Unidimensional continuous-variable measurement-device-independent quantum key distribution

Abstract

Continuous-variable (CV) measurement-device-independent (MDI) quantum key distribution (QKD) is immune to imperfect detection devices, which can eliminate all kinds of attacks on practical detectors. Here we first propose a CV-MDI QKD scheme using unidimensional modulation (UD) in general phase-sensitive channels. The UD CV-MDI QKD protocol is implemented with the Gaussian modulation of a single quadrature of the coherent states prepared by two legitimate senders, aiming to simplify the implementation compared with the standard, symmetrically Gaussian-modulated CV-MDI QKD protocol. Our scheme reduces the complexity of the system since it ignores the requirement in one of the quadrature modulations as well as the corresponding parameter estimations. The security of our proposed scheme is analyzed against collective attacks, and the finite-size analysis under realistic conditions is taken into account. UD CV-MDI QKD shows a comparable performance to that of its symmetric counterpart, which will facilitate the simplification and practical implementation of the CV-MDI QKD protocols.

Appendix A: The derivation of displacement parameters of the two quadratures

After the eavesdropper performs one-mode attack, modes $A'$ and $B'$ can be obtained as:

$$\begin{aligned} A'= & {} \sqrt{\eta _{A}}A_{2} + \sqrt{1 - \eta _{A}}E_{A}, \end{aligned}$$

(29)

$$\begin{aligned} B'= & {} \sqrt{\eta _{B}}B_{2} + \sqrt{1 - \eta _{B}}E_{B}. \end{aligned}$$

(30)

where we assume $\eta _{A,x}, \eta _{A,p} = \eta _{A}$ and $\eta _{B,x}, \eta _{B,p} = \eta _{B}$. Modes $A'$ and $B'$ interfere on the balanced BS with output modes C and D as:

$$\begin{aligned} C= & {} \frac{1}{\sqrt{2}}(A' - B') = \frac{1}{\sqrt{2}}(\sqrt{\eta _{A}}A_{2} - \sqrt{\eta _{B}}B_{2}) + \frac{1}{\sqrt{2}}(\sqrt{1 - \eta _{A}}E_{A} - \sqrt{1 - \eta _{B}}E_{B}), \end{aligned}$$

(31)

$$\begin{aligned} D= & {} \frac{1}{\sqrt{2}}(A' + B') = \frac{1}{\sqrt{2}}(\sqrt{\eta _{A}}A_{2} + \sqrt{\eta _{B}}B_{2}) + \frac{1}{\sqrt{2}}(\sqrt{1 - \eta _{A}}E_{A} + \sqrt{1 - \eta _{B}}E_{B}). \end{aligned}$$

(32)

After the displacement operation, the quadratures of mode $B^{'}_{1}$ can be described as $B^{'}_{1,x} = B_{1,x} + gC_{x}$ and $B^{'}_{1,p} = B_{1,p} + gD_{p}$.

The variance of mode $B^{'}_{1,x}$ can be calculated as:

$$\begin{aligned} \langle |B^{'}_{1,x}|^{2} \rangle= & {} V_{B} + \frac{g_{x}^{2}\eta _{B}}{2}V_{B}^{2} - 2g_{x}\frac{\eta _{B}}{\sqrt{2}}\sqrt{V_{B}(V_{B}^{2} - 1)} \nonumber \\&+ \frac{g_{x}^{2}\eta _{A}}{2} + \frac{g_{x}^{2}}{2}(\eta _{A}\chi _{A} + \eta _{B}\chi _{B}) \end{aligned}$$

(33)

Let $T_{A,x} = \frac{g_{x}^{2}\eta _{A}}{2}$, Eq. (33) can be rewritten as:

$$\begin{aligned} \langle |B^{'}_{1,x}|^{2} \rangle= & {} T_{A,x}(V_{A}^{2} - 1) + 1 + T_{A,x}\left( 1 + \frac{\eta _{A}\chi _{A} + \eta _{B}\chi _{B}}{\eta _{A}}\right) \nonumber \\&+ \frac{2T_{A,x}}{\eta _{A}}\left( \frac{V_{B} - 1}{g_{x}^{2}} + \frac{\eta _{B}V_{B}^{2}}{2} - \frac{\sqrt{2\eta _{B}V_{B}(V_{B}^{2} - 1)}}{g_{x}}\right) . \end{aligned}$$

(34)

To minimize the equivalent excess noise from mode $B^{'}_{1,x}$, we can choose the displacement parameter $g_{x}$ as:

$$\begin{aligned} g_{x} = \sqrt{\frac{2}{\eta _{B}V_{B}}}\cdot \sqrt{\frac{V_{B} - 1}{V_{B} + 1}}. \end{aligned}$$

(35)

with the minimized equivalent excess noise $\epsilon _{x,\hbox {mim}} = 1 + \chi _{A} + \frac{\eta _{B}}{\eta _{A}}(\chi _{B} - V_{B})$.

Similarly, the variance of mode $B^{'}_{1,p}$ can be calculated as:

$$\begin{aligned} \langle |B^{'}_{1,p}|^{2} \rangle= & {} V_{B} + g^{2}_{p}\cdot \frac{\eta _{B}}{2} - 2g_{p}\sqrt{\frac{\eta _{B}(V^{2}_{B} - 1)}{2V_{B}}} \nonumber \\&+ g^{2}_{p}\frac{\eta _{A}}{2} + \frac{g^{2}_{p}}{2}(\eta _{A}\chi _{A} + \eta _{B}\chi _{B}). \end{aligned}$$

(36)

Let $T_{A,p} = \frac{g^{2}_{p}\eta _{A}}{2}$, Eq. (36) can be rewritten as:

$$\begin{aligned} \langle |B^{'}_{1,p}|^{2} \rangle= & {} 1 + T_{A,p}\left( 1 + \frac{1}{\eta _{A}}(\eta _{A}\chi _{A} + \eta _{B}\chi _{B})\right) \nonumber \\&+ \frac{2T_{A,p}}{\eta _{A}}\left( \frac{V_{B}-1}{g^{2}_{p}} + \frac{\eta _{B}}{2} - \frac{\sqrt{\frac{2\eta _{B}(V^{2}_{B}-1)}{V_{B}}}}{g_{p}} \right) . \end{aligned}$$

(37)

To minimize the equivalent excess noise from mode $B'_{1,p}$, we can choose the displacement parameter $g_{p}$ as:

$$\begin{aligned} g_{p} = \sqrt{\frac{2V_{B}(V_{B} - 1)}{\eta _{B}(V_{B} + 1)}}. \end{aligned}$$

(38)

with the minimized equivalent excess noise $\epsilon _{p,\hbox {min}} = 1 + \chi _{A} + \frac{\eta _{B}}{\eta _{A}}\left( \chi _{B} - \frac{1}{V_{B}}\right)$.