Proof-of-principle demonstration of semi-quantum key distribution based on the Mirror protocol

Semi-quantum key distribution (SQKD) is used to establish a string of shared secret keys between a quantum party and a classical party. Here, we report the first proof-of-principle experimental demonstration of SQKD based on the Mirror protocol, which is the most experimentally feasible SQKD protocol, and equipped with time-phase encoding scheme employing the method of selective modulation. The experiment was performed at a repetition frequency of 62.5 MHz and a high raw key rate arrived at 69.8 kbps, and the average quantum bit error rate was found to be 4.56% and 2.78% for the “SWAP-x-Z” (x∈{01,10}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathrm{x}\in \{01,10\}$\end{document}) and the “CTRL-X”, respectively. The results demonstrate the feasibility of our system, and this study is helpful for future research on SQKD experiments.

Among the various branches mentioned above, QKD is a very important technology that is currently the most developed and fastest to be incorporated into practical applications. For secure key distribution, is it possible that only one party is quantum, yet and the other has only classical capabilities? Of course the answer is"yes" [17][18][19], and this new kind of key distribution is called semi-quantum key distribution (SQKD).
In 2007, the concept of SQKD was introduced originally by Boyer et al. and the BKM07 protocol -"QKD with classical Bob" was presented [17]. Subsequently, "QKD with classical Alice" protocol [18,19] as well as other SQKD protocols [20][21][22][23][24] have been proposed in succession, and new kinds of protocols were derived from crossover with other fields, such as semi-quantum secret sharing (SQSS) [25], semi-quantum secure direct communication (SQSDC) [26], semi-quantum private comparison (SQPC) [27] and so on. Some of these protocols (e.g., Single-State SQKD protocol) have been proven to be secure in the "perfect qubit scenario" [28][29][30][31]. For the SQKD protocols, there are two operations can be chosen by the classical party Alice: "CTRL" and "SIFT". "CTRL" means to return the photon to the quantum party Bob undisturbed, while "SIFT" is commonly achieved as follows: Alice is required to measure the photon from Bob in the Z basis {|0 , |1 } and re-transmit the photon to Bob according to the measurement results. However, given the current experimental technology, the re-generated photon cannot be the same as the original photon, thus the photons will be attacked by using the "tagged" method in the mock protocol presented in [17], and information will be leaked to Eve [32,33]. Therefore, based on the Single-state SQKD protocol, Boyer et al. proposed the experimentally feasible Mirror protocol [34] to avoid the security vulnerability introduced by regenerating new photons. Nevertheless, the Mirror protocol is more complicated than other SQKD protocols, and the classical operations (CTRL, SWAP-10, SWAP-01 and SWAP-ALL) pose technical challenges to its experimental implementation. Before that, Gurevich used the time encoding scheme to partially implement the experiment based on the SQKD protocol [35]. However, as described by the author, the error rates of this system are relatively high: the quantum bit error rate of "CTRL-X" is 16.7%, and that of "SWAP-x-Z" (x ∈ {01, 10}) is 23.59%. Meanwhile, because of the incomplete implementation of the classical operations (i.e., the lack of SWAP-ALL), attacks can take place in the current implementation [36]. Strictly speaking, it is not a successful feasibility verification of the SQKD protocol.
Here, we have implemented the first proof-of-principle experimental demonstration of SQKD based on the Mirror protocol, which uses the time-phase encoding scheme with the method of selective modulation to avoid the security problem of regenerating new photons. Moreover, our experiment obtains a higher raw key rate and lower bit error rate. The method of selective modulation we presented may promote the research progress of theory and experiments in the semi-quantum field and open new ideas for future researchers.
The remainder of this paper is organized as follows. In Sect. 2, the Mirror protocol is briefly introduced. In Sect. 3, we propose the time-phase coding scheme based on the Mirror protocol, and then the experimental setup is described in detail in Sect. 4. The experimental results are showed and discussed in Sect. 5. We conclude in Sect. 6.

The Mirror protocol
In the Mirror protocol, the operations of the classic party Alice change from the original operations "CTRL" and "SIFT" [18,19]  The Mirror protocol is as follows: the sender Bob sends the quantum state |+ to Alice. Then, Alice randomly selects one of the operations "CTRL, " "SWAP-ALL, " "SWAP-01, " "SWAP-10", and records whether it receives a "click". If Alice chooses "SWAP-01, " no photon is detected with a one-half probability, and then the quantum state is projected into the state |1 . If Alice chooses "SWAP-10, " no photon is detected with a one-half probability, and then the quantum state is projected into the state |0 .
Bob then randomly chooses Z basis {|0 , |1 } or X basis {|+ , |-} to measure the qubit sent back from classical Alice. After N qubits are measured and sent, Alice declares her operation choices (CTRL, SWAP-x, or SWAP-ALL; she keeps x ∈ {01, 10} in secret) and Bob announces his basis choices. For the combination chosen by both parties, if Bob chooses the Z basis, Alice chooses to "SWAP-x" (which is called "SWAP-x-Z"), and the photon is not detected, they will use the bits of this position as the raw keys. That is, Bob's measurement result is |i (i ∈ {0, 1}), and the opposite result of Alice's measurement result is also |i . Hence, their shared secret key is coded as i. If Alice selects "CTRL" and Bob chooses measurement in X basis, this kind of operation will be called "CTRL-X". In this case, the position of the bits can be used to detect Eve's eavesdropping. Alice and Bob then check the bit error rate in "CTRL-X" and "SWAP-x-Z, " and if the bit error rate is too high, the protocol will be aborted. Alice and Bob also check whether other errors exist (for example, they verify if Bob detects no photons in case Alice uses "SWAP-ALL"). Alice and Bob discard the other mismatched bits of "CTRL-Z" and "SWAP-x-X" because of the uncertainty in the measurement results. Finally, Alice and Bob perform error correction and privacy amplification on the code location "SWAP-x-Z, " and both parties obtain the secret security keys.

Time phase coding scheme based on the Mirror protocol
A proof-of-principle experimental demonstration of SQKD based on the Mirror protocol is implemented by using time-phase encoding. The time-phase encoding rule is shown in Fig. 1 [37]. In this rule, the qubits in the Z basis and the X basis are composed of the two temporal modes, denoted as early (t E ) and late (t L ), and the phase difference of the  two temporal modes. When the photons are located at the two different time windows, t E and t L , the Z basis is determined by the time window in which the photon resides, and the X basis is determined by the phase difference between the optical pulses in the two time windows. In the Z basis, if the photon is only located at the time window t E , this situation represents |ϕ 0 , which is encoded as "0". If it is only located in the time window t L , this situation stands for |ϕ 1 , which is encoded as "1". Considering the case where both photons are located on two time windows (t E and t L ), we define it as the encoding under X basis. In the X basis, the state |ϕ + or |ϕ -is encoded as "0" or "1", depending on phase difference ϕ = 0 or π .
The scheme based on the Mirror protocol with time-phase encoding is as follows (see Fig. 2). The quantum Bob prepares the qubit state |ϕ + in the X basis by an unbalanced interferometer and then sends it to Alice through the quantum channel. For the qubit state |ϕ + from Bob, Alice randomly selects one of the classical operations: (1) CTRL: the intensity modulator (IM) is not operated. It also means to return |ϕ + directly; (2) SWAPx: IM randomly modulates the intensity of one of the pulses to zero, to return the state |ϕ 0 or |ϕ 1 in the Z basis to Bob; (3) SWAP-ALL: IM modulates the intensity of both pulses to zero. Thus, Alice realizes the four classical operations by selective modulation.
When the qubit comes back to Bob again, Bob randomly measures it either in the Z or X basis. The Z basis measurement is a direct measurement of the arrival time of the photons, and the X basis is the measurement of the phase difference carried by the temporal modes. For the measurement results, the single-photon detectors (SPDs) may receive photons in the three time windows t 0 , t 1 , and t 2 . The time pattern of the single-photon detector response is shown in Fig. 3. If the CTRL-X used for eavesdropping detection is selected by both communication parties, there is a corresponding constructive or destructive interference at the t 1 of SPD1 and SPD2. If the SWAP-x-Z used for distilling the key bits is selected, there are two cases: If SPD1 or SPD2 detects photons in the first time window t 0 , the result is the "0" code in the Z basis; If SPD1 or SPD2 detects photons in the third time window t 2 , the result is the "1" code in the Z basis. Finally, Bob announces the chosen basis and Alice declares "CTRL, " "SWAP-x" or "SWAP-ALL" (she keeps x ∈ {01, 10} secretly). Then, after error correction and privacy amplification, Alice and Bob both obtain the shared secure key.

Experimental setup
The experimental setup, which is based on the protocol and scheme mentioned in the previous section, is depicted in Fig. 4. At Bob's site, it is composed of three modules: laser source module, Mach-Zehnder Interferometer (MZI) module and detection module. And at Alice's site, there is the Sagnac loop module.
In the optical source module, the optical pulse is generated by a picosecond pulse laser (LD, manufactured by QuantumCTek, QCL-102). Its wavelength centers at 1550.15 nm, and the pulse width is 60 ps, with the system frequency of 62.5 MHz. Behind the LD, an isolator (ISO) is used to reduce the effect of reflected light. We then use a programmable attenuator (ATT) to reduce the average photon number of the optical pulse to the singlephoton level. After leaving the ATT, the optical pulse enters the circulator (CIR) through the Port 1, and exits from the Port 2 into the MZI module.
The setup between two 50:50 beam splitter (BS1 and BS2) is the MZI module, which contains a short arm and a long arm. The short arm consists of a polarization-independent phase modulator (PM, manufactured by KANGGUAN, KG-PM-15-10G-PP-FP), which is driven by an electrical pulse generator (PG, manufactured by SIGLENT, SDG6052X-E) to load the phase for the second pulse in reverse transmission, and a polarization controller (PC1) that adjusts the polarization state of the backward optical pulse for PM to work in the single mode. In the long arm, there are a PC2 and a variable fiber optic attenuator (VOA). The role of the VOA is used to modify the optical attenuation of the long arm, so that the intensities of the optical pulses output from the two arms are equal. The detection module is composed of two single photon detectors (SPD1 connected to the Port 4 of BS1, SPD2 connected to the Port 3 of CIR).
At the classical site, the symmetric Sagnac loop, which is used to modulate optical pulses for classical operations, is constructed by a polarization-maintaining polarization beam splitter (PMPBS) and a polarization-maintaining IM (PMIM) [38]. PGs and SPDs are synchronized by an electrical delay generator (DG) at Bob's end.
In the SQKD experiment, the critical technical challenge is to completely implement the "CTRL-X", "SWAP-x" and "SWAP-ALL" operations. To solve this problem, we utilize the above-mentioned apparatus in combination with the method of selective modulation to continue characterizing the path of the optical pulse. Upon entering the MZI module, the optical pulse is split in two and the state |ϕ + with a time interval of t = 5.86 ns is prepared due to the presence of the arm length difference of L = 119.8 cm. For conve- nience of description, the optical pulses of the short and long arms are denoted as P S and P L , respectively. Subsequently, the optical pulses are sent to Alice via the quantum channel. After the classical operations are performed in the Sagnac loop, the optical pulses are returned to Bob via the quantum channel. Once again, they pass through the MZI module and eventually the response will occur at the SPDs.
Next, we describe in detail the implementation of the classical operations. If Alice chooses "CTRL" operation, IM does nothing. The optical pulses (P S and P L ) emitted from Alice are sent to Bob and are split into four pulses by BS2. Along the lines of the above representation, we record the four pulses as P S+S , P L+L , P S+L and P L+S . For example, a pulse that passes through the short arm of MZI in the forward propagation and the long arm of MZI in the backward propagation, is denoted as P S+L . In the reverse optical path, PC1 adjusts the polarization of the optical pulses so that the PM works in the single mode. And the PM, which is driven by the PG1 with the repetition frequency of 62.5 MHz and the FWHM of 3.3 ns, modulate voltage 0 V or half-wave voltage V π = 4.82V which loads the corresponding phase 0 or π on the P L+S randomly. Meanwhile, by adjusting the polarization state of P S+L with PC2 to the same as that of P L+S , interference will occur on the second time window t 1 at BS1. The forward and backward optical path of the "CTRL-X" operation is illustrated in Fig. 4(a).
If Alice chooses "SWAP-10" (or "SWAP-01") operation, the PG2, with the frequency of 62.5 MHz, the voltage of 6.3 V, the FWHM of 3.3 ns, and the corresponding delay of 13.8 ns (or 7.6 ns), drives IM to modulate optical pulses P L (or P S ) reducing light intensity to be close to zero, where the extinction ratio of IM is 18.4 dB. In the backward propagation, the pulse P S (or P L ) is then divided into two pulses by BS2. After the optical pulses are combined by BS1, SPD1 and SPD2 will have the same response probabilities. When SPD1 or SPD2 responds in the first time window t 0 , the code "0" in the Z basis is obtained. When SPD1 or SPD2 responds in the third time window t 2 , the code "1" in the Z basis is obtained. The forward and backward optical path of the "SWAP-01-Z" operation is illustrated in Fig. 4(b).
If Alice chooses "SWAP-ALL" operation, the PG2, with the pulse width of 7 ns, drives IM to modulate the both optical pulses P S and P L for reducing the intensity of light to zero and there are no photons returned to Bob.

Results and discussion
For the sake of the verification of experimental feasibility, we can evaluate the quantum bit error rates, which are obtained from the photon counts (within 10 mins) by the measurement of SPDs. In this proof-of-principle experiment, the employed SPDs are commercial InGaAs/InP detectors working in gated mode, with 300-ps-gate windows, 1.25-GHz repetition frequency, average dark count probability per gate of 1× 10 -6 , 50 ns dead time, and a detection efficiency of approximately 16.91%. For the "CTRL-X" operation, the PM modulates the phase difference between the two optical pulses to 0 or π so that the SPDs detect the interference signal in the second time window t 1 eventually, and the average interference contrast is 94.45% as shown in Fig. 5(a). Moreover, this position can be used to monitor noise and error rates. For "SWAP-10" and "SWAP-01"operations, IM modulates one of the two light pulses corresponding to the codes "0" and "1", respectively. For"SWAP-ALL" operation, IM modulates both the pulses, so that no light pulses return. When the average number of photons per pulse is attenuated to μ = 0.1, depending on the photon counts in different cases, we obtain the quantum bit error rate of "CTRL-X" which is 2.78%, and that of "SWAP-x-Z"which is 4.56%(see Fig. 5(b)). While the raw key rate is 69.8 kbps. It can be seen that, compared with the experiment of the SQKD protocol implemented by Gurevich [35], our experiment has better performance. The core of our experiment is adopting the method of selective modulation, which can take full advantage of the high-speed characteristics of the LiNbO3 intensity modulator to achieve high-speed switching among "SWAP-ALL, " "SWAP-10, " and "SWAP-01" operations. In future work, we intend to improve the secure key rate by increasing the repetition rate of the system and the detection efficiency of the SPDs. The quantum bit error rate can be reduced by increasing the extinction ratio of the IM and using lower-loss devices. Once the SQKD field breaks through in practical applications, experimental secure key rate evaluation based on weakly coherent states will be an important research topic, which is what our group will concentrate on for the next.

Conclusion
In conclusion, we performed the first proof-of-principle demonstration of semi-quantum key distribution based on the Mirror protocol, proving that the implementation of the Mirror protocol employing the method of selective modulation is feasible. Compared with the previous experimental version of the Mirror protocol, our experiment complements the "SWAP-ALL" classical operation, and avoids the "Full attack" and the "Weak attack". In addition, our scheme dramatically reduces the quantum bit error rate. In this proof-ofprinciple experiment, we adopt the method of the selective modulation to avoid preparing identical photons. This research indicates that this method will be available for other experimental demonstration of SQKD protocols to drive the development of SQKD experiments.
In addition, it can be noted that the physical structure of QSDC [8] based on single photons has similarities with that of SQKD, such as the selection of control mode and encode mode, so our experimental approach may be applied to the implementation of QSDC based on single photons to improve its performance in the future as well.