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Abstract
Quantum identity authentication serves as a crucial technology for secure quantum communication, but its security often faces challenges due to quantum hacking of measurement devices. This study introduces a measurement-device-independent mutual quantum identity authentication (MDI MQIA) scheme capable of ensuring secure user authentication, despite the use of measurement devices vulnerable to quantum hacking. To realize the MDI MQIA scheme, we proposed and applied a modified Bell state measurement based on linear optics, enabling the probabilistic measurement of all Bell states. Furthermore, the proposed experimental setup adopted a plug-and-play architecture, thus efficiently establishing the indistinguishability of two photons prepared by the communication members. Finally, we successfully performed a proof-of-principle experimental demonstration of the proposed scheme using a field-deployed fiber, achieving quantum bit error rates of less than 3%.
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1. Introduction
The explosive development of quantum computing technology [1] has significantly threatened the security of modern cryptography [2,3], urgently necessitating a paradigm shift in cryptosystems for secure communication. In particular, realizing quantum-safe communication has become essential for future quantum internet environments [4–9]. Quantum cryptography is a representative secure communication technology, providing unconditional security even in a quantum computing environment [10]. Thus, to provide the security services of quantum cryptography such as confidentiality, integrity, authentication, and non-repudiation, studies on quantum key distribution (QKD) [11–24], quantum message authentication [25–30], quantum identity authentication (QIA) [31–42], and quantum signature [43–48] are being conducted. Among these, successfully implementing QIA, which verifies the legitimacy of entities, is the first step in achieving secure quantum communication [49]. However, research on QIA has mainly focused on theoretical proposals rather than its practical realization.
Theoretically, several QIA realization methods have been proposed to design a QIA protocol using easily implementable elements such as single qubits, single-qubit operations, and single-qubit measurements. However, QIA designs using these elements for practical realization have remained theoretical and have not been completely implemented so far. Additionally, these methods have not adequately addressed potential implementation vulnerabilities, such as optical device imperfections. Such loopholes could lead to severe security issues, including quantum hacking [50–57]. In particular, it is reasonably expected that quantum hacking of measurement devices can pose a substantial threat to QIA implementation, as in QKD. To address this crucial security concern, a theoretical proposal for a measurement-device-independent mutual quantum identity authentication (MDI MQIA), which offers security against quantum hacking, was proposed [38]. In MDI MQIA, users Alice and Bob prepare and transmit a quantum state to a third party, Charlie who performs a Bell state measurement (BSM) and announces the results to Alice and Bob. Charlie can only know the correlation between the quantum states of Alice and Bob and cannot estimate the encoded authentication information. Therefore, the security of the MDI MQIA does not rely on the measurement device, thereby eliminating loopholes of measurement device in practical implementation.
In this study, we successfully implemented MDI MQIA and verified its robustness through field demonstrations using a deployed optical fiber. We utilized a plug-and-play (PnP) architecture which resulted in a simpler experimental setup with fewer active controls for MDI MQIA. Furthermore, we proposed a modified BSM capable of measuring all Bell states using linear optics, applying it to the efficient implementation of MDI MQIA. We conducted our experiment using a 30.88-km deployed optical fiber as the quantum channel (QC) and confirmed the robustness of the MDI MQIA implementation, achieving a reasonable quantum bit error rate (QBER).
2. Experimental setup
The methodology employed to authenticate the users in MDI MQIA [38] is as follows. Alice and Bob pre-share a secret key ${k_{AB}}$ for authentication, and each generate decoy qubit information ${D_{A(B )}}$. Subsequently, both generate authentication and decoy qubits corresponding to ${k_{AB}}$ and ${D_{A(B )}}$, which are transmitted to Charlie. Charlie performs a BSM and publicly announces the BSM outcomes $CI$ to Alice and Bob. Finally, Alice and Bob simultaneously authenticate each other using ${k_{AB}}$, ${D_{A(B )}}$, and $CI$. Throughout the entire process, the experiments primarily focused on demonstrating the qubit preparation conducted by Alice and Bob as well as the modified BSM performed by Charlie.
We adopted a PnP architecture [14,16,58], which is well suited for QKD implementations. The PnP MDI architecture allows Alice and Bob to automatically achieve optical-mode matching, except for arrival timing. This is due to the fact that the photons are produced by a single laser located at Charlie, resulting in identical wavelengths. Moreover, the photons generated at Charlie are transmitted to Alice and Bob and then returned to Charlie, inducing polarization auto-calibration due to the round-trip nature of the PnP architecture via Faraday rotator mirrors (FMs) [59]. Alice and Bob share a common interferometer at Charlie, eliminating the need for complicated controllers to align the polarization, wavelength, and interferometer, in contrast to the one-way MDI scheme in which Alice and Bob use separate lasers and interferometers [60,61]. Thus, only timing control is required to achieve indistinguishability of two photons in the PnP MDI scheme.
As shown in Fig. 1, our experimental setup involves two authentication parties, Alice and Bob, along with a measuring center node named Charlie. Charlie consists of three main parts: optical pulse generation, interferometry, and measurement.
Fig. 1. Experimental setup of plug-and-play (PnP) measurement-device-independent mutual quantum identity authentication (MDI MQIA). The PnP architecture is adopted to efficiently achieve indistinguishability of two photons, which is crucial for performing Bell state measurements (BSMs). Additionally, a modified BSM setup is implemented to detect both $ {|{\Phi ^ \pm }} \rangle $ and $ {|{\Psi ^ \pm }} \rangle $ states without requiring photon-number-resolving single-photon detectors. To eliminate the polarization dependence of the interferometer, a Michelson interferometer is used instead of a Mach–Zehnder interferometer. Two polarization beam splitters are connected to each IM of Alice and Bob, decreasing the influence of the polarization-dependent characteristics of the IMs by allowing only the passage of vertically polarized pulses. LD: laser driver; SOA: semiconductor optical amplifier; VOA: variable optical attenuator; BS: beam splitter; PBS: polarization beam splitter; DWDM: dense wavelength division multiplexer; IM: intensity modulator; PM: phase modulator; PC: polarization controller; FM: Faraday rotator mirror; DL: delay line; SL: storage line; QC: quantum channel; SNSPD: superconducting nanowire single-photon detector.
The optical pulse generation part comprises several components: a continuous-wave (CW) distributed feedback (DFB) laser, a beam splitter (BS), semiconductor optical amplifiers (SOAs), and a polarization beam splitter (PBS). These components are utilized to generate polarization multiplexed optical pulse trains. To distinguish the signals from the backscattering noise in the time domain, optical pulses are generated in the form of trains rather than continuous pulses. The connector keys of the PBS are aligned to the slow axis, naturally enabling the implementation of the polarization-division multiplexing method [16]; that is, following the PBS, the trains from the upper and lower SOAs are vertically and horizontally polarized, respectively. The SOA is primarily used to amplify the CW seed light in the form of a pulse (1 MHz with a pulse width of 2.5 ns (full width at half maximum, FWHM)). Thus, the output light from the SOA inherits the optical characteristics of the input light, such as the wavelength and polarization state [62]. This allows the exploitation of passive mode-matching from a single light source, which is similarly possible under an IM-based scheme [16]. Furthermore, unlike IM, SOA does not rely on a phase interferometer, which exhibits high sensitivity to the ambient environment. Consequently, the on/off extinction ratio of the SOA can be stably maintained without an auto bias controller [63–68], approximately one order of magnitude higher than that of the IM.
The interferometer part consists of a BS, two dense wavelength division multiplexers (DWDMs), a delay line, and two FMs. We utilized an asymmetric Michelson interferometer to implement time-bin phase encoding. We selected a Michelson interferometer instead of the commonly used Mach–Zehnder interferometer to avoid a propagation delay mismatch of the orthogonal trains in the polarization-maintaining fiber [16]. To filter the noise from the SOAs, two DWDMs are placed in the interferometer, which naturally compensates for the polarization disturbance via the round trip of the path by the FMs.
The measurement part comprises three PBSs, three BSs, four polarization controllers (PCs), and four superconducting nanowire single-photon detectors (SNSPDs). In MDI MQIA [38], unlike in MDI-QKD [13], verifying all Bell states including $ {|{\varPhi ^ \pm }} \rangle $ is essential for user identification. Therefore, we implemented a modified BSM scheme that can measure these states without relying on photon-number-resolving SPDs [69,70].
The proposed modified BSM scheme is illustrated in Fig. 2; PBSs and PCs, which are used to configure the optical path and couple the input polarization state to that of the SNSPDs, respectively, are not included. In this scheme, the input states $ {\hat{a}_l^\dagger |0} \rangle $ and $ {\hat{b}_r^\dagger |0} \rangle $ interfere at the central BS [71], and the corresponding output states are measured by four SPDs; $SP{D_{ll}}$, $SP{D_{lr}}$, $SP{D_{rl}}$, and $SP{D_{rr}}$. Here, ${\hat{a}^\dagger }$ and ${\hat{b}^\dagger }$ are the creation operators of Alice and Bob, respectively. The subscripts l and r indicate the direction of the qubit, and $ {|0} \rangle $ denotes the null state. In the conventional BSM, it is not possible to detect Bell states $ {|{\varPhi ^ \pm }} \rangle$, output states corresponding to the input states ${ {|H} \rangle _A}{ {|H} \rangle _B} = {\hat{a}_{H,l}^\dagger \hat{b}_{H,r}^\dagger |0} \rangle $ and ${ {|V} \rangle _A}{ {|V} \rangle _B} = {\hat{a}_{V,l}^\dagger \hat{b}_{V,r}^\dagger |0} \rangle $, without utilizing photon-number-resolving SPDs [69,70]. In the modified BSM, these states can be measured by incorporating two additional BSs that split the bunching photons into individual photons, allowing their distinction. For instance, the output state corresponding to the input state ${ {|H} \rangle _A}{ {|H} \rangle _B} = {\hat{a}_{H,l}^\dagger \hat{b}_{H,r}^\dagger |0 \ge } \rangle $ is expressed as:
Fig. 2. Modified BSM setup. The modified BSM can probabilistically measure $ {|{\varPhi ^ \pm }} \rangle $ without utilizing photon-number-resolving SPDs. Additionally, the BSM results of $ {|{\Psi ^ \pm }} \rangle $ can be obtained in an identical manner to those of conventional BSMs. $l$: left-hand side; $r$: right-hand side.
As shown in Eq. (1), the Bell states $ {|{\varPhi ^ \pm }} \rangle $ can be measured with a probability of $0.5$ though the outputs of ${ {|H} \rangle _{ll}}{ {|H} \rangle _{lr}}$ and ${ {|H} \rangle _{rl}}{ {|H} \rangle _{rr}}$ (see Appendix A for the output states corresponding to all input states). As in the conventional BSM, ${ {|H,\; H} \rangle _{ll}}$, ${ {|H,\; H} \rangle _{lr}}$, ${ {|H,\; H} \rangle _{rl}}$, and ${ {|H,\; H} \rangle _{rr}}$ cannot be distinguished from single photon detections. Nevertheless, detecting the Bell states $ {|{\varPhi ^ \pm }} \rangle $ using commercial SPDs without the photon-number-resolving SPDs [69,70] is an obvious advantage.
Both Alice and Bob consist of simpler devices for bit encoding because complicated devices, such as light sources, detectors, and interferometers, are installed at Charlie. The simplicity and uniformity of the structures of Alice and Bob provide a significant advantage for expanding to network systems in the future. An IM and phase modulator (PM) are used to perform time-bin phase encoding on the qubit, and a double-phase modulation method [72] is applied to mitigate the polarization-dependent modulation efficiencies of the IM and PM. As the two PBSs with slow-axis-aligned connector keys only allow vertically polarized pulses to pass through the IM, the polarization-dependent loss of the IM can be passively eliminated [17,73]. An FM with a 45° rotation angle is used to form a round-trip route. A 15-km fiber spool is used as a storage line to separate the forward signal pulses and backward scattering noises in the time domain [59], and a variable optical attenuator is employed to attenuate the intensity of the optical pulse to the single-photon level.
The signal flow in the setup is as follows. Initially, Charlie generates two orthogonally polarized optical pulse trains via the optical pulse generation part. Each optical pulse in the trains is then divided into time-bin pulses (fast and slow pulses) using an asymmetric Michelson interferometer. Following the generation of the time-bins, the pulse trains are transmitted to Alice and Bob via the PBSs and QCs, depending on their polarization states. The vertically and horizontally polarized trains are sent to Alice and Bob, respectively. Subsequently, Alice and Bob independently perform time-bin phase encoding [60,61,74] on the incoming trains using PMs and IMs. They then reflect the encoded pulse trains as orthogonally polarized ones using FMs with a 45° rotation angle. The pulse trains returning from Alice and Bob are guided to the modified BSM setup by the PBSs and interfere at the BS. The results of this interference are measured using the SNSPDs. Finally, Alice and Bob compare the interference results with their pre-shared secret key information for mutual authentication, as outlined above.
We conducted a proof-of-concept experimental demonstration of the proposed MDI MQIA scheme using a field-deployed fiber (30.88 km) between Alice and Charlie (Korea Research Environment Open Network). The experimental setup and field deployment are shown in Figs. 3(a) and (b), respectively. As shown in Fig. 3(a), the QC between Alice and Charlie in Fig. 1 was replaced by the deployed fiber. Accordingly, for experimental convenience through symmetry, the length of the QC between Bob and Charlie was set to 30.84 km (Table 1). As indicated in Fig. 3(b), all users were placed at the Korea Advanced Nano Fab Center (KANC) and the field-deployed channel consisted of two optical fiber links connecting the KANC and Sungkyunkwan University (SKKU). To establish a round-trip optical path, the two connector terminations on the SKKU side were interconnected, enabling the optical pulse to enter port RED (in KANC) and exit port BLUE (in KANC), and vice versa.
Fig. 3. Full experimental setup of a PnP MDI MQIA for the field test. (a) Experimental setup. All the users are located at the Korea Advanced Nano Fab Center (KANC). A field-deployed fiber, specifically the Korea Research Environment Open Network (30.88 km) is utilized as the QC connecting Alice and Charlie; a fiber spool (30.84 km) is employed as the QC between Bob and Charlie. (b) Field deployment of the optical links. The field-deployed channel consists of two optical fiber links that connect KANC and Sungkyunkwan University (SKKU) through three intermediary nodes. To establish a round-trip optical path, the two connector terminations on the SKKU side are interconnected, allowing the optical pulse to enter port RED (in KANC) and exit port BLUE (in KANC), and vice versa. Map data: © 2022 Google, Airbus, Maxar Technologies, and TMap Mobility.
Table 1. Experimental conditions.
In the experiment, we configured the pulse specifications, including the frequency and width, to readily available levels. As listed in Table 1, we set the optical wavelength to 1550.12 nm, which corresponding to ITU DWDM channel 34. We set the quantum efficiencies of the SNSPDs to 37%. Although the SNSPDs could support quantum efficiencies of approximately 80%, we intentionally selected low efficiencies to prevent the latching effect of the detectors. Table 1 provides a comprehensive list of the experimental conditions.
The following active devices were used for implementation: DFB (FITEL FRL15DCWA-A81-19340), SOA (InPhenix IPSAD1513C), IM (EOSpace Inc. AZ-0S5-10-PFU-PFU), PM (EOSpace Inc. PM-DK5-12-PFU-LV-UL), and Single Quantum EOS 810 CS (SNSPD). The DFB was driven by the laser driver MODEL6100 from Newport, whereas the SOAs were driven by the SOA drivers SOA-std from AeroDiODE. The IMs and PMs at Alice and Bob were modulated using waveform generators 33500B from Keysight. To implement a coincidence count unit and synchronize the trigger signals of the active devices, a field-programmable gate array (FPGA) board (TR4 FPGA Development Kit from Terasic) was used. As a proof-of-principle experiment, we achieved synchronization among Alice, Bob, and Charlie by connecting the FPGA board and active devices using electrical cables. However, in a practical system, it is advisable to employ a synchronization method that is commonly used in PnP QKD systems [14,73,75–79] (see Discussion for further information).
3. Experimental results
Under the given conditions, we initially measured the Hong-Ou-Mandel visibility, which is a key evaluation parameter for the MDI scheme. As a result, we achieved an interference visibility of 44.75%. This is quite a reasonable result considering that our scheme does not require real-time controllers for wavelength and reference frame adjustments and that the theoretical interference visibility with a weak coherent source can reach up to 50%. Subsequently, we demonstrated the proposed MDI MQIA by measuring the QBERs of the modified BSM. The measured BSM counts corresponding to the eight encoding cases for Alice and Bob are shown in Fig. 4. The QBERs were estimated from these results using the same formula as MDI QKD (see Appendix B for the detailed formulas). On the Z basis, QBERs of 1.04 and 25.62% were achieved for $ {|{\Psi ^ \pm }} \rangle $ and $ {|{\varPhi ^ \pm }} \rangle $, respectively, while on the X basis, a QBER of 27.32% was obtained. Considering the theoretical existence of an offset error rate of 25% in regards to QBERZ for $ {|{\varPhi ^ \pm }} \rangle $ and QBERX due to the pulses containing multi photons of the weak coherent source [80], the effective QBERs for all the cases were less than 3%. These low QBERs in the noisy field-deployed fiber verified the feasibility of the proposed MDI MQIA scheme implemented with a modified BSM scheme.
Fig. 4. Bell state measurement counts corresponding to eight encoding cases of Alice and Bob. Each detection result is accumulated for 408 s. Despite the utilization of time-bin phase encoding, the encoding bits are represented using the polarization notations (H, V, D, and A) to improve understanding and adhere to the widely used representations of Bell states; encoding a qubit as a H or V polarization state corresponds to selecting either the fast or slow time-bin pulse using an intensity modulator. On the other hand, encoding a qubit as a D or A polarization state corresponds to modulating the relative phase of the time-bin pulses to 0 or π using a phase modulator.
Furthermore, experiments were conducted using mean photon numbers of 0.48 and 0.10. As shown in Table 2, QBERs comparable to those from the experiments with a mean photon number of 0.62 were obtained. Notably, the QBER has shown an increasing trend; the smaller the mean photon number, the greater is the susceptibility of the quantum system to noise. We remark that the QBER in MDI MQIA scales linearly with the transmission loss, similar to MDI QKD. This occurs because the dark count of the SPD becomes more prominent as the transmission loss increases.
Table 2. Experimental results of measuring the quantum bit error rates (QBERs) with three different mean photon numbers.
Upon analyzing the experimental results, we identified an issue. As shown in Fig. 4, the ratios of $ {|{\Psi ^ + }} \rangle $:$ {\; |{\Psi ^ - }} \rangle $ differed slightly from the theoretical values. In the time-bin phase encoding scheme, the coincident click when both the fast and slow pulses are measured in one of the two arms of the interfering BS (the first BS of the modified BSM setup) is labeled $ {|{\Psi ^ + }} \rangle $, and the coincident click when both pulses are measured in different arms of the BS is labeled $ {|{\Psi ^ - }} \rangle $. Theoretically, the ratios of $ {|{\Psi ^ + }} \rangle $:$\; {|{\Psi ^ - }} \rangle $ for the ${ {|D} \rangle _A}{ {|D} \rangle _B}$ and ${ {|A} \rangle _A}{ {|A} \rangle _B}$ encoding cases and the ${ {|D} \rangle _A}{ {|A} \rangle _B}$ and ${ {|A} \rangle _A}{ {|D} \rangle _B}$ encoding cases are 3:1 and 1:3, respectively. However, in our experiments, the ratios were measured as 2.2:1 for the ${ {|D} \rangle _A}{ {|D} \rangle _B}$ and ${ {|A} \rangle _A}{ {|A} \rangle _B}$ encoding cases and 1:3.2 for the ${ {|D} \rangle _A}{ {|A} \rangle _B}$ or ${ {|A} \rangle _A}{ {|D} \rangle _B}$ encoding cases, resulting in an increase in the QBERs. The decreased effective efficiency of the detector is attributable to the dead time and latching effect of the detector. Thus, to address this issue, it is necessary to reduce the dead time and latching effect or to design a new scheme that does not require multiple clicks on a single detector within a short period of time. We remark that these imperfect detector properties may reduce the performance but do not provide a loophole for other quantum hacking, as the MDI scheme ensures unconditional security for the measurement devices. Refer to Ref. [38] for a comprehensive security analysis of MDI MQIA against certain specific attacks, such as intercept-and-resend and impersonation attacks.
4. Discussion
In this study, we propose an experimental setup for the MDI MQIA scheme with a modified BSM that can probabilistically detect $ {|{\varPhi ^ \pm }} \rangle $ as well as $ {|{\Psi ^ \pm }} \rangle $ without photon-number-resolving SPDs [69,70]. Moreover, we have described several approaches that improve the practicality of implementation, such as the adoption of the PnP architecture, replacement of the IM with the SOA, double-phase modulation, and polarization-independent intensity modulation. Furthermore, we experimentally demonstrated the feasibility of using a 30.88-km field-deployed fiber between Alice and Charlie. We conducted QBER measurements using the proposed setup with the mean photon numbers of 0.10, 0.48, and 0.62. We obtained reasonable QBERs, considering that a theoretical offset error rate of 25% exists in QBERZ for $ {|{\varPhi ^ \pm }} \rangle $ and QBERX owing to the use of the weak coherent source. For the mean photon number of 0.62, QBERs of 1.04 and 25.62% were obtained for $ {|{\Psi ^ \pm }} \rangle $ and $ {|{\varPhi ^ \pm }} \rangle $, respectively, on the Z basis. Meanwhile, a QBER of 27.32% was achieved on the X basis. Table 2 shows the results corresponding to the other mean photon numbers.
In this study, certain aspects of the implementation load were simplified for experimental convenience. However, several factors should be considered when implementing a more comprehensive system in the future. First, to improve the authentication rate, increasing the repetition rate of optical pulse generation is essential. In this study, the repetition rate was set relatively low at 1 MHz, considering the round-trip time (>100 ns) between the IM and FM of Alice and Bob. However, to enhance the authentication rate for future implementation, increasing the repetition rate by reducing the round-trip path length is crucial. The round-trip time can be decreased by either shortening the lengths of the optical fibers coupled to the devices of Alice and Bob or by implementing chip-based Alice and Bob systems. Second, to achieve long-term stable operation, adopting a timing auto-calibration system [17,76,78] that can compensate for the drift in photon arrival time in real time is essential. Fortunately, implementing such a system in the proposed architecture, where the laser and detectors are jointly installed at Charlie, is relatively straightforward. Third, to improve the extinction ratio of the Z basis, from which QBERZ is obtained, SOAs can be employed as substitutes for the IMs [81] of Alice and Bob. Moreover, this reduces the system complexity because the SOA does not require an auto-bias controller. Fourth, synchronization between Alice, Bob, and Charlie was implemented by directly connecting them to the FPGA board. However, for a practical system, implementing a synchronization method [14,73,76–79] commonly used in PnP QKD systems to synchronize remote nodes is essential. Given that the architecture of the transmitters in our scheme involving Alice and Bob is almost identical to that of a typical PnP system, the synchronization method can be directly incorporated into our scheme. Finally, with the advent of rigorous security verifications [77,79,82–87] considering the implementation of power and timing monitoring, the PnP architecture has become widely accepted as a secure and practical structure [73,76,77,85–88]. Nevertheless, it is necessary to consider at least the following attacks:
In the PnP architecture, in the worst-case scenario, Eve could attempt to use a stronger pulse and analyze the reflected signal to estimate the phase value sent by Alice and Bob. Alice and Bob should implement pulse power monitoring for incoming pulses to prevent such an attack. If Eve can manipulate the arrival times of the pulses, they will pass through the phase modulator at different times, resulting in distinct phase modulations. This phase remapping process enables Eve to execute an intercept-and-resend attack. However, users can detect this attack by monitoring the time-shifted pulses and observing the QBERs.Based on the results of this study, we outline our plans for future research. First, we intend to implement an MDI MQIA setup with Alice, Bob, and Charlie remotely located from each other. This allows exploring the practical feasibility of long-distance quantum communication using the proposed scheme. Furthermore, we plan to implement an MDI quantum cryptosystem by combining MDI QKD and MDI MQIA. This integration can address the security vulnerabilities associated with QKD and pave the way for establishing a more robust and secure quantum communication system. Finally, we aim to extend our research to network communication. At present, we anticipate that we can implement polarization, wavelength, and time division multiplexing networks through structural modifications such as those in Refs. [17,73,76]. In addition to these modifications, it is essential to explore the structural properties of the MDI architecture and develop strategies for constructing larger quantum networks with enhanced functionality. By pursuing these future research areas, we aim to support advancements in the field of quantum communication and contribute to the development of practical quantum technologies.
Appendix A. Formulas for all output states in the modified BSM scheme
In total, there are eight formulas for the output states in the modified BSM scheme, including Eq. (1) in the manuscript. Equations $({\textrm{A}1} )$–$({\textrm{A}4} )$ represent the outputs when Alice and Bob choose the Z basis, while Eqs. (A5–A8) correspond to the X basis.
As indicated in Eq. (A1), $ {|{\varPhi ^ \pm }} \rangle $ can be measured with a probability of $0.5$ though the outputs of ${ {|H} \rangle _{ll}}{ {|H} \rangle _{lr}}$ and ${ {|H} \rangle _{rl}}{ {|H} \rangle _{rr}}$.
As indicated in Eq. (A2), $ {|{\varPhi ^ \pm }} \rangle $ can be measured with a probability of $0.5$ though the outputs of ${ {|V} \rangle _{ll}}{ {|V} \rangle _{lr}}$ and ${ {|V} \rangle _{rl}}{ {|V} \rangle _{rr}}$.
As indicated in Eq. (A3), $ {|{\psi^ + }} \rangle $ can be measured with a probability of $0.5$ though the outputs of ${|{H,{\; }V\rangle} _{ll}}$, ${|{H,{\; }V\rangle} _{lr}}$, ${|{H,{\; }V\rangle} _{rl}}$, ${|{H,{\; }V\rangle} _{rr}}$, ${|H\rangle _{ll}}{|V\rangle_{lr}}$, ${|H\rangle_{lr}}{|V\rangle_{ll}}$, ${|H\rangle_{rl}}{|V\rangle_{rr}}$, and ${|H\rangle_{rr}}{|V\rangle_{rl}}$. Furthermore, $ {|{\psi^ - }} \rangle $ can be measured with a probability of $0.5$ though the outputs of ${|H\rangle_{ll}}{|V\rangle_{rl}}$, ${|H\rangle_{ll}}{|V\rangle_{rr}}$, ${|H\rangle_{lr}}{|V\rangle_{rl}}$, ${|H\rangle_{lr}}{|V\rangle_{rr}}$, ${|H\rangle_{rl}}{|V\rangle_{ll}}$, ${|H\rangle_{rl}}{|V\rangle_{lr}}$, ${|H\rangle_{rr}}{|V\rangle_{ll}}$, and ${|H\rangle_{rr}}{|V\rangle_{lr}}$.
As indicated in Eq. (A4), $ {|{\psi^ + }} \rangle $ can be measured with a probability of $0.5$ though the outputs of ${|{H,{\; }V\rangle} _{ll}}$, ${|{H,{\; }V\rangle} _{lr}}$, ${|{H,{\; }V\rangle} _{rl}}$, ${|{H,\; V\rangle}_{rr}}$, ${|H\rangle_{ll}}{|V\rangle_{lr}}$, ${|H\rangle_{lr}}{|V\rangle_{ll}}$, ${|H\rangle_{rl}}{|V\rangle_{rr}}$, and ${|H\rangle_{rr}}{|V\rangle_{rl}}$. Furthermore, $ {|{\psi^ - }} \rangle $ can be measured with a probability of $0.5$ though the outputs of ${|H\rangle_{ll}}{|V\rangle_{rl}}$, ${|H\rangle_{ll}}{|V\rangle_{rr}}$, ${|H\rangle_{lr}}{|V\rangle_{rl}}$, ${|H\rangle_{lr}}{|V\rangle_{rr}}$, ${|H\rangle_{rl}}{|V\rangle_{ll}}$, ${|H\rangle_{rl}}{|V\rangle_{lr}}$, ${|H\rangle_{rr}}{|V\rangle_{ll}}$, and ${|H\rangle_{rr}}{|V\rangle_{lr}}$.
As indicated in Eq. (A5), $ {|{\varPhi ^ \pm }} \rangle $ can be measured with a probability of $0.25$ though the outputs of ${|H\rangle_{ll}}{|H\rangle_{lr}}$, ${|H\rangle_{rl}}{|H\rangle_{rr}}$, ${ {|V} \rangle _{ll}}{ {|V} \rangle _{lr}}$ and ${ {|V} \rangle _{rl}}{ {|V} \rangle _{rr}}$. Furthermore, $ {|{\psi^ + }} \rangle $ can be measured with a probability of $0.5$ though the outputs of ${|{H,{\; }V\rangle} _{ll}}$, ${|{H,{\; }V\rangle} _{lr}}$, ${|{H,{\; }V\rangle} _{rl}}$, ${|{H,\; V\rangle} _{rr}}$, ${|H\rangle_{ll}}{|V\rangle_{lr}}$, ${|H\rangle_{lr}}{|V\rangle_{ll}}$, ${|H\rangle_{rl}}{|V\rangle_{rr}}$, and ${|H\rangle_{rr}}{|V\rangle_{rl}}$.
As indicated in Eq. (A6), $ {|{\varPhi ^ \pm }} \rangle $ can be measured with a probability of $0.25$ though the outputs of ${|H\rangle_{ll}}{|H\rangle_{lr}}$, ${|H\rangle_{rl}}{|H\rangle_{rr}}$, ${ {|V} \rangle _{ll}}{ {|V} \rangle _{lr}}$ and ${ {|V} \rangle _{rl}}{ {|V} \rangle _{rr}}$. Furthermore, $ {|{\psi^ + }} \rangle $ can be measured with a probability of $0.5$ though the outputs of ${|{H,{\; }V\rangle} _{ll}}$, ${|{H,{\; }V\rangle} _{lr}}$, ${|{H,{\; }V\rangle} _{rl}}$, ${|{H,\; V\rangle} _{rr}}$, ${|H\rangle_{ll}}{|V\rangle_{lr}}$, ${|H\rangle_{lr}}{|V\rangle_{ll}}$, ${|H\rangle_{rl}}{|V\rangle_{rr}}$, and ${|H\rangle_{rr}}{|V\rangle_{rl}}$.
As indicated in Eq. (A7), $ {|{\varPhi ^ \pm }} \rangle $ can be measured with a probability of $0.25$ though the outputs of ${|H\rangle_{ll}}{|H\rangle_{lr}}$, ${|H\rangle_{rl}}{|H\rangle_{rr}}$, ${ {|V} \rangle _{ll}}{ {|V} \rangle _{lr}}$ and ${ {|V} \rangle _{rl}}{ {|V} \rangle _{rr}}$. Furthermore, $ {|{\psi^ - }} \rangle $ can be measured with a probability of $0.5$ though the outputs of ${|H\rangle_{ll}}{|V\rangle_{lr}}$, ${|H\rangle_{lr}}{|V\rangle_{ll}}$, ${|H\rangle_{rl}}{|V\rangle_{rr}}$, ${|H\rangle_{rr}}{|V\rangle_{rl}}$, ${|V\rangle_{ll}}{|H\rangle_{lr}}$, ${|V\rangle_{lr}}{|H\rangle_{ll}}$, ${|V\rangle_{rl}}{|H\rangle_{rr}}$, and ${|V\rangle_{rr}}{|H\rangle_{rl}}$.
As indicated in Eq. (A8), $ {|{\varPhi ^ \pm }} \rangle $ can be measured with a probability of $0.25$ though the outputs of ${|H\rangle_{ll}}{|H\rangle_{lr}}$, ${|H\rangle_{rl}}{|H\rangle_{rr}}$, ${ {|V} \rangle _{ll}}{ {|V} \rangle _{lr}}$ and ${ {|V} \rangle _{rl}}{ {|V} \rangle _{rr}}$. Furthermore, $ {|{\psi^ - }} \rangle $ can be measured with a probability of $0.5$ though the outputs of ${|H\rangle_{ll}}{|V\rangle_{lr}}$, ${|H\rangle_{lr}}{|V\rangle_{ll}}$, ${|H\rangle_{rl}}{|V\rangle_{rr}}$, ${|H\rangle_{rr}}{|V\rangle_{rl}}$, ${|V\rangle_{ll}}{|H\rangle_{lr}}$, ${|V\rangle_{lr}}{|H\rangle_{ll}}$, ${|V\rangle_{rl}}{|H\rangle_{rr}}$, and ${|V\rangle_{rr}}{|H\rangle_{rl}}$.
Appendix B. Formulas for QBER calculation
In this study, the following equations were used for calculating the QBER as the ratio of the number (count) of wrong events and total number (count) of detected events for different mean photon numbers, $\mu = 0.6,\; 0.4,\; 0.1$.
In the X basis, QBERX (denoted as $E_{psi \pm }^{X.\mu \; }$) of MQIA is equivalent to that of MDI QKD. However, in the Z basis, the QBERZ (denoted as $E_j^{Z.\mu \; }$ for $j = psi \pm $, $phi \pm $) of MQIA can be derived because $ {|{\mathrm{\varPhi }^ \pm }} \rangle $ should be measured to confirm the user’s identity according to the pre-shared secret key [38]. Specifically, $E_{psi \pm }^{z.\mu }$ in the Z basis is calculated as the ratio of abnormal events $(C_{psi \pm }^{HH} + C_{psi \pm }^{VV})\; $ to the total count $(C_{psi \pm }^{HH} + C_{psi \pm }^{VV} + C_{psi \pm }^{HV} + C_{psi \pm }^{VH})$ for $ {|{\Psi ^ \pm }} \rangle $. Furthermore, with the modified BSM scheme that we proposed, it becomes possible to measure $ {|{\mathrm{\varPhi }^ \pm }} \rangle $. Therefore, $E_{phi \pm }^{z.\mu }$ in the Z basis can be estimated using the measurement results of $ {|{\mathrm{\varPhi }^ \pm }} \rangle $ in the verification step. It is calculated as the ratio of abnormal events $(C_{phi \pm }^{HH} + C_{phi \pm }^{VV})\; $ to the total count $(C_{phi \pm }^{HH} + C_{phi \pm }^{VV} + C_{phi \pm }^{HV} + C_{phi \pm }^{VH})$ for $ {|{\mathrm{\varPhi }^ \pm }} \rangle $. The details of the abnormal events $(C_{psi \pm }^{HH} + C_{psi \pm }^{VV})$ and $(C_{phi \pm }^{HH} + C_{phi \pm }^{VV})$ are presented in Appendix A.
Funding
Korea Research Environment Open Network (Advanced Research Program Grant from KISTI); Korea Institute of Science and Technology (2E31531, 2E32801); Commercializations Promotion Agency for R and D Outcomes (2022SCPO_B_0210); Institute for Information and Communications Technology Promotion (2020-0-00890); National Research Foundation of Korea (2021M1A2A2043892, 2022M3K4A1097119).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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