# Sending-or-Not-Sending Twin-Field Quantum Key Distribution with Light Source Monitoring
## 2 SNS Protocol with LSM  
### Security Analysis under UPC  
In the SNS protocol [22; 27], [PERSON] ([PERSON]) prepares a coherent state with an intensity \(\mu_{A}\) (\(\mu_{B}\)), a random modulated phase \(\delta_{A}\) (\(\delta_{B}\)) and a global phase \(\gamma_{A}\) (\(\gamma_{B}\)). Only the states where [PERSON] and [PERSON] choose the same intensity (\(\mu_{A}=\mu_{B}=\mu\)) will be retained for discussion, thus the joint state sent by [PERSON] and [PERSON] is \(|\sqrt{\mu}e^{i(\delta_{A}+\gamma_{A})}\rangle|\sqrt{\mu}e^{i(\delta_{B}+\gamma _{B})}\rangle\). In [PERSON]'s view, it has the convex form  
\[\rho_{AB}=\sum_{k}p_{k}(\mu)|\psi_{k}\rangle\langle\psi_{k}|, \tag{1}\]  
where \(|\psi_{k}\rangle\) refers to a joint state with total photon numbers \(k\), and \(p_{k}(\mu)\) refers to the probability that the joint state contains \(k\) photons totally [22]. Specifically, the single-photon part of the state has the form of \[|\psi_{1}\rangle=\frac{1}{\sqrt{2}}[e^{i(\delta_{B}+\gamma_{B})}|0\rangle_{A}|1 \rangle_{B}+e^{i(\delta_{A}+\gamma_{A})}|1\rangle_{A}|0\rangle_{B}]. \tag{2}\]  
\(|\psi_{1}\rangle\) plays an important role during the whole security analysis of the SNS protocol [22]. In fact, the equivalent virtual protocols discussed in the security analysis focus on the single-photon part and the security analysis could remain valid when the phase difference between the two parts of the single state \(|1\rangle_{A}|0\rangle_{B}\) and \(|0\rangle_{A}|1\rangle_{B}\) remains to be  
\[\Delta\phi=\delta_{A}+\gamma_{A}-\delta_{B}-\gamma_{B}. \tag{3}\]  
Under UPC, the quantum state prepared by [PERSON] ([PERSON]) is no longer an ideal coherent state \(|\sqrt{\mu}e^{i(\delta_{A}+\gamma_{A})}\rangle\) (\(|\sqrt{\mu}e^{i(\delta_{B}+\gamma_{B})}\rangle\)), but a state with arbitrary PND, which can be written as \(|\psi_{A}\rangle=\sum_{k}e^{ik(\delta_{A}+\gamma_{A})}\sqrt{P_{k,A}(\mu)}|k \rangle_{A}\) (\(|\psi_{B}\rangle=\sum_{k}e^{ik(\delta_{B}+\gamma_{B})}\sqrt{P_{k,B}(\mu)}|k \rangle_{B}\)) after the modulation process, where \(P_{k,A}(\mu)\) (\(P_{k,B}(\mu)\)) is completely unknown. As a result, with the analysis shown in Appendix A, the joint state sent by [PERSON] and [PERSON] can still have the convex form  
\[\rho_{AB}=\sum_{n}p_{n}(\mu)|\psi^{\prime}_{n}\rangle\langle\psi^{\prime}_{n}| \tag{4}\]  
in [PERSON]'s view, with the \(n\)-photon state  
\[|\psi^{\prime}_{n}\rangle=\frac{1}{\sqrt{p_{n}(\mu)}}\sum_{k=0}^{n}\sqrt{P_{k, A}(\mu)P_{n-k,B}(\mu)}e^{ik\Delta\phi}|k\rangle_{A}|n-k\rangle_{B}, \tag{5}\]  
and the probability of \(n\)-photon  
\[p_{n}(\mu)=\sum_{k=0}^{n}P_{k,A}(\mu)P_{n-k,B}(\mu). \tag{6}\]  
Specifically, the single photon part of \(\rho_{AB}\) under UPC can be written as  
\[|\psi^{\prime}_{1}\rangle=\frac{1}{\sqrt{2}}[|0\rangle_{A}|1\rangle_{B}+e^{i \Delta\phi}|1\rangle_{A}|0\rangle_{B}] \tag{7}\]  
with a symmetric condition  
\[P_{k,A}(\mu)=P_{k,B}(\mu)=P_{k}(\mu), \tag{8}\]  
which is also assumed in the original SNS protocol [22]. This result indicates that the original security analysis can be directly applied to UPC, since the phase difference between \(|1\rangle_{A}|0\rangle_{B}\) and \(|0\rangle_{A}|1\rangle_{B}\) is still \(\Delta\phi\), that is, \(|\psi^{\prime}_{1}\rangle\) only has a global phase difference with \(|\psi_{1}\rangle\) in Equation (2). Secret Key Rate  
Unlike the original TF-QKD, the SNS protocol divides all the transmitted light pulses into two categories, which are called signal windows and decoy windows [20; 22]. In the signal window (equivalent to Z-basis), [PERSON] ([PERSON]) randomly chooses to send or not send a signal pulse, and the data are used to generate keys. In the decoy window (equivalent to X-basis), [PERSON] ([PERSON]) sends decoy states with different intensities and the data are used to estimate the count rate and phase error rate of the signals' single photon part. In addition, there is a post-selection process for the random phases \(\delta_{A},\delta_{B}\) to ensure more accurate estimation results, which originates from TF-QKD.