## 2 Theoretical Analysis  
When vacuum noise and excess noise produced by the effect of the depolarized GAWBS is considered, the output (\(\hat{a}_{out}\)) and input (\(\hat{a}_{in}\)) relation of optical mode transmitted in the fiber is given by [25; 30]:  
\[\hat{a}_{out}=\sqrt{\eta}\hat{a}_{in}+\sqrt{1-\eta}\hat{a}_{v}+\sqrt{\eta_{2}} \hat{a}_{g}, \tag{1}\]  
where \(\hat{a}_{g}\) is the mode of GAWBS extra noise, \(\hat{a}_{v}\) is the mode of vacuum noise induced by the transmission loss. \(\eta=\eta_{1}\eta_{2}\) is the total transmission efficiency in the fiber, which includes the coupling efficiency of fiber coupler (\(\eta_{1}\)) and the fiber transmission efficiency (\(\eta_{2}=10^{-al}\)), \(a\) denotes the transmission loss in fiber per kilometer (0.36 dB/km @ 1342 nm) and \(l\) denotes the transmission distance. The noise variances of quadrature amplitude (\(\hat{x}_{aut}=\hat{a}_{out}+\hat{a}_{out}^{+}\)) and quadrature phase [\(\hat{p}_{aut}=(\hat{a}_{out}-\hat{a}_{out}^{+})/i\)] of the mode \(\hat{a}_{out}\) over the fiber channel can be given by[25]:  
\[\Delta^{2}\hat{x}_{aut} = \eta\Delta^{2}\hat{x}_{a_{in}}+1-\eta+\eta elI_{LO},\] \[\Delta^{2}\hat{p}_{aut} = \eta\Delta^{2}\hat{p}_{a_{in}}+1-\eta+\eta elI_{LO}, \tag{2}\]  
where \(elI_{LO}\) is the excess noise induced to the signal beam by the GAWBS processing on \(\hat{a}_{g}\), \(\varepsilon\) is the noise coefficient of GABWS in the fiber, \(I_{LO}\) is the power of the corresponding LO beam before the fiber coupler. When the signal beam is a coherent state, i.e., \(\Delta^{2}\hat{x}_{a_{in}^{C}}=\Delta^{2}\hat{p}_{a_{in}^{C}}=1\), the variances of both two quadratures of the output state are the same:  
\[\Delta^{2}\hat{x}_{a_{out}^{C}}=\Delta^{2}\hat{p}_{a_{out}^{C}}=1+\eta elI_{LO}. \tag{3}\]  
When the signal beam is changed to a quadrature amplitude squeezed state, i.e., the noise variance of quadrature amplitude is below the corresponding QNL. Here, we suppose that \(\Delta^{2}\hat{x}_{a_{in}^{S}}=e^{-2r}\) and \(\Delta^{2}\hat{p}_{a_{in}^{S}}=e^{2r}\), where \(r\) is the squeezing factor. The variances of the output state from the fiber become:  
\[\Delta^{2}\hat{x}_{a_{out}^{S}} = \eta e^{-2r}+1-\eta+\eta elI_{LO},\] \[\Delta^{2}\hat{p}_{a_{out}^{S}} = \eta e^{2r}+1-\eta+\eta elI_{LO}. \tag{4}\]  
It is obvious that the squeezing property can be greatly influenced in this processing. In some situation, the squeezing even can vanish due to the existance of GAWBS extra noise. In the usual transmission processing through single-mode fiber, the signal beam and the corresponding LO beam are sent to the fiber with orthogonal polarization [24; 25; 28; 38]. The LO beam stimulates additional noise on the polarization of the signal beam due to GAWBS when the LO beam and the signal beam are simultaneously transmitted. But if the LO beam is chopped to the periodic pulsed beam, the spurious GAWBS extra noise loaded on the signal beam only appears when the LO beam exists in the fiber, i.e., the noise of the signal beam in the time interval without the existing of LO beam maintain a normal level without the influence of GAWBS. In this condition, if the LO beam pulse sequence is properly time-delayed, it could overlap with the signal beam without the GAWBS extra noise. When these two light beams are injected into the BHD system, the noise of the pure signal beam without the influence of GAWBS can be measured. At the same time, the part of the signal beam with GAWBS extra noise is not measured because there is no LO beam when it is incident on the photodiodes of BHD system. There are two schemes to choose for measuring the noise of quantum state with pulsed LO beam as shown in Figures 1a,b. Figure 1a shows the amplitude noise at 3.0 MHz analysis frequency of a coherent state when the LO pulse repetition rate is 200 kHz, where the red trace denotes the sequence of LO beam and the blue trace is the measured quadrature amplitude noise, which means that the pulse time is longer than the handling time for analysing at a certain frequency.