Continuous-variable measurement-device-independent quantum key distribution via quantum catalysis

Abstract

The continuous-variable measurement-device-independent quantum key distribution (CV-MDI-QKD) is a promising candidate for the immunity to side-channel attacks, but unfortunately seems to face the limitation of transmission distance in contrast to discrete-variable (DV) counterpart. In this paper, we suggest a method of improving the performance of CV-MDI-QKD involving the achievable secret key rate and transmission distance by using zero-photon catalysis (ZPC), which is indeed a noiseless attenuation process. The numerical simulation results show that the transmission distance of ZPC-based CV-MDI-QKD under the extreme asymmetric case is better than that of the original protocol. Attractively, in contrast to the previous single-photon subtraction (SPS)-based CV-MDI-QKD, the proposed scheme enables a higher secret key rate and a longer transmission distance. In particular, the ZPC-based CV-MDI-QKD can tolerate more imperfections of detectors than both the original protocol and the SPS-based CV-MDI-QKD.

Appendix A: The secret key rate of the SPS-based CV-MDI-QKD under one-mode collective Gaussian attacks

To make a comparison of the proposed ZPC-based CV-MDI-QKD scheme, here we present the SPS-based CV-MDI-QKD protocol, as shown in Fig. 10, where Alice and Bob respectively generates a two-mode squeezed vacuum state, $\mathrm{EPR}_{1}$ and $\mathrm{EPR}_{2}$, with the same variance $V_{A}=V_{B}$. Then, one half of $\mathrm{EPR}_{1}$ is kept by Alice to perform the heterodyne detection, the other half of $\mathrm{EPR}_{1}$ after the SPS operation is sent to Charlie through an insecure quantum channel with a length $L_{AC}$. Independently, one half of $\mathrm{EPR}_{2}$ is retained by Bob to perform the heterodyne detection, the other half of $\mathrm{EPR}_{2}$ is directly sent to Charlie through the quantum channel with a length $L_{BC}$. For the sake of discussions, we assume that both channels are same as the cases of our proposed scheme, and Eve implements one-mode collective Gaussian attacks on the quantum channels. In addition, it should be noted that, different from the ZPC operation, the SPS is a common non-Gaussian operation, which leads to make it impossible for directly using typical Gaussian methods to derive the secret key rate. Fortunately, according to the Gaussian optimality theorem [48,49,50], the lower bound of asymptotic secret key rate $K_{\mathrm{asy}}$ of reverse reconciliation under one-mode collective attack is

$$\begin{aligned} K_{\mathrm{asy}}=P_{1}\left\{ \beta \widetilde{I}\left( A{:}B\right) -\widetilde{ \chi }\left( B{:}E\right) \right\} , \end{aligned}$$

(13)

where $P_{1}$ represents the success probability of implementing the SPS, $\beta ,\widetilde{I}\left( A{:}B\right)$ and $\widetilde{\chi }\left( B {:}E\right)$ are the same definitions as Eq. (8).

Fig. 10

(Color online) Schematic diagram of the CV-MDI-QKD protocol using the SPS operation (purple box). a EB scheme of the SPS-based CV-MDI-QKD. b Equivalent one-way protocol of the EB scheme of the SPS-based CV-MDI-QKD under the assumption that Eve is aware of Charlie, and Bob’s EPR$_{2}$ state and displacement except for heterodyne detection. EPR$_{1}$ and EPR$_{2}$: Alice’s and Bob’s two-mode squeezed state, respectively. Het: heterodyne detection. Hom: homodyne detection. $\left\{ X_{A},P_{A}\right\}$ and $\left\{ X_{B},P_{B}\right\}$: Alice’s and Bob’s measurement results of heterodyne detection, respectively. $X_{C_{1}},P_{C_{2}}$: measurement results of homodyne detection of measuring the X and P quadrature, respectively. PNRD: photon number resolving detector; $\left| 1\right\rangle \left\langle 1\right|$: single-photon Fock state; $T_{A}( \varepsilon _{A})$, $T_{B}\left( \varepsilon _{B}\right)$: channel parameters for Alice-Charlie and Bob-Charlie. $T_{c}( \varepsilon _{\mathrm{th}})$: equivalent channel transmittance (excess noise). $D\left( \beta \right)$: displacement operation of Bob

In order to obtain the exact form of Eq. (13), we first make a review about the SPS operation (purple box) shown in Fig. 10, which can be viewed as an equivalent operator

$$\begin{aligned} \varTheta&=_{C}\left\langle 1\right| B\left( T\right) \left| 0\right\rangle _{C} \nonumber \\&=\frac{1-T}{T}a_{2}\exp \left[ a_{2}^{\dagger }a_{2}\ln \sqrt{T}\right] . \end{aligned}$$

(14)

Thus, the state $\left| \varPsi \right\rangle _{A_{1}\widetilde{A}_{2}}$ after operating single-photon subtraction is expressed as

$$\begin{aligned} \left| \varPsi \right\rangle _{A_{1}\widetilde{A}_{2}}&=\frac{1}{\sqrt{ P_{1}}}\varTheta \left| \mathrm{EPR}_{1}\right\rangle _{A_{1}A_{2}} \nonumber \\&=\frac{W_{1}W_{2}}{\sqrt{P_{1}}}\exp \left[ W_{2}a_{1}^{\dagger }a_{2}^{\dagger }\right] a^{\dagger }\left| 00\right\rangle _{A_{1} \widetilde{A}_{2}}, \end{aligned}$$

(15)

with

$$\begin{aligned} W_{1}&=\sqrt{\frac{\left( 1-\lambda ^{2}\right) \left( 1-T\right) }{T}}, \nonumber \\ W_{2}&=\lambda \sqrt{T}, \end{aligned}$$

(16)

and

$$\begin{aligned} P_{1}=\frac{W_{1}^{2}W_{2}^{2}}{1-2W_{2}^{2}+W_{2}^{4}}. \end{aligned}$$

(17)

According to the equivalent one-way protocol, after the state $\left| \varPsi \right\rangle _{A_{1}\widetilde{A}_{2}}$ passing through the quantum channel characterized by the transmission efficiency $T_{c}$ and excess noise $\varepsilon _{\mathrm{th}}$, the covariance matrix $\varGamma _{A\widetilde{B} _{1}}$ can be given by

$$\begin{aligned} \varGamma _{A\widetilde{B}_{1}}= & {} \left( \begin{array}{cc} aII &{} c\widetilde{Z}\sigma _{z} \\ c\sigma _{z} &{} bII \end{array} \right) \nonumber \\= & {} \left( \begin{array}{cc} \widetilde{X}II &{} \sqrt{T_{c}}\widetilde{Z}\sigma _{z} \\ \sqrt{T_{c}}\widetilde{Z}\sigma _{z} &{} T_{c}\left( \widetilde{Y}+\chi _{\mathrm{tot}}\right) II \end{array} \right) , \end{aligned}$$

(18)

where $\chi _{\mathrm{tot}}$ is the same definition as Eq. (9), and

$$\begin{aligned} \widetilde{X}&=\frac{4W_{1}^{2}W_{2}^{2}}{P_{1}\left( 1-W_{2}^{2}\right) ^{3}}-1, \nonumber \\ \widetilde{Y}&=\frac{2W_{1}^{2}W_{2}^{2}\left( 1+W_{2}^{2}\right) }{ P_{1}\left( 1-W_{2}^{2}\right) ^{3}}-1, \nonumber \\ \widetilde{Z}&=\frac{4W_{1}^{2}W_{2}^{3}}{P_{1}\left( 1-W_{2}^{2}\right) ^{3}}. \end{aligned}$$

(19)

Similar to the derivation of Eq. (10), by using Eq. (18), one can obtain the Shannon mutual information $\widetilde{I}\left( A{:}B\right)$ as the following form

$$\begin{aligned} \widetilde{I}\left( A{:}B\right) =\log _{2}\frac{\left( a+1\right) \left( b+1\right) }{\left( a+1\right) \left( b+1\right) -c^{2}}. \end{aligned}$$

(20)

To derive the Holevo bound $\widetilde{\chi }\left( B{:}E\right)$, we assume that Eve purifies the whole system $\rho _{A_{1}\widetilde{B}_{1}E}$, so that

$$\begin{aligned} \widetilde{\chi }\left( B{:}E\right)= & {} \widetilde{S}\left( E\right) - \widetilde{S}\left( E|B\right) \nonumber \\= & {} \widetilde{S}\left( A_{1}\widetilde{B}_{1}\right) -\widetilde{S}\left( A_{1}|\widetilde{B}_{1}^{m_{B}}\right) , \nonumber \\= & {} \underset{i=1}{\overset{2}{\sum }}G\left( \frac{\lambda _{i}-1}{2}\right) -G\left( \frac{\lambda _{3}-1}{2}\right) , \end{aligned}$$

(21)

where the von Neumann entropy $G\left( \varsigma \right)$ has been given in Eq. (12), and

$$\begin{aligned} \lambda _{1,2}^{2}= & {} \frac{\varDelta \pm \sqrt{\varDelta ^{2}-4\xi ^{2}}}{2},\nonumber \\ \lambda _{3}= & {} a-{c^{2}}/({b+1}), \end{aligned}$$

(22)

with

$$\begin{aligned} \varDelta&=a^{2}+b^{2}-2c^{2}, \nonumber \\ \xi&=ab-c^{2}. \end{aligned}$$

(23)