}{{x}_{\text {B}_{2}^{'}}}\rangle =-\frac{{{g}_\text {p}}}{\sqrt{2}}\sin \theta \sqrt{{{T}_\text {B}}(V_\text {B}^{2}-1})\nonumber \\&+\frac{{{g}_\text {x}}{{g}_\text {p}}}{2}\sin \theta ({{T}_\text {B}}({{V}_\text {B}}+{{\varepsilon }_\text {B}}-1)-{{T}_\text {A}}({{V}_\text {A}}+{{\varepsilon }_\text {A}}-1)). \end{aligned}$$

(A4)

Lastly, the covariance matrix between Alice and Bob is

$$\begin{aligned} {{C}_{{\text {A}_{2}}\text {B}_{2}^{'}}}=\left( \begin{array}{ll} \sqrt{{{T}_\text {x}}(V_\text {A}^{2}-1)} &{} -\sin \theta \sqrt{{{T}_\text {p}}(V_\text {A}^{2}-1)} \\ 0 &{} -\cos \theta \sqrt{{{T}_\text {p}}(V_\text {A}^{2}-1)} \\ \end{array} \right) . \end{aligned}$$

(A5)

Here, we get the matrix ${{r}_{{\text {A}_{2}}\text {B}_{2}^{'}}}$

$$\begin{aligned} {{r}_{{\text {A}_{2}}\text {B}_{2}^{'}}}=\left( \begin{matrix} {{r}_{{\text {A}_{2}}}} &{} {{C}_{{\text {A}_{2}}\text {B}_{2}^{'}}} \\ C_{{\text {A}_{2}}\text {B}_{2}^{'}}^\text {T} &{} {{r}_{\text {B}_{2}^{'}}} \\ \end{matrix}\right) . \end{aligned}$$

(A6)

Appendix B Modulation curve of phase modulator

In order to analyze the measurement angle error in the practical application of CV-MDI QKD, we need to know the modulation voltage corresponding to different angles of the phase modulator, which can be obtained by measuring the modulation curve of the phase modulator. We utilized a CETC 44th, GC15PMTC7813 phase modulator in our experiment. At a temperature of 22 $^\circ$C, we constructed a coherent structure using two phase modulators. A ramp waveform with a frequency of 3 MHz was applied to the modulator under test, and we performed homodyne detection. We record the output optical power and modulation voltages and then draw the modulation curve according to the records. The drawing results are shown in Fig. 5. The equation of fitted modulation curve is

$$\begin{aligned} \eta =0.4738\sin \left( {\frac{U+\text {1}\text {.8078}}{\text {4}\text {.0537}}\pi }\right) +0.4996, \end{aligned}$$

(B7)

Fig. 5

The modulation curve of the phase modulator. The black dot represents the data point of the transmittance corresponding to the modulation voltage, and the red solid line represents the fitting curve of the data point

where U is the modulation voltage and $\eta$ is the transmittance. The fitting degree of this curve is 0.9995, which shows that the fitting curve is very consistent with the actual data. According to Eq. B7, we get the half wave voltage is ${{V}_{\pi }}=4.0516V$, and the voltage required for perfect orthogonal measurement is ${{V}_{\pi /2}}=2.0258V$. Then, we can convert the angle error to voltage deviation for calculation and measurement:

$$\begin{aligned} {{\theta }_{\text {dev}}}=\frac{{{\theta }_{\text {mou}}}-{{\theta }_{\pi /2}}}{{{\theta }_{\pi /2}}}=\frac{{{V}_{\text {mou}}}-{{V}_{\pi /2}}}{{{V}_{\pi /2}}}, \end{aligned}$$

(B8)

where ${{\theta }_{\text {dev}}}$ is the angle deviation, $\theta {{(V)}_{\text {mod}}}$ is the modulation angle (voltage), and $\theta {{(V)}_{\pi /2}}$ is the ideal angle (voltage). Based on Eq. B8, we can conclude that a modulation voltage change of 0.0225V will result in a phase change of 1$^\circ$. In addition, it is worth noting that the voltage deviation (angle error) here is a relative value with respect to the ideal value, so this method is effective for different experimental environments and phase modulators.