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In this paper, we derive a closed form expression for the output state of aCV-QKD protocol in the presence of zero-photon catalysis (ZPC) and quantumscissor (QS). Then, based on this closed form solution, we use direct searchalgorithm to find the appropriate values of input state and QS parameters,which considerably enhance the range and the fidelity of a CV-QKD protocol. Inthe special case of pure loss channel, the largest range of the protocol isonly 6.5% less than the fundamental limit of repeaterless quantumcommunication. In addition, examination of the protocol for different values ofexcess noise, reveals that their is a trade-off between range and fidelity, andhigh value of fidelity can be obtained at the cost of a slight reduction inprotocol range.

Long-distance high-fidelity continuous-variable quantum key distribution with non-Gaussian operations: An exact closed form solution