Plug-and-play dual-phase-modulated continuous-variable quantum key distribution with photon subtraction

Abstract

Plug-and-play dual-phase-modulated continuous-variable quantum key distribution (CVQKD) protocol can effectively solve the security loopholes associated with transmitting local oscillator (LO). However, this protocol has larger excess noise compared with one-way Gaussian-modulated coherent-states scheme, which limits the maximal transmission distance to a certain degree. In this paper, we show a reliable solution for this problem by employing non-Gaussian operation, especially, photon subtraction operation, which provides a way to improve the performance of plug-and-play dual-phase-modulated CVQKD protocol. The photon subtraction operation shows experimental feasibility in the plug-and-play configuration since it can be implemented under current technology. Security results indicate that the photon subtraction operation can evidently enhance the maximal transmission distance of the plug-and-play dual-phase-modulated CVQKD protocol, which effectively makes up the drawback of the original one. Furthermore, we achieve the tighter bound of the transmission distance by considering the finite-size effect, which is more practical compared with that achieved in the asymptotic limit.

Graphical abstract

Keywords

plug-and-play / dual-phase-modulated / continuous variable / quantum key distribution / photon subtraction

1 Introduction

Quantum key distribution (QKD) allows two remote parties, Alice and Bob, to share a sequence of random secure key, even in the presence of eavesdroppers (Eve) [1-4]. The security of a key is guaranteed by the laws of quantum physics [5, 6]. Generally speaking, QKD has two main approaches, namely, discrete-variable (DV) QKD [7, 8] and continuous-variable (CV) QKD [9-13]. In the first approach, the information of key bit is carried by the properties of a single photon [14, 15]. For example, Ref. [16] utilized the transverse momentum of single photons to encode high-dimensional secure keys and demonstrated it with the existing telecommunication infrastructure. While in CVQKD, the key bits are encoded on the quadrature variables of the optical field, and the detection is realized through high-efficiency homodyne or heterodyne detection techniques [17-20].

At present, CVQKD has been implemented by using standard telecommunication technology [21], which provides more simple and cost-effective implementation compared with its DV counterpart. In addition, CVQKD has been proved to be secure against arbitrary collective attacks not only in the asymptotic limit [22-25] but also the finite-size scenario [26-28].

Recently, several experiments of CVQKD have been carried out in the laboratory [10, 11, 29-31] and most of them were demonstrated in view of the one-way Gaussian-modulated coherent-states (GMCS) protocol. In the one-way GMCS CVQKD experiments, quantum signals encoded in two quadratures of coherent states were transmitted together with a strong local oscillator (LO) through a single-fiber channel. Unfortunately, the nonlocal arrangement of LO would cause wavelength attacks [32, 33], saturation attacks [34] and LO fluctuation attacks [35], which are closely related to the security loopholes of LO. In order to solve this problem, self-referenced CVQKD protocols without sending a LO is proposed [36-38]. Such protocols can effectively deal with the security loopholes of LO by employing an independent laser source at the receiver’s side. However, a major problem, common to those protocols is that it is difficult to guarantee the frequency instabilities of two independent laser sources in real-life experiments. Besides, environmental perturbations can lead to the fiber channel length fluctuations, the polarization drifts and phase diffusion [39-41], which also have negative effects on the security and performance of those protocols.

Different from the aforementioned protocols, in the preliminary experiment of plug-and-play configuration [42], it uses a single laser source instead of two separate lasers to generate a local LO for legitimate parties. However, this plug-and-play scheme is more sensitive to excess noise compared with one-way GMCS protocol and suffers the Trojan-horse attack [43, 44]. More recently, CVQKD based on a plug-and-play dual-phase-modulated coherent-states (DPMCS) protocol is proposed and experimentally demonstrated [45]. In this protocol, Bob prepares quantum states and Alice generates the LO locally for quantum state measurement, which is in contrast to the one-way GMCS protocol. Therefore, this protocol can remove the security loopholes of LO and automatically compensate the polarization drifts. What is more, a tight security bound can be derived against collective attacks since the plug-and-play DPMCS protocol considers the untrusted source noise. Unfortunately, this proposed protocol has larger excess noise compared with one-way GMCS QKD scheme, which limits the maximal transmission distance to a certain degree.

Currently, non-Gaussian operations, including photon subtraction and photon addition operations, have been theoretically and experimentally demonstrated to be able to significantly enhance the transmission distance of CVQKD protocols in view of the fact that these operations can be employed to increase and distill the entanglement in Gaussian entangled states [46-49]. What the photon subtraction operation attracts us is that it can be easily implemented in practice with existing technologies, thus providing more cost-effective solution than photon addition operation. Therefore, in this paper, we propose a method to improve the performance of plug-and-play DPMCS protocol by mainly considering the photon subtraction operation. Through performing a suitable photon subtraction operation with the entangled source at Bob’s side, the protocol can extend the maximum transmission distance under a relatively high key rate compared with the original protocols in consideration of the untrusted source noise. In addition, we also consider the influence of finite-size effect on the plug-and-play DPMCS protocol with photon-subtraction operation, which is more accordant with practical circumstances than the asymptotic limit. That is to say, our proposal can provide a feasible method for the demonstration of the plug-and-play DPMCS protocol in the real experimental implementation.

This paper is structured as follows. In Section 2, we first introduce the original plug-and-play DPMCS protocol, then present the model of plug-and-play DPMCS QKD protocol with photon subtraction and photon addition operations, respectively. In Section 3, we show numeric simulation and performance analysis. Finally, conclusions are drawn in Section 4.

2 Non-Gaussian operations in the plug-and-play DPMCS protocol

In this section, we first review the original plug-and-play DPMCS protocol, particularly the entanglement-based (EB) scheme [50]. After that, we show the model of plug-and-play DPMCS protocol with non-Gaussian operations (based on EB scheme), namely, photon subtraction and photon addition.

2.1 Description of plug-and-play DPMCS protocol

In practice, the implementation of the plug-and-play DPMCS protocol is based on the prepare-and-measure (PM) scheme. However, the PM scheme cannot provide a powerful description to establish security proofs. Therefore, it is necessary to introduce the EB scheme (see Fig. 1) to conveniently analyze the security of plug-and-play DPMCS protocol, which is fully equivalent to the PM version. Here we do not introduce the PM scheme of plug-and-play DPMCS protocol since it has been analyzed in detail in Ref. [45].

In the EB version, the general collective attack is taken into consideration, which is viewed as the optimality of Gaussian attack [51]. A entanglement state denoted as |{\text{Ψ}}_{ABF}\rangle is prepared by Fred. The state (denoted as B) is kept by Bob and another state (denoted as A₀) is sent to Alice through the quantum channel. Note that the quadratures of modes A₀ and B are, respectively, denoted as \langle {X^{\prime }}^2\rangle =\langle {P^{\prime }}^2\rangle =V+\zeta and \langle X^2\rangle =\langle P^2\rangle =V, where V=V_B+1 and \zeta =g-1+(g-1)V_I. Here V_B represents the modulation variance of mode B, V_I is the noise variance of a vacuum state, and g (g\ge 1) is the gain of a phase-insensitive amplifier (PIA), which is used to characterize the untrusted source noise in the PM scheme of plug-and-play DPMCS protocol [45]. Besides, the quantum channel features a transmittance T and an excess noise ξ. Bob applies a heterodyne detection on mode B to prepare the coherent state and Alice performs homodyne detection to measure the incoming mode A₁. Once enough quantity of correlated data has been collected, Alice and Bob take advantage of an authenticated public channel for parameter estimation. Finally, they obtain the final secret key by information reconciliation and privacy amplification. Here, the total channel-added noise (denoted as {\chi }_{line}) referred to the channel input can be calculated as {\chi }_{line}=1/T-1+\xi.

It is remarkable that Fred here is not a neutral party and is assumed to be controlled by Eve. In other words, the entanglement state |{\text{Ψ}}_{ABF}\rangle in the EB scheme is used to model the noisy coherent source in the PM scheme. Therefore, the entanglement state |{\text{Ψ}}_{ABF}\rangle is virtually a two-mode state and the calculation of the entanglement state is similar to the traditional two-mode squeezed vacuum state in CVQKD protocol. Under this assumption, we can obtain a tight security bound. While, if Fred is assumed to be a neutral party, the entanglement state |{\text{Ψ}}_{ABF}\rangle becomes a three-mode state since Fred is not controlled by Eve and can be considered equal to legitimate parties, Alice and Bob. Detailed analysis has been shown in Ref. [52].

2.2 Plug-and-play DPMCS protocol with photon subtraction

As shown in Fig. 2, we suggest the plug-and-play DPMCS protocol with photon subtraction operation applied at Bob’s station. In this new version of the protocol, Bob employs a BS with transmission μ to split the mode A₀ and the vacuum state C₀ into modes A₁ and C. Consequently, we can obtain the yielded tripartite state {\rho }_{FB{A}_{1}C}, which can be expressed as

\begin{array}{}\text{(1)}& {\rho }_{FB{A}_{1}C}=U_{BS}{[|\text{Φ}\rangle }_{FB{A}_0}\langle \text{Φ}{|}_{FB{A}_0}\otimes |0\rangle \langle 0|]U_{BS}^{†}\mathrm{.}\end{array}

Through using the positive operator-valued measurement (POVM) \left\{{\text{ϒ}}_0^m\mathrm{,}{\text{ϒ}}_1^m\right\}, the photon-number-resolving detector (PNRD) can be applied to measure mode C [53, 54]. The photon number of subtraction operation m depends on {\text{ϒ}}_1^m=|m\rangle \langle m|. Note that Bob can successfully perform the photon subtraction operation only when the POVM element {\text{ϒ}}_1^m clicks. The photon-subtracted state {\rho }_{FB{A}_1}^{PS} can be given by

\begin{array}{}\text{(2)}& {\rho }_{FB{A}_1}^{PS}=\frac{{\text{tr}}_C({\text{ϒ}}_1^m{\rho }_{FB{A}_{1}C})}{P^{{\text{ϒ}}_1^m}(m)}\mathrm{,}\end{array}

where {\text{tr}}_X(Y) stands for the partial trace of the multimode quantum state. P^{{\text{ϒ}}_1^m}(m) stands for the success probability of subtracting m photons, which can be calculated as

\begin{array}{}\text{(3)}& \begin{array}{ll}{P}^{{\text{ϒ}}_1^m}(m)\hfill & ={\text{tr}}_{FB{A}_{1}C}({\text{ϒ}}_1^m{\rho }_{FB{A}_{1}C})\hfill \\ \hfill & =(1-{\vartheta }^2){\displaystyle \sum _{n=m}^{\infty }}{C}_n^m{\vartheta }^{2n}{(1-\mu )}^m{\mu }^{n-m}\hfill \\ \hfill & =(1-{\vartheta }^2){\left(\frac{1-\mu }{\mu }\right)}^m{\displaystyle \sum _{n=m}^{\infty }}{C}_n^m{({\vartheta }^2\mu )}^n\hfill \\ \hfill & =(1-{\vartheta }^2){\vartheta }^{2m}\frac{{(1-\mu )}^m}{{(1-{\vartheta }^2\mu )}^{m+1}}\mathrm{,}\hfill \end{array}\end{array}

where C_n^m is a combinatorial number and n\ge m and \vartheta =\sqrt{\frac{V-1}{V+1}}. Here, we assume {\rho }_{FB{A}_{1}C}=|\alpha \rangle \langle \alpha |, where

\begin{array}{}\text{(4)}& \begin{array}{ll}|\alpha \rangle \hfill & =U_{BS}|\text{Φ}{\rangle }_{FB{A}_0}\otimes |0\rangle \hfill \\ \hfill & =\sqrt{1\text{}-\text{}{\vartheta }^2}{\displaystyle \sum _{n=0}^{\infty }}{\vartheta }^n(U_{BS}|n\mathrm{,0}\rangle )\otimes |n{\rangle }_B\hfill \\ \hfill & =\sqrt{1\text{}-\text{}{\vartheta }^2}{\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{t=0}^n}{\vartheta }^n\sqrt{C_n^t{(1\text{}-\text{}\mu )}^t{\mu }^{n-t}}|n\mathrm{,}t\mathrm{,}n\text{}-\text{}t\rangle \mathrm{.}\hfill \end{array}\end{array}

Based on Eq. (2) and Eq. (3), the covariance matrix of {\rho }_{FB{A}_1}^{PS} can be expressed as

\begin{array}{}\text{(5)}& {\text{Γ}}_{FB{A}_1}^{PS}=\left(\begin{array}{cc}U{I}_2& S{\sigma }_z\\ S{\sigma }_z& R{I}_2\end{array}\right)\mathrm{,}\end{array}

where I₂ and {\sigma }_z are 2×2 identity matrix and \text{diag}(\mathrm{1,}-1), respectively, and

\begin{array}{}\text{(6)}& \begin{array}{l}U=\frac{2(1+m)}{1-\mu {\vartheta }^2}-\mathrm{1,}\\ R=\frac{2(1+m{\vartheta }^2\mu )}{1-\mu {\vartheta }^2}-\mathrm{1,}\\ S=\frac{\sqrt{\mu }\vartheta (1+m)}{1-\mu {\vartheta }^2}\mathrm{.}\end{array}\end{array}

According to above analysis, the POVM {\text{ϒ}}_1^m becomes {\text{ϒ}}_1^1 and clicks with a probability P^{{\text{ϒ}}_1^1}(1) if we want to subtract 1 photons in practice. Similarly, the POVM {\text{ϒ}}_1^m becomes {\text{ϒ}}_1^2 and clicks with a probability P^{{\text{ϒ}}_1^2}(2) if we want to subtract 2 photons. By using this way, we can subtract more photons in real implementation. It is remarkable that the afore-derived bipartite state {\rho }_{FB{A}_1}^{PS} is not Gaussian anymore. However, its entangle degree can be increased through introducing the photon-subtraction operation [46].

2.3 Plug-and-play DPMCS protocol with photon addition

We depict the EB scheme of the plug-and-play DPMCS protocol with photon addition operation in part of Fig. 3. Here we only show the part of Bob’s side since the part of Alice’s side is the same as Fig. 2. Different from the implementation of the photon subtraction operation shown above, in this subsection, we implement the photon addition operation at both sides of the entanglement state |{\text{Ψ}}_{ABF}\rangle. It has been proved that this arrangement can improve both the quality of distilled entanglement and the success probability for a weak two-mode squeezed vacuum state [49]. In practice, to effectively implement the photon addition operation, conditional spontaneous parametric down-conversion (SPDC) in a nonlinear crystal is needed. Besides, a careful mode match between the signal mode and input quantum mode is required to achieve the successful photon addition.

As shown in Fig. 3, we introduce the photon addition scheme, which can be accomplished by using beam splitters and on-off photon detectors. Here Bob uses two beam splitters with transmittance η, namely, B{S}_1 and B{S}_2, to respectively split modes A₀, X₀ and B, Y₀ into modes A₁, X and B₁, Y. Bob then performs heterodyne detection on the mode B₁ and sends mode A₁ to Alice through the quantum channel. The photons |1\rangle in modes X₀ and Y₀ are projected into vacuum state |0\rangle \langle 0|. The unitary coupling between modes A_0{X}_0 and B{Y}_0 is expressed by

\begin{array}{}\text{(7)}& \begin{array}{l}|\text{Θ}{\rangle }_{F{A}_{0}B{X}_0{Y}_0}\\ =\text{}{U}_{F{A}_0{X}_0}(\eta )\text{}\otimes \text{}{U}_{B{Y}_0}(\eta )(|\varphi {\rangle }_{F{A}_{0}B}\text{}\otimes \text{}|1{\rangle }_{X_0}\text{}\otimes \text{}|1{\rangle }_{Y_0})\mathrm{.}\end{array}\end{array}

In this scheme, we employ on-off photon detectors as the measuring device, which are usually used in quantum optics experiments. Note that the on-off photon detectors D₁ and D₂ have only two measurement results: on (one or more photons) and off (no photon). We can use positive operators \left\{{\widehat{\text{∏}}}^{on},{\widehat{\text{∏}}}^{off}\right\} to express these two results in Fock-state space, namely,

\begin{array}{}\text{(8)}& {\widehat{\text{∏}}}^{off}=|0\rangle \langle 0|,{\widehat{\text{∏}}}^{on}=I-|0\rangle \langle 0|.\end{array}

It is worth mentioning that a successful photon addition operation can be obtained when both detectors register “off” results. Note that {\widehat{\text{∏}}}^{off}=|0\rangle \langle 0| stands for a projection operation into a vacuum state, thus the unnormalized entangled state is given by

\begin{array}{}\text{(9)}& \begin{array}{l}|{\varphi }_{unnorm}{\rangle }_{F{A}_{0}B}\\ {=}_{X_0}\langle 0|\underset{E}{{\displaystyle \otimes }}{\langle 0|\text{Θ}\rangle }_{F{A}_{0}B{X}_0{Y}_0}\\ ={\displaystyle \sum _{n=0}^{\infty }}\sqrt{1-{\delta }^2}{(\delta \eta )}^n(n\text{}+\text{}1)(1\text{}-\text{}\eta )|n\text{}+\text{}1{\rangle }_{F{A}_0}|n\text{}+\text{}1{\rangle }_B\mathrm{,}\end{array}\end{array}

where \delta =tanh(r) is the squeezing parameter in the two-mode squeezed state.

Since the success probability of photon addition operation is determined by the norm of state |{\varphi }_{unnorm}{\rangle }_{F{A}_{0}B}, which can be given by

\begin{array}{}\text{(10)}& \begin{array}{ll}{P}_{add}\hfill & {=}_{F{A}_{0}B}{\langle {\varphi }_{unnorm}|{\varphi }_{unnorm}\rangle }_{F{A}_{0}B}\hfill \\ \hfill & =\frac{{(1-\eta )}^2(1-{\delta }^2)\left(1+{\eta }^2{\delta }^2\right)}{{\left(1-{\eta }^2{\delta }^2\right)}^3}\mathrm{.}\hfill \end{array}\end{array}

We can derive the normalized entangled state, which follows as

\begin{array}{}\text{(11)}& |\varphi {\rangle }_{F{A}_{0}B}\text{}=\text{}{\displaystyle \sum _{n=0}^{\infty }}\frac{{(1\text{}-\text{}{\eta }^2{\delta }^2)}^{\mathrm{3/2}}}{\sqrt{1\text{}+\text{}{\eta }^2{\delta }^2}}(n\text{}+\text{}1)(\eta \delta {)}^n|n\text{}+\text{}1{\rangle }_{F{A}_0}|n\text{}+\text{}1{\rangle }_B\mathrm{.}\end{array}

The entangled state shown in Eq. (11) happens to be in the form of Schmidt decomposition [55]. In addition, we can employ logarithmic negativity to estimate the entanglement [56], which is given by

\begin{array}{}\text{(12)}& \begin{array}{ll}{E}_{add}\hfill & =2{{\displaystyle \log }}_2\left[{\displaystyle \sum _{n=0}^{\infty }}\frac{(n+1)(\eta \delta {)}^2{(1-{\eta }^2{\delta }^2)}^{\mathrm{3/2}}}{\sqrt{1+{\eta }^2{\delta }^2}}\right]\hfill \\ \hfill & ={{\displaystyle \log }}_2\left[\frac{{(1+\eta \delta )}^2}{(1-\eta \delta )(1+{\eta }^2{\delta }^2)}\right]\mathrm{.}\hfill \end{array}\end{array}

It is important to point out that the photon addition operation is more sensitive to detection efficiency compared with photon subtraction operation. This problem could be solved by taking advantage of a superconducting single-photon detector (SSPD) [49]. However, the cost of SSPD is high and its efficiency is low. Besides, the implementation of photon addition operation is more complex than that of photon subtraction operation based on above analysis. From a practical point of view, the photon subtraction operation is more attractive than the photon addition operation because of its relative simple and cost-effective implementation. Therefore, in the following, we focus our attention on the performance analysis and calculation of the photon subtraction since those of the photon addition can be performed in similar way.

3 Performance analysis and discussion

In this section, we show the the theoretical security simulation results of plug-and-play DPMCS protocol with photon subtraction compared with the original plug-and-play DPMCS protocol. We first consider the relationship between the success probability P^{{\text{ϒ}}_1^m}(m) of subtracting m (m=\mathrm{1,2,3,4}) photons and the transmittance μ since the success probability is very important in calculating the secret key rate. As shown in Fig. 4, it is clear that whether gain g=1 or g=2, the success probability of subtracting one photon is higher than subtracting other numbers of photon. In addition, the optimal success probability decreases with increasing photon number of subtraction. Thus, it cannot guarantee that only 1 or 2 photons are subtracted, the 3 or 4 photons may also be subtracted with lower probability. We can also observe from Fig. 4 that the success probability is slightly affected by the source noise which is weighted by parameter g.

Considering the fact that the change of the transmittance μ of Bob’s beam splitter will lead to the change of the covariance matrix {\rho }_{FB{A}_1}^{PS} and the success probability P^{{\text{ϒ}}_1^m}(m) when Bob performs the m-photon subtraction. Therefore, we may find an optimal μ to maximize the secret key rate for each channel loss. Here we show the maximal secret key rate as a function of channel loss for optimal μ in Fig. 5(a) and show the optimal choice of μ for each channel loss in Fig. 5(b). As illustrated in Fig. 5(a) and Fig. 5(b), the solid lines and dash lines stand for two different scenes where g=1 and g=1.005, respectively. The blue lines denote the case of the original plug-and-play DPMCS protocol, whose performance is worse than the plug-and-play DPMCS protocol with photon subtraction in the high channel-loss range. That is to say the photon-subtraction operation contributes to expand the maximal transmission distance in the usual case. However, in the low channel loss range, even we choose the optimal μ, the secret key rate of the plug-and-play DPMCS protocol with photon subtraction is still worse than that of the original plug-and-play DPMCS protocol. The reason can be found from Fig. 4 that the success probability of subtracting m photons is relatively low in low channel-loss range. We also observe that the plug-and-play DPMCS protocol with one-photon subtraction operation can tolerate more channel losses while the secret key rates of the protocol are relatively high. In addition, the source noise weighted by parameters g (dash lines) can reduce the tolerance in channel losses and decrease the secret key rates.

It is remarkable that Fig. 5 shown above is simulated using the optimal transmittance μ of Bob’s beam splitter and considering the effect of Fred’s behaviors. In the following, to provide more general analysis in the proposed protocol, Fig. 6 shows the secret key rate of plug-and-play DPMCS protocol with optimal one-photon subtraction operation as a function of μ at each channel loss under different g. The values of parameter g in Fig. 6(a), Fig. 6(b), Fig. 6(c) and Fig. 6(d) are, respectively, set to g=1, g=1.005, g=1.01 and g=1.015. We can observe that the unsecure region (blank area) enlarges with increasing the parameter g. In unsecure region, it shows the unsuitable values of transmittance μ can lead to the unphysical negative secret key rate, as illustrated in Fig. 6(a), Fig. 6(b), Fig. 6(c). However, when g=1.015, the maximum channel-loss tolerance of the plug-and-play DPMCS protocol with one-photon subtraction is decrease even using the optimal transmittance μ. Therefore, it is very important to control the untrusted source noise to ensure the security of the proposed protocol. Detailed calculation of the asymptotic secret key rate is shown in Appendix A.

In addition, from a practical point of view, it is necessary for us to consider the finite-size effect since the number of exchanged signals is impossibly unlimited in practice. Generally speaking, in the asymptotic case, it assumes that the quantum channel is perfectly known before the transmission is performed. However, in the finite-size scenario, one does not know in advance the characteristics of the quantum channel. The reason is that a portion of exchanged signals needs to be employed to estimate parameter. Since the finite-size scenario takes into account the loss of parameter estimation, the analysis is more realistic than the asymptotic regime. The main plots in Fig. 7 illustrate the finite-size secret key rate of plug-and-play DPMCS protocol with photon subtraction operation as a function of channel loss. Here Fig. 7(a), Fig. 7(b), Fig. 7(c) and Fig. 7(d) show the proposed schemes with one-photon subtraction, two-photon subtraction, three-photon subtraction and four-photon subtraction, respectively. We notice that whether in the proposed schemes or in the original scheme, the asymptotic scenario tolerates more channel losses than the finite-size scenario. In other words, the maximal transmission distance in the the asymptotic scenario is longer than that in the finite-size scenario. Besides, the curves of the finite-size scenario are more and more close to the curve of asymptotic case with increasing the number of exchanged signals N. The reason is that we can use the more signals to perform the parameter estimation procedure and thus make it closer to perfection when increasing the number of exchanged signals. It is worth mentioning that the one-photon subtraction operation still achieves the optimal performance in the finite-size scenario. Detailed calculation of secret key rate in the finite-size regime is shown in Appendix B.

4 Conclusions

We have suggested a method to improve the performance of the plug-and-play DPMCS protocol by applying non-Gaussian operations in Bob’s entangled source, mainly considering the photon subtraction operation. The proposed method can be easily implemented with existing technologies and thus shows better experimental feasibility in the plug-and-play configuration. We analyze the impact of the transmittance μ of the beam splitter on the photon-subtraction operation and the untrusted source noise weighted by parameter g on the performance of our proposal. The numerical simulations show that the photon subtraction operation can remarkably enhance the maximal transmission distance in the plug-and-play DPMCS protocol, which effectively makes up the drawback of the original plug-and-play DPMCS protocol. However, the unsuitable transmittance μ can lead to the unphysical negative secret key rate and the increase of g can reduce the tolerance in channel losses and decrease the secret key rates. Therefore, it is very important to control these two parameters to improve the performance in practical implementation. Furthermore, we can achieve the tighter bound of the transmission distance by considering the finite-size effect, which is more practical compared with that of achieved in the asymptotic limit.

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