## 4 Bounds on Excess Noise
### Excess Noise Correction
PM, phase modulation; AM, Amplitude modulation; PD, photodiode; PBS, polarization beam splitter; ADC, analog-to-digital; AMP, amplifier. where \(C,F^{2}\) denotes the coupling coefficient and the maximum coupling ratio, respectively. \(A=\frac{Cw}{F}\), \(X=\lambda^{5/2}\). Take the \(50:50\) fused biconical taper BS produced by Thorlabs (center wavelength is 1550 nm) as an example, when \(\lambda_{0}^{1}=1550\) nm, \(T_{\lambda_{0}}^{1}=0.5\). The above formula can be rewritten as:  
\[X=\frac{\arcsin\sqrt{T_{1}(\lambda)}}{\arcsin(\sqrt{0.5})}X_{0}^{1}. \tag{17}\]  
When center wavelength \(\lambda_{0}^{2}\) is 1310 nm, the relationship between transmittance and wavelength of the BS can be expressed as:  
\[X=\frac{\arcsin\sqrt{T_{2}(\lambda)}}{\arcsin(\sqrt{0.5})}X_{0}^{2}. \tag{18}\]  
According to Equations (17) and (18), the relationship between their coupling ratios and wavelength with center wavelength of 1550 nm and 1310 nm are shown in Tables 2 and 3, respectively. Only a set of false signal state and false LO cannot escape the real-time detection at [PERSON]'s side. Another set of signal pulse and LO must be added to maintain the output of shot noise at a normal level. The two sets of wavelengths and their transmittance are expressed as: \(\lambda_{1}^{s}\), \(\lambda_{1}^{lo}\), \(\lambda_{2}^{s}\), \(\lambda_{2}^{lo}\), \(T_{1}^{s}\), \(T_{1}^{lo}\), \(T_{2}^{s}\), \(T_{2}^{lo}\). In order to make \(T_{1}^{s}\approx T_{1}^{lo}\), \(T_{2}^{s}\approx T_{2}^{lo}\), the wavelength and its corresponding transmittance are:  
\[\lambda_{1}^{s}=1309\text{ nm},T_{1}^{s}=0.4982, \tag{19a}\] \[\lambda_{1}^{lo}=1548.6\text{ nm},T_{1}^{lo}=0.4982;\] (19b) \[\lambda_{2}^{s}=1311.2\text{ nm},T_{2}^{s}=0.5017,\] (19c) \[\lambda_{2}^{s}=1551.4\text{ nm},T_{2}^{lo}=0.5017; \tag{19d}\]  
The corresponding differential currents are expressed as:  
Figure 7:0 Figure 7:1 (20b) Figure 7:2 (20c) Figure 7:3  
where \(I_{i}^{j}\) denotes the differential currents of corresponding wavelength, \(\eta_{i}^{j}\) is the detector efficiency of different wavelengths. In order to make the average value of \(D^{lo}\) equal to 0, the simplest method is to select the wavelength corresponding to the transmittance of \(T_{2}^{lo}\approx 1-T_{1}^{lo}\). Then choose the right light intensity to make \(D_{2}^{lo}=D_{1}^{lo}\). In order to keep the variance of [PERSON]'s measured grainy noise as \(N_{0}\), let \(D_{lo}^{2}=1/3N_{0}\). When [PERSON] measures the signal state, it is assumed that the differential currents generated by \(I^{s}\) and \(I^{lo}\) can cancel each other, i.e., [PERSON] selects two wavelengths with the same probability and satisfies \(D_{1}^{s}=-D_{1}^{lo}\), \(D_{2}^{s}=-D_{2}^{lo}\). Figure 7:4  
Figure 7:5  
Through comprehensive analysis, the following relationship can be obtained:  
Figure 7:6  
According to Equation (5), the excess noise \(\epsilon\) can be estimated as  
Figure 7:7Suppose the real shot noise is \(\gamma N_{0}\). Therefore, the estimation of shot noise level and excessive noise can be expressed as (see Appendix A for more details):  
Figure 7:8  
Figure 7:9  
To correct the noise caused by Trojan horse attacks and the impefection of BS at the same time, \(\epsilon^{*}\) must satisfy:  
\[T=F^{2}\text{sin}^{2}(\frac{C\lambda^{5/2}w}{F})=\text{sin}^{2}(AX)\equiv T( \lambda), \tag{16}\]0  
By selecting the appropriate \(I^{s}\), \(I^{lo}\) and \(\gamma\), the shot noise and excess noise under the joint attack can be corrected simultaneously, so that \(\widetilde{N_{0}}=N_{0}\) and \(\widetilde{\epsilon}\) is arbitrarily close to 0. Take \(T_{ch}=0.032\) (i.e., fiber length \(L=50\) km), \(\epsilon_{0}=0.01\) as an example in this article. The original LO intensity, the original shot noise unit and the detector efficiency of different wavelength are set to \(I_{lo}=1\times 10^{8}\), \(N_{0}=0.5I_{lo}\), \(\eta_{L}^{j}=0.5\). Calculated by the above formula, \(I_{1}^{s}=I_{1}^{lo}=2.268\times 10^{6}\), \(I_{2}^{s}=I_{2}^{lo}=2.40\times 10^{6}\).