Impact of Imbalanced Modulation on Security of Continuous-Variable Measurement-Device-Independent Quantum Key Distribution

Abstract

Continuous variable measurement-device-independent quantum key distribution (CV-MDI-QKD) removes all known or unknown side-channel attacks on detectors. However, it is difficult to fully implement assumptions in the security demonstration model, which leads to potential security vulnerabilities inevitably existing in the practical system. In this paper, we explore the impact of imbalanced modulation at transmitters on the security of the CV-MDI-QKD system mainly using a coherent state and squeezed state under symmetric and asymmetric distances. Assuming two different modulation topologies of senders, we propose a generalized theoretical scheme and evaluate the key parameter achievable of the protocol with the mechanism of imbalanced modulation. The presented results show that imbalanced modulation can achieve a relatively nonlinearly higher secret key rate and transmission distances than the previous protocol which is the balanced modulation variance used by transmitters. The advantage of imbalanced modulation is demonstrated for the system key parameter estimation using numerical simulation under different situations. In addition, the consequences indicate the importance of imbalanced modulation on the performance of CV-MDI-QKD protocol and provide a theoretical framework for experimental implementation as well as the optimal modulated mode.

1. Introduction

Quantum key distribution (QKD) can build the bridge between two legitimate communicating terminals (Alice and Bob), which are generally linked by a precariously quantum channel threaten by eavesdropper (Eve) and an authenticated classical channel [1,2,3,4,5,6]. Its unconditional security proof is guaranteed by primary principles of quantum mechanics with one-time pad protocol that any stealing activities for enciphered massage using the shared secure keys by Eve can be detected by both sides of communication, even though the more powerful quantum computing is employed. Continuous-variable (CV) QKD is a major category of QKD protocol, which encodes information on the amplitude or phase quadratures of coherent or squeezed state, respectively, and receivers perform the homodyne or heterodyne detection. Owing to the advantage of promising potential high secure key rate at medium and short distances, as well as high compatibility with the current optical communication networks and simpler cost-effective practical implementations, CV-QKD systems have drawn scientists and engineer’s attention over the past decade [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]. CV-QKD using a Gaussian quantum resource modulated state, such as the coherent state and squeezed state, has been demonstrated in their security level based on the procedure of parameter estimation, data reconciliation, and privacy amplification. Based on the rapid progress in theory and experiment, the CV-QKD protocol has achieved tremendous advances in practical applications and optical networks [8,25,26,27,28,29].

At present, the theoretical security analyses of the Gaussian modulated coherent state (GMCS) and the modulated squeezed state-based CV-QKD protocol have been initially proved under the all possible attacks and later expanded to consider the finite-size effect and composable security [30,31,32,33,34,35]. However, the non-negligible gap between idealized hypothesis models and the process of practical experiment implementation for QKD, inevitably leads to some potential security vulnerabilities exploited by a variety of targeted attacks from Eve [4,5,6,36]. For the purpose of filling the gap between the theoretical models and practical experiments as soon as possible, the measurement-device-independent (MDI) QKD protocol has been proposed [37,38,39,40,41,42], in which both transmitting terminal Alice and Bob send modulation data to an untrusted third party, who are introduced to perform Bell-state measurement (BSM) and announce the result through an authenticated classical channel used for generating the secure keys. MDI-QKD can naturally eliminate all side-channel attacks against the detectors, whether we already know it or not.

In previous work [15,20,43,44,45,46], Alice and Bob generally perform a symmetry modulated coherent state or squeezed state during the process of theoretical analysis and experimental implementation based on the CV-MDI-QKD protocol. The results of the above literature indicate that when the distances from Alice to the untrusted third party are equal to the distances from Bob to the untrusted third party (the symmetric case), the performance of the protocol, in terms of both maximum secure transmission distances and secret key rate, is lower than in the asymmetric case, where these distances are different. Furthermore, the secure analysis of CV-MDI-QKD protocol is completed considering the worst-case scenario where the two links are subject to a two-mode Gaussian attack based on both sides utilizing balanced modulation. These results pave the way towards a quantum network and promote the progress of MDI security. Based on the theoretical analytical framework, the finite-key size effect is considered against general attacks with a rigorous security proof [46,47,48,49] that has shown the robustness security of MDI-QKD in asymmetric unstable channels. In addition, the secure bound of the ultimate secure key rate is acquired under the statistical fluctuations, which can provide practical guidance to threshold detectors with highly lossy channels and against environmental disturbances [50,51,52].

Motivated by [15,43,44,45], we mainly extend the advantages of the MDI framework and explore the impact of imbalanced modulation at both transmitting sides on the security of CV-MDI-QKD protocol in the paper. Both the asymmetric case and symmetric case have been demonstrated in the presence of an arbitrary two-mode Gaussian attack of the links, which has been proven to be the optimal attack for CV-MDI-QKD protocols [15]. Our results show that the imbalanced modulation can improve the performance of the protocol for both the secret key rate and transmission distances compared with the previous balanced modulation.

The rest of this paper is organized as follows. In Section 2, we describe the details of the CV-MDI-QKD protocol with imbalanced modulation using coherent state and squeezed state. In Section 3 and Section 4, we present the simulation results under the case of symmetric and asymmetric transmitted distances respectively. In Section 5, we give a discussion, and in Section 6, we summarize our results.

2. CV-MDI-QKD Protocol with Imbalanced Modulated Using Squeezed State and Coherent State

In this section, we first review the conception of the CV-MDI-QKD protocol and theoretically establish equivalent entanglement-based (EB) scheme of the protocol using the coherent state and squeezed state [7,44,45,53]. We then present the influence of imbalanced modulation on key parameters against a two-mode Gaussian attack in the protocol under symmetric and asymmetric cases. We then elucidate the different performance between balanced modulation and imbalanced modulation.

In the case of CV-MDI-QKD with squeezed-state or coherent-state preparation, Alice and Bob independently generate randomly Gaussian-distributed values $x_A$($p_A$) and $x_B$($p_B$), satisfying the zero mean and variances $V_A$ and $V_B$, respectively. During one of the crucial processes of the experimental realization of CV-MDI-QKD protocol for encoding the key information, the amplitude and phase modulator or IQ modulator are utilized to modulate x or p quadrature for the purpose of achieving various modulation types. In general, either coherent or squeezed state is prepared as the modulated source for the prepare-and-measure (PM) scheme. Then, these modulated-squeezed states or coherent states are transmitted through two different unsecured and loss standard quantum channels that can be controlled by the eavesdropper (Eve) to the third party (Charlie), respectively. The third party, Charlie, receives two independent quantum signals and mixes each of them with a 50:50 beam splitter (BS). The output port of the interfered mode signal is measured with x(p) quadrature by two realistic homodyne detectors: one for amplitude quadrature and the other for phase quadrature, respectively. The detection results are publicly announced to Alice and Bob through an authenticated classical channel. The knowledge of public outcomes can enable each party (Alice or Bob) to infer the measurement results of the other party by data post-processing. In particular, the performance of the modulated-squeezed state can aggrandize the resistant capacity against a two-mode Gaussian attack when Alice and Bob generate an x or p quadrature-squeezed-state. If we assume that an x quadrature-squeezed state is prepared, the modulation variances $V_{PM}$ and $V_{AM}$ should satisfy the equation $s+V_{AM}=1/s+▵V_0+V_{PM}$, where s and $1/s+▵V_0$ denote the squeezing and anti-squeezing variances of the states, respectively. $▵V_0$ is the excess noise of the anti-squeezing [45,53].

For convenience, to analyze the security of the protocol, an entanglement-based (EB) scheme is proposed to provide comprehensive proof and equivalent to the PM scheme even more important. The EB scheme of CV-MDI-QKD protocol equivalence to the PM scheme described above is presented in Figure 1.

Alice and Bob generate a two-mode squeezed vacuum state with the variance $V_A$ and $V_B$ independently (generally, $V_A=V_B$, but in this paper, the main analysis focuses on the case of $V_A\ne V_B$), and keep one of the modes on each side, another mode is sent to the third party through the two different quantum channels at the same time. Taking the modulated squeezed state as an example, an additional Einstein–Podolsky–Rosen (EPR) mode I is appended to the mode A through BS with a coupling ratio of 50:50, respectively. We can implement the homodyne detection on mode A and the other mode $A_0$ is projected to the squeezed state. At the same time, Bob performs the same thing.

However, during primary studies [15,20,43,44,45,46], Alice and Bob keep the same operations, including squeezed parameters and modulated variance under two different conditions with symmetric and asymmetric transmission distances. This work emphasizes the point of imbalance modulation in the CV-MDI-QKD protocol using the coherent and squeezed state. The third-party, Charlie, performs a CV Bell-state measurement. On this station, the two received imbalanced modulated modes $A_2$ and $B_2$ are interfered at a 50:50 beam splitter, then the output two modes C and D are transformed into the modes C′ and D′ to model the realistic homodyne detector with detection efficiency $\eta$ and electronic noise $v_{el}$. The measurement results are publicly announced to both sides, Alice and Bob, through the authenticated classical channel.

Next, based on the knowledge from the third party, Charlie, each party (Alice or Bob) can infer the measurement results from the other party. Finally, Alice and Bob can distill a common secret key over a quantum channel by implementing parameter estimation, information reconciliation, and privacy amplification procedures with the aid of an authenticated classical channel. We can calculate the secure key rate of the imbalance CV-MDI-QKD protocol based on the EB scheme. According to ref. [15], the asymptotic secret key rate against collective attacks is given by

$$\begin{array}{c}\hfill R=\beta I_{ab}-{\chi }_E,\end{array}$$

(1)

where $\beta$ is reconciliation efficiency, $I_{ab}$ is Shannon’s mutual information between Alice and Bob after Charlie published the measurements. ${\chi }_E$ is the Holevo bound between Alice and Eve, representing the maximum information that Eve has stolen theoretically.

In this paper, we mainly consider a joint two-mode Gaussian attack. From the perspective way of eavesdropper, the “negative EPR attack” has proven to be an optimal correlated attack, which is the most general eavesdropping strategy for the CV-MDI-QKD protocol (see supplementary information of ref. [15] for more details). The main process of the eavesdropping program is appended two correlative ancillary modes $E_1$ and $E_2$ on each channel. Two ancillary modes have the covariance matrix of the form ${\sigma }_{E_1{E}_2}=\left[\begin{array}{cc}{\omega }_A\mathbf{I}& \mathbf{G}\\ \mathbf{G}& {\omega }_B\mathbf{I}\end{array}\right]$, $\mathbf{G}=\left[\begin{array}{cc}g& 0\\ 0& g^{\prime }\end{array}\right]$, $\mathbf{I}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$. Here, ${\omega }_A$ is the variance of the thermal noise introduced by $E_1$ in Alice’s link, while ${\omega }_B$ is the variances of thermal noise introduced by $E_2$ in Bob’s link. g and ${\mathrm{g}}^{\prime }$ represent the quantum correlation between the two modes, and must satisfy a set of bona-fide conditions.

Then, the Shannon’s mutual information between the trusted parties can be expressed as

$$\begin{array}{c}\hfill I_{ab}=\frac{1}{2}{log}_2\frac{V_b}{V_b^a},\end{array}$$

(2)

where $V_b$ and $V_b^a$ denote the variance of the covariance matrices ${\gamma }_b$ and ${\gamma }_b^a$, which can be calculated in CV-MDI-QKD protocol (see Appendix B of ref. [45] for more details), respectively.

The Holevo bound ${\chi }_E$ is given by

$$\begin{array}{c}\hfill {\chi }_E=S\left({\rho }_E\right)-S\left({\rho }_E^a\right),\end{array}$$

(3)

where $S\left(\rho \right)$ is the von Neumann entropy of the quantum state $\rho$. During the process of calculation, we assume that Eve has a capacity that can purify the whole system within the physical range. This means that the system $ABE$ is pure after Charlie’s measurement announcement. So, the equation between two Holevo bounds should satisfy the relation $S\left({\rho }_E\right)=S\left({\rho }_{AB}\right)$. On account of these assumptions, the Holevo bound can be represented as

$$\begin{array}{c}\hfill {\chi }_E=\sum _{i=1}^{2}g\left(\frac{{\lambda }_i-1}{2}\right)-\sum _{i=3}^{5}g\left(\frac{{\lambda }_i-1}{2}\right),\end{array}$$

(4)

where ${\lambda }_i$ is the symplectic eigenvalues of the corresponding covariance matrix and $g\left(x\right)$ should satisfy the relationship $g\left(x\right)=\left(x+1\right){log}_2\left(x+1\right)-x{log}_2\left(x\right)$.

3. Imbalanced Modulation Influence on the Protocol under the Symmetric Distances

First, we consider the performances of the protocol under the symmetric distance, which means the transmittance distances $L_{AC}=L_{BC}$. The total transmission distance is $L=L_{AC}+L_{BC}$ (we assume that the channel loss is 0.2 dB per km hereafter). The secure key rate and maximum transmission distances using the imbalanced pure squeezed state and coherent state are demonstrated in this section respectively. Without loss of generality, we describe the performance of the protocol in the presence of an arbitrary two-mode Gaussian attack of the links.

We demonstrate the performance of the protocol using coherent state under symmetric distances. Figure 2 shows the contour of the secure key rate versus imbalanced modulation variance of $V_A$ and $V_B$ using the coherent state under symmetric distances and ideal detection. Different from the case of balanced modulation, we can obtain the same secure key rate by means of permutation and combination between $V_A$ and $V_B$. Meanwhile, the consequence of protocol with balanced modulation is not optimal in comparison to the imbalanced modulation. Without loss of generality, the consequence of balanced modulation is a subset of the imbalanced modulation virtually. In the course of calculating the secret key rate, the equivalent excess noise due to the introduction of thermal noise from Eve in both quantum channels is not symmetric. If Alice is the encoder, Bob can modify his data while Alice keeps hers unchanged during the postprocessing step. This means that the symmetric case cannot result in an optimal system performance and imbalanced variance modulation cannot result in a symmetric secret key rate topological distribution. According to the actual experiment condition, imbalanced modulation variance can improve the performance of the protocol using coherent state. Based on the different modulated modes, we compare the consequence of coherent state protocol between optimal imbalanced modulation and optimal balanced modulation under ideal and practical detection efficiency. As illustrated in Figure 3, it is clear that the secure key rate and maximum transmission distances with imbalanced modulation are always larger than the balanced modulation under different detection efficiency even though just slightly. The dashed lines demonstrate that the imperfection of the detectors decreases the total transmission distances dramatically, but the imbalanced modulation mode is still better than the balanced modulation mode. These results reveal the advantages of the imbalanced modulation variance and provide a modulation method on the premise of guaranteeing the performance parameter.

For the purpose of comparing the performance of the coherent state preparation protocol, the pure squeezed-state protocol using balanced and imbalanced modulation is analyzed under the symmetric distance. As illustrated in Figure 4, we calculate the maximum transmission distance $L_{max}$, where the border region of secret key rate ${10}^{-4}$ bit per pulse is applied to the results as a function of imbalanced squeezed parameters $s_A$ and $s_B$. We find that multiform arrangements and compositions between imbalanced squeezed parameters $s_A$ and $s_B$ can achieve the same maximum transmission distances. It is clear that the few squeezed-parameter states prepared can cause a more maximum distance of the protocol closer to the bound PLOB [54]. When squeezed parameter s tends to 1, the maximum transmission distances are close to the performance of coherent state protocol. Explicitly, the influence of squeezed parameters $s_A$ and $s_B$ on the secret key rate is nonlinear. In some certain regions, different imbalanced squeezed states have the same effect on maximum transmission distances and secret key rate. In order to achieve the purpose that is as as far a distance as possible, Figure 4 also provides proof that as few squeezed parameters are applied to the experiment as possible. In the case of the secret key rate, there is a same similar conclusion, that different combinations of squeezed parameters in both modulation terminals can obtain the same secret key rate, as shown in Figure 5. The solid and dashed lines represent different detection efficiency $\eta =1,v_{el}=0$ and $\eta =0.97,v_{el}=0.031$, respectively.

4. Imbalanced Modulation Influence on the Protocol under the Asymmetric Distances

In this section, we analyze the performance of the protocol using coherent state and squeezed state under the asymmetric distances, because it has been proved that the performance of asymmetric case ($L_{AC}\ne L_{BC}$) is superior to the symmetric case ($L_{AC}=L_{BC}$) [43]. If the terminal of Charlie located close to Alice, the total transmission distance L will increase significantly as $L_{AC}$ decreases. The extreme asymmetric distance case, in which the two-mode Gaussian attack degenerates into two independent Gaussian attacks, is investigated using both the coherent state and squeezed state.

As depicted in Figure 6, we plot the secret key rate R as a function of the total transmission distances for the coherent state with optimal balanced modulation variance and imbalanced modulation under the case of asymmetric distances. It is clear that the optimal imbalanced modulation variance is always larger than the optimal balanced modulation variance for the secret key rate. When the total distances are less than 15 km, the performance of optimal imbalanced modulation approaches the balanced modulation. As the total distances exceed the 20 km, the secret key rate under the optimal imbalanced modulation is always slightly larger than the balanced modulation. Explicitly, the total maximal transmission distances of imbalanced modulation increase by 2 km compared to the balanced modulation. Furthermore, these results provide proof that imbalanced modulation mode improves the performance of the protocol under the case of asymmetric distances. We can also observe the impact of imperfections of the detectors on the performance of the protocol. Figure 6 also shows that when $L_{AC}$ increases, the transmission distance decreases dramatically. If the position of Charlie is located close to Alice, the total transmission distance can be a relatively longer distance. For instance, if the $L_{AC}$ increases to 0.1 km, the total transmission distance decrease dramatically under both modulation mode. As a contrast, we consider a practical detector that has imperfections such as finite efficiency and electronic noise. These imperfections will increase the thermal noise, which reduces the detection efficiency of Bell state measurements. Thus, the secret key rate and total transmission distances will decrease when the detector is imperfect. For instance, the transmission distance reduces to 12 km when the detector’s efficiency is setting to 0.90 depending on certain system parameters.

As mentioned above, we also analyze the available key rate and maximum transmission distances with respect to the imbalanced squeezed degree prepared for the extreme asymmetric distances. The simulation results are shown in Figure 7, where we calculate the maximum transmission distances vs. the imbalanced squeezed state prepared for extreme asymmetric distances. It is clear that there are optimal maximum transmission distances by means of choosing different combinations between squeezed parameters. With the increase in s, the performance of the protocol is not better as the same traditional thinking method. In some tops of curve regions, different imbalanced squeezed degree states prepared can reach the same relatively longer distances. Even though there is a relatively larger difference between $s_A$ and $s_B$, the maximum transmission distances are influenced slightly. Under the case of extreme asymmetric distances, different combinations of squeezed parameters can achieve different amounts of Shannon’s mutual information and Holevo bound ${\chi }_E$ describing Eve’s information obtained from the entangling cloner. Asymmetric combinations between squeezed parameters $s_A$ and $s_B$ can obtain a relatively higher secret key rate and transmission distances comparing the commonly used symmetric modulation. The squeezed state reduces the quantum noise in one quadrature of the light field at the expense of increasing the noise in the orthogonal quadrature. By optimizing the squeezed parameters, we minimize the noise in the quadrature that is most sensitive to transmission losses. The chosen squeezed parameters represent an optimal balance between noise reduction and the loss of signal. This character provides a guide for the experiment for squeezed state, which is relatively difficult to prepare in comparison to the coherent state to achieve longer distances and a higher secure key rate. According to this result, we can establish the experiment scheme that compromises between the maximum transmission distances and the squeezed parameter, which is difficult to prepare for the experiment.

5. Discussion

As the theoretical analysis mentioned above, it may be insufficient to guarantee the realistic implement, which the practical device is imperfect for achieving the ideal protocol performance. Considering the detection noise and source noise, imperfect detector efficiency can be reformulated into quantum efficiency, vacuum electronic noise and source noise. According to the imperfect detector results, it has a guiding significance to experimental preparations. It is possible that these imperfections can be compensated by optical phase-sensitive amplifiers (PSA) [55]. Based on the simulation results for imbalanced modulation, the performance of the CV-MDI-QKD protocol with discrete modulation can be derived, taking the finite-size effect and composable security analysis into account at the same time.

6. Conclusions

In this paper, we have theoretically investigated and analyzed the performance of the protocol using coherent and squeezed states with balanced and imbalanced modulation. Security analysis shows that imbalanced modulation outperforms balanced modulation in both symmetric and extreme asymmetric distances against two-mode Gaussian attack. Furthermore, these simulation results show how to design the modulation mode of the experimental scheme for the purpose of reducing cost and complexity. It is found that there is slight enhancement in the transmission distances with imbalanced modulation using coherent state prepared. In terms of the imbalanced squeezed state prepared, there are optimal squeezed-parameter compositions for maximum transmission distances, and the advantage of the imbalanced state preparation is shown. In practice, imperfect detection efficiency and source noise can influence the performance of the protocol; nevertheless, PSA can be applied to the experiment procedure of the CV-MDI-QKD protocol.