One-Photon-Interference Quantum Secure Direct Communication

Abstract

Quantum secure direct communication (QSDC) is a quantum communication paradigm that transmits confidential messages directly using quantum states. Measurement-device-independent (MDI) QSDC protocols can eliminate the security loopholes associated with measurement devices. To enhance the practicality and performance of MDI-QSDC protocols, we propose a one-photon-interference MDI QSDC (OPI-QSDC) protocol which transcends the need for quantum memory, ideal single-photon sources, or entangled light sources. The security of our OPI-QSDC protocol has also been analyzed using quantum wiretap channel theory. Furthermore, our protocol could double the distance of usual prepare-and-measure protocols, since quantum states sending from adjacent nodes are connected with single-photon interference, which demonstrates its potential to extend the communication distance for point-to-point QSDC.

1. Introduction

Quantum communication uses physical principles to ensure the security of communication. Bennett and Brassard proposed the quantum key distribution (QKD) protocol in 1984 (BB84) [1] which provides a secure way for key agreement. Long and Liu proposed quantum secure direct communication (QSDC) [2] in 2000, which provides secure and reliable communication in a channel with both noise and eavesdropping [2]. Rapid inevitable developments have been made in the fields of QKD and QSDC [1,2,3,4,5,6,7,8,9,10]. QSDC and QKD are designed for different application scenarios: QKD uses quantum states for key negotiation, which is used in classical processes to transmit information. QSDC does not require the use of keys and can directly load information onto quantum states for secure transmission, eliminating some processes and the related hidden security vulnerabilities. QSDC is communication in the traditional communication sense, namely it is the transmission of information from one place to another place. Therefore, QSDC can perform the distribution of a predetermined key in classical cryptography. Because QSDC is a type of classical communication, it is compatible with the existing Internet naturally. Therefore, the secure repeater network formed by QSDC leverages QSDC to transmit classical encrypted messages and relay them using classical repeaters [11,12], can construct quantum networks with end-to-end security using existing technologies, and thus can accelerate the development of quantum networks. Another advantage of QSDC is that it can perform stealth communication, offering secrecy and protection of the users because quantum signals are weak and difficult to be found, and furthermore the users can perceive their exposure by inspecting a sudden change in their communication error rate.

QSDC transmits information directly through a quantum channel, using entanglement [2,4,5,13,14] or single photons [3,13]. The security of these protocols has been completed [15,16,17] based on the quantum wiretap channel theory [18,19,20]. Due to imperfections in measurement devices, practical systems possess security loopholes [21,22,23]. In order to fix these loopholes, measurement-device-independent QSDC (MDI-QSDC) [24,25] protocols have been developed. In 2021, Long et al. proposed a simple and powerful method to increase channel capacity using masking (INCUM) [26], which increases the channel capacity and communication distance of QSDC. There have been several successful demonstrations of QSDC in fiber [15,27,28] and in free space [29,30]. In 2020, a secure-classical repeater QSDC network have been proposed and experimentally demonstrated [11,12]. A 15-user QSDC network with direct links has been reported [31]. In 2022, Zhang et al. reported a QSDC system with mixed coding of phase and time-bin states and achieved a communication distance of 102.2 km in fiber, setting up a new world record [32].

Traditional QSDC protocols [2,3,24,25] require two-way transmission, which leads to double loss of signal in the channel and also limits the transmission distance by half compared to one-way protocols in principle. Remarkably, with the help of the quantum-memory-free (QMF) technique, one-way QSDC protocols have been proposed [33,34] and QMF-QSDC with sophisticated coding has also been designed [35,36]. In addition, there are some one-way QSDC protocols using hyperentanglement [37,38,39].

However, the MDI-QSDC [24,25] protocols face several practical limitations, including reliance on immature quantum memory, the assumption of ideal entangled or single-photon light sources, limited transmission distances and low secrecy rates. Single-photon-memory MDI-QSDC [40] utilizes QMF technique to eliminate the dependency of MDI-QSDC protocol on high-performance quantum memory. To further address other issues, we propose a one-photon-interference MDI QSDC (OPI-QSDC) protocol in this paper. Our OPI-QSDC protocol is a new one-way MDI-QSDC protocol that operates without relying on quantum memory. Moreover, we harnesses single-photon interference, as utilized in twin-field QKD protocols [41,42,43,44,45,46,47,48,49] that have broken the repeaterless quantum communications bound known as the Pirandola–Laurenza–Ottaviani–Banchi (PLOB) bound [50]. This bound has been extended into bipartite quantum interactions [51] and quantum network universal limitations [52]. In the OPI-QSDC protocol, Alice and Bob are able to achieve single-photon interference at the intermediate node Charlie by utilizing phase-locking techniques [53], when using weak coherent light sources. This doubles the transmission distance compared to other one-way QSDC protocols [34]. We analyze the security of our protocol using quantum wiretap channel theory and simulate its performance, demonstrating its ability to break the PLOB bound.

The remainder of this paper is organized as follows. In Section 2, we describe the detailed steps of the proposed OPI-QSDC protocol, while in Section 3, we analyze its security. Then in Section 4, we present our numerical analysis of performance. Finally, we give a conclusion in Section 5.

2. Details of Protocol

In this section, we propose our OPI-QSDC protocol. As illustrated in Figure 1a, we suppose that Alice and Bob utilize phase locking techniques to lock the frequency and global phase of their lasers. They then simultaneously send light pulses to Charlie, an untrusted third party right in the middle between them. We use $M\in {\{0,1\}}^m$ to represent the message that Alice wants to transmit to Bob, and $C\in {\{0,1\}}^c$ represents the ciphertext. In particular, The detailed steps of our protocol are as follows.

Step 1: encoding. Alice encodes the message M to be transmitted to form the codeword C. Encoding operations use forward error correction codes, secure codes [36] and INCUM [26]. Details of the encoding process can be found in the Appendix A.

Step 2: mode preparation. Alice and Bob randomly select the coding mode with a probability of $1-p$ and the multi-intensity mode with a probability of p, where $0<p<<1$. The quantum states sent in each mode are as follows.

Coding mode: a weak coherent state $|\alpha \rangle$ for logical 0 or a weak coherent state $|-\alpha \rangle$ for logical 1, where ${\left|\alpha \right|}^2=u$ is the intensity of the coherent state. In this mode, Alice selects the coherent state to be sent based on the encoding result. For example, if the encoding result is 0, Alice sends a $|\alpha \rangle$ state; otherwise, she sends a $|-\alpha \rangle$ state. Bob randomly chooses to send these two states.

Multi-intensity mode: three different intensity and phase-randomized weak coherent states ${\widehat{\rho }}_{{\beta }_a}=|{\beta }_a{e}^{i{\varphi }_a}\rangle$ and ${\widehat{\rho }}_{{\beta }_b}=|{\beta }_b{e}^{i{\varphi }_b}\rangle$, where ${\beta }_a$ and ${\beta }_b$ are randomly chosen from the set of $\{\sqrt{{\nu }_1},\sqrt{{\nu }_2},0\}$, while $u>>{\nu }_1>{\nu }_2>0$, and ${\varphi }_a$, ${\varphi }_b$ are randomly chosen from $[0,2\pi )$.

Step 3: measurement. Charlie measures the pulses sent by Alice and Bob using single-photon interferometer, and publishes measurement results. Let $m_C$ and $m_D$ denote the measurement outcome of ${\mathrm{D}}_0$ and ${\mathrm{D}}_1$, where value “0” indicates a no-click event and value “1” indicates a click event. As shown in Figure 1c, Alice and Bob discard the no-click events and two-click events, and retain the one-click events, namely $m_C\oplus m_D=1$.

Step 4: mode matching. After all measurements are completed, Alice and Bob publish the modes information. They retain the measurement results of the same modes, and discard the measurement results of different modes, as shown in Figure 1b. Note that there is a probability of $2p(1-p)$ for a mode mismatch, resulting in the loss of information transmitted by Alice. However, Alice and Bob can utilize error correcting codes in Step 1 to recover the lost information. If they both send the multi-intensity mode, they publish the intensity and phase of the weak coherent state. Then they retain pulses with ${\beta }_a={\beta }_b$ and $|{\varphi }_a-{\varphi }_b|=0$ or $\pi$.

Step 5: parameter estimation. Alice and Bob randomly publish the bit values in some coding modes to estimate the quantum bit error rate (QBER), and then use different intensity values in multi-intensity modes to estimate the amount of information leakage. Based on these results of parameter estimation, they proceed to step 6.

Step 6: decoding. Bob decodes the message M from the codeword C. Details of the decoding process can be found in the Appendix A.

3. Security Analysis

In order to complete our security proof, we introduce an equivalent entanglement-based OPI-QSDC protocol, as detailed in Appendix B. In this protocol, we transform the laser source into an entanglement-photon source that can be analyzed conveniently. This way, the security of the entanglement-based protocol will imply the security of OPI-QSDC.

According to quantum wiretap channel theory [16,17,18,54,55], there is a secrecy capacity $C_s=C_M-C_W$ that enables us to reliably and securely transmit the message to recipients by using a forward encoding with a coding rate R lower than it, where $C_M$ and $C_W$ are the main channel’s capacity and wiretap channel’s capacity, respectively. We discard the case where there is no detector click and both detectors click, then first consider the case where only detector ${\mathrm{D}}_0$ clicks. In this case, the achievable secrecy rate is

$$R^C=I^C(A:B)-I^C(A:E),$$

(1)

where $I^C(A:B)$ is the mutual information of Alice and Bob when only detector ${\mathrm{D}}_0$ clicks, while $I^C(A:E)$ is the mutual information of Alice and Eve when only detector ${\mathrm{D}}_0$ clicks. We assume the channel of Alice and Bob is a symmetric channel, thus

$$I^C(A:B)=1-h\left(e\right),$$

(2)

where $h\left(x\right)$ is the binary entropy function, i.e., $h\left(x\right)=-x{log}_2\left(x\right)-(1-x){log}_2(1-x)$, and e is quantum bit error rate (QBER). In our protocol, we use X basis to transmit information, so $e=E_u^{X,C}$, where $E_u^{X,C}$ is the X-basis error rate when only detector ${\mathrm{D}}_0$ clicks.

To calculate $I^C(A:E)$, we can analyze the process of eavesdropping and then use the results of the parameter estimation. See Appendix B for details of the derivation. The upper bound on $I^C(A:E)$ is

$$\begin{array}{cc}\hfill I^C(A:E)& \le h\left(E_u^{Z,C}\right)\hfill \\ \hfill & =h[{\displaystyle \frac{1}{Q_u^C}}\hfill & \hfill {\left(\sum _{n=0}^{\infty }\sqrt{|C_{2n}{|}^2{Y}_{2n}^C}\right)}^2]\end{array}$$

(3)

where $Q_u^C$ is the total gain, i.e., the conditional probability of only detector ${\mathrm{D}}_0$ clicks when Alice and Bob send pulses with the intensity of u, while $|C_{2n}{|}^2$ is the probability when there are $2n$ photons in the channel. $Y_{2n}^C$ is the yield of the $2n$-photon state, i.e., the conditional probability of only detector ${\mathrm{D}}_0$ clicks when there are $2n$ photons in the channel. Hence, the achievable rate is

$$\begin{array}{cc}\hfill R^C=& I^C(A:B)-I^C(A:E)\hfill \\ \hfill \ge & q·Q_u^C·[1-fh\left(E_u^{X,C}\right)-h\left(E_u^{Z,C}\right)],\hfill \end{array}$$

(4)

where $q=1-2p(1-p)$ is the mode matching rate, and $f\ge 1$ is an inefficiency function for forward coding. We skip the discussion for only ${\mathrm{D}}_1$ clicks; however, it holds in a similar manner, which is

$$\begin{array}{c}\hfill R^D\ge q·Q_u^D·[1-fh\left(E_u^{X,D}\right)-h\left(E_u^{Z,D}\right)].\end{array}$$

(5)

Finally, the total secrecy rate formula is given by

$$\begin{array}{cc}\hfill R=& max\{R^C,0\}+max\{R^D,0\}.\hfill \end{array}$$

(6)

4. Performance Analysis

4.1. Comparison with Other QSDC Protocols

We performed numerical simulations for characterizing the performance of the proposed OPI-QSDC and other QSDC protocols [3,24]. The key parameter settings for our simulations are shown in

Table 1. Key parameter settings of simulation.

Parameter	Value	Description
$\zeta$	0.2 dB/km	the attenuation coefficient
${\eta }_d$	15%	the efficiency of detectors
$p_d$	$8\times {10}^{-8}$	the probability of dark count
$\delta$	1.5%	the misalignment probability
f	1.2	the inefficiency function for forward coding
u	0.046	the light intensity

[32,42].

As shown in Figure 2, the OPI-QSDC protocol is able to exceed the PLOB bound when $d>$ 228 km with practical parameters. The Appendix C contains derivation details of simulation formulas. We also simulate the performance of MDI-QSDC [24] and DL04 [3] protocol with the method of INCUM [26]. The MDI-QSDC and DL04 protocols are two-way protocols which suffer double channel loss over a certain transmission distance. To be more precise, the MDI-QSDC protocol detects twice to complete the transmission of information, while DL04 protocol detects only once. Therefore, the explicit secrecy rate equations used to draw the idealized protocol performance in Figure 2 are:

$$\begin{array}{cc}\hfill & R_{PLOB}^{ideal}=-{log}_2(1-{\eta }_c),\hfill \\ \hfill & R_{OPI-QSDC}^{ideal}={\eta }_d\sqrt{{\eta }_c},\hfill \\ \hfill & R_{DL04}^{ideal}={\eta }_d{\eta }_c^2,\hfill \\ \hfill & R_{MDI-QSDC}^{ideal}={\left({\eta }_d{\eta }_c\right)}^2,\hfill \end{array}$$

(7)

where ${\eta }_d$, ${\eta }_c$ are the detection efficiencies and the channel losses function, respectively. The DL04 protocol’s beginning rates will be higher than the OPI-QSDC, since we did not account for the effects of real light sources on its performance when we parameterized the protocols. The Appendix D contains further simulation details of the DL04 and MDI-QSDC protocols.

4.2. Effect of Light Intensity

The effects of light intensity u on the OPI-QSDC protocol have been explored separately. The data in Table 1 are utilized to determine the parameters of the numerical simulations in this subsection.

In terms of light intensity, there are two aspects to consider. Firstly, as the light intensity increases, the mean photon number in the channel also increases, leading to a corresponding increase in the gain $Q_u$. Secondly, the even-photon-number component in the channel becomes more prominent, resulting in a higher phase-error rate $E_u^Z$. As illustrated in Figure 3, the OPI-QSDC protocol demonstrates its optimal performance when the light intensity is set to $u=0.046$. With these parameters, our OPI-QSDC protocol achieves a maximum transmission distance of 443.5 km. This finding underscores the significance of employing a relatively weaker coherent state light source to enhance the protocol’s performance in practical applications. However, it is worth noting that an excessively weak light source may also lead to a low transmission distance.

5. Conclusions

In summary, we have proposed an OPI-QSDC protocol, and its security has been analyzed when using practical coherent light sources. This protocol has the following advantages compared to previous QSDC protocols: (1) It has a longer transmission distance and a higher secrecy rate. It provides a new option for the application scenarios of QSDC; for example, the OPI-QSDC protocol can play a role in long-distance intercity quantum communication. (2). The property of MDI further improve the safety of the system when using devices with imperfections. (3). It is more practical because it uses coherent state light sources.

The performance analysis of our protocol shows that, compared to the DL04 and MDI-QSDC protocols, it has a longer transmission distance which could achieve 443.5 km with a light intensity of 0.046 when using standard optical fiber. Additionally, it surpasses the PLOB bound when the transmission distance exceeds 228 km. In the future, the OPI-QSDC protocol has the potential to be put into practical application and may find applications in the global quantum Internet.