# Performance Analysis of Continuous-Variable Quantum Key Distribution with Multi-Core Fiber
## 2 The MCF-Based CVQKD
Therefore, assuming the wavelength of quantum channels in core \(m\) and core \(n\) are equal, the excess noise contributed by the inter-core crosstalk from core \(m\) to core \(n\) referred to the input can be given by  
\[\xi_{XT}=\xi^{ref}_{XT}+\xi^{sig}_{XT} \tag{3}\]  
with  
\[\xi^{ref}_{XT} = \frac{1}{2}\frac{\eta_{F}\eta\cdot 2\langle\hat{N}^{out}_{ref} \rangle\tanh(\overline{h}_{nm}L)}{T\eta_{F}\eta}\sum_{k=-1}^{1}e^{-\frac{( \Delta^{r}_{t}-\frac{k}{2})^{2}}{2\overline{\epsilon}_{T}^{2}}}=\frac{\langle \hat{N}^{out}_{ref}\rangle\tanh(\overline{h}_{nm}L)}{T}\sum_{k=-1}^{1}e^{- \frac{(\Delta^{r}_{t}-\frac{k}{2})^{2}}{2\overline{\epsilon}_{T}^{2}}}, \tag{4}\] \[\xi^{sig}_{XT} = \frac{1}{2}\frac{\eta_{F}\eta\cdot 2\langle\hat{N}^{out}_{sig} \rangle\tanh(\overline{h}_{nm}L)}{T\eta_{F}\eta}\sum_{k=-1}^{1}e^{-\frac{( \Delta^{s}_{t}-\frac{k}{2})^{2}}{2\overline{\epsilon}_{T}^{2}}}=\frac{\langle \hat{N}^{out}_{sig}\rangle\tanh(\overline{h}_{nm}L)}{T}\sum_{k=-1}^{1}e^{- \frac{(\Delta^{s}_{t}-\frac{k}{2})^{2}}{2\overline{\epsilon}_{T}^{2}}}, \tag{5}\]  
Figure 1: Multi-core fiber and the illustration of inter-core crosstalk. (**a**) A common 7-core MCF and the transverse distribution of the optical field in a single core. (**b**) The inter-core crosstalk between two cores. (**c**) The time delay of pulses between two cores. \(\Delta^{r}_{t}\) is the time delay between the phase reference pulse in core \(m\) and the most nearby quantum signal pulse in core \(n\). \(\Delta^{s}_{i}\) is the time delay between the quantum signal pulse in core \(m\) and the most nearby quantum signal pulse in core \(n\). \(t_{d}\) is the time delay between quantum signal pulse and phase reference pulse in the same core and \(t_{d}=\Delta^{s}_{t}-\Delta^{r}_{t}\). We assume \(\Delta^{r}_{t}<0\) for the case in (**c**). where \(\Delta_{t}^{r}\) is the time delay between the phase reference pulse in core \(m\) and the most nearby quantum signal pulse in core \(n\), \(\Delta_{t}^{s}\) is the time delay between the quantum signal pulse in core \(m\) and the most nearby quantum signal pulse in core \(n\), \(\sigma_{t}\) is the half-width of the pulse (at \(1/\mathrm{e}\)-intensity point), \(e^{-(\Delta_{t}^{r}-k/R)^{2}/2\sigma_{t}^{2}}\) and \(e^{-(\Delta_{t}^{s}-k/R)^{2}/2\sigma_{t}^{2}}\) are the overlapping factor [34], \(\langle\hat{N}_{ref}^{out}\rangle\) is the mean photon number of the phase reference pulse in the core \(m\), \(\langle\hat{N}_{sig}^{out}\rangle\) is the mean photon number of the quantum signal pulse in the core \(m\), \(\xi_{XT}^{ref}\) is the excess noise contributed by the phase reference pulses, \(\xi_{XT}^{sig}\) is the excess noise contributed by the quantum signal pulses, \(R\) is the repetition rate of CVQKD system, \(\eta_{F}\) is the insert loss of the FIFO, \(\eta\) is the detector quantum efficiency. We only consider the adjacent 3 quantum signal pulses and phase reference pulses (corresponding to \(k=-1,0,1\)), and others' impact can be neglected as the overlapping factor will be too low. The phase reference pulse would be the LO pulse or the weak reference pulse [8; 9; 10]. We assume that all the pulses are the Gaussian pulse shape with the same pulse width. Note, as the coupling between cores occur randomly, the polarization state of the crosstalk will be randomized and chaotic. Consequently, there is a factor of \(1/2\) in Equations (4) and (5) due to the polarization selection of the LO. According to Equation (4), we show the excess noise contributed from the weak reference pulses in Figure 2, where the pulse width is \(50\) ns, \(\alpha\) is \(0.2\) dB/km, \(R\) is \(1\) MHz and the coupling coefficient takes an ultra-low value of \(-90\) dB/km [31]. Usually, the mean photon number of the weak reference pulse is about \(100\)\(\sim\)\(1000\) at the fiber output. Therefore, the resulting excess noise contributed from the weak reference pulse cannot be neglected when \(\Delta_{t}^{r}\) approaches to zero.