# Performance Analysis of Continuous-Variable Quantum Key Distribution with Multi-Core Fiber
## 2 The MCF-Based CVQKD  
The MCF constrains several cores in the same cladding, where each core can be seen as a single SMF transmitting signals independently. Compared with the SMF, all the cores in the MCF can be exploited as parallel SMFs for independent signals so that the communication capacity can grow exponentially. Figure 1a shows a common 7-core MCF. However, this multi-core structure will bring out some challenges. For example, extra optical devices, fan-in and fan-out, must be inserted at the input and output of the MCF. The insert loss of these components should not be neglected. In addition, the electric field of mode in each core is not completely confined to the core, but part of the cladding, as shown in Figure 1a. Consequently, as all the cores are confined in the same cladding, the electric field at each wavelength in the individual core of MCF will transfer into adjacent cores through evanescent field coupling, as shown in Figure 1b. As a result, the inter-core crosstalk, which has the same wavelength with the launched signal, is generated. Usually, the CVQKD system is sensible with the magnitude of excess noise. The inter-core crosstalk between cores may increase the excess noise and then degrade the performance of the CVQKD system. Therefore, the impact contributed from the inter-core crosstalk should be taken into account. It will be accumulated linearly along the fiber length, which can be attributed to randomly perturbation caused by fiber imperfection [31]. Its magnitude depends on fiber length and coupling coefficient, which is determined by fiber bend, fiber twist, the electric field differential propagation constant \(\delta\beta\) of two cores, core pitch, mode coupling coefficient and correlation length [31; 32; 33]. As described in [32], the inter-core crosstalk \(XT_{mn}\) from core \(m\) to core \(n\) can be evaluated by  
\[XT_{mn}=\tanh(\overline{h}_{mn}L), \tag{1}\]  
where \(\overline{h}_{mn}\) is the coupling coefficient between core \(m\) and core \(n\), and \(L\) is the transmission distance. \(XT_{mn}\) should be understood as the output power ratio of core \(n\) to core \(m\). The leakage power contributed from the inter-core crosstalk from core \(m\) to core \(n\) can be estimated in power by \(P_{XT}^{mn}(L)=XT_{mn}TP_{m}(0)\) with \(T=10^{-aL}\), where \(P_{m}(0)\) is the input power of core \(m\), \(T\) is the transmittance of the fiber, and \(\alpha\) is the fiber loss coefficient. In mean photon number, the formula above can be rewritten as  
\[\langle\hat{N}^{mn}_{XT}(L)\rangle=XT_{mn}T\langle\hat{N}^{m}(0)\rangle=\tanh( \overline{h}_{nm}L)\langle\hat{N}^{m}(L)\rangle, \tag{2}\]  
where \(\langle\hat{N}^{m}(0)\rangle\) is the input mean photon number of the core \(m\), and \(\langle\hat{N}^{m}(L)\rangle\) is the output mean photon number of the core \(m\). If the above inter-core crosstalk has the same wavelength with the quantum signals, it will contribute to the in-band excess noise as it is in the same spatiotemporal mode as the LO of the core \(n\), where the LO is generated locally or transmitted with quantum signal. The CVQKD systems at present are all pulse system. In order to estimate the above excess noise, we consider the impact of the adjacent cores' quantum signal pulses and phase reference pulses, which are overlapped temporally with the quantum signal pulse in the interested core. Figure 1c shows the time delay between one core's quantum signal pulse and the adjacent core's quantum signal pulse and phase reference pulse.