Finite-key Analysis Of Continuous-Variable Quantum Key Distribution With Post-Selection

As we know, the number of signals quantum key distribution(QKD) which can be exchanged is finite in practice, therefore the finite-key effects becomes a vital issue in the physical implementation of QKD. So far, the study of finite-key effects has been focused on discrete-variable(DV) protocols. However, continuous-variable(CV) protocols are relatively easy to implement in contrast to DV protocols. Hence the finite-key effects for CV protocols are analyzed here. Meanwhile, the post-selection is applied to CV QKD to improve the reconciliation efficiency and to achieve high secret key rate. The concrete research content is as follows:(1) Gaussian states and Gaussian unitaries involved in CV QKD are introduced. The quantum communication procedure of CV protocols is addressed in detail. In the prepare-and-measure(P&M) scheme considered here, the sender’s state preparation step can be described into an entanglement-based(EB) representation in principle, the receiver performs a heterodyne measurement which can obtain higher secret key than homodyne measurement does, and the eavesdropper’s collective Gaussian attacks in the lossy Gaussian noise channel from the receiver’s point of view correspond to tapping off part of the sent signal and injecting a thermal state in a beam splitter. The classical communication procedure of CV protocols is also described. In the error correction(EC) phase, the reverse reconciliation is considered to exceed the3dB loss limit faced by direct reconciliation and increase the transmission distance. The Gaussian protocols and the discrete-modulated protocols which are the two families of CV protocols studied here are briefly presented and compared.(2) The finite-key effects for no-switching protocol belonging to Gaussian protocols are studied and the advantage of the protocol combing post-selection and reverse reconciliation is analyzed. Two main finite-key effects for Gaussian protocols are the imperfect reconciliation efficiency and the parameter estimation. The former prevents Gaussian protocols from reaching long transmission distances. The post-selection is considered here to improve the reconciliation efficiency of EC step by retaining the closely correlated variables. The latter chiefly estimate the transmission and excess noise of the quantum channel which are required to compute the secret key rate. The finite-key effect would decrease the transmission and increase the excess noise, thereby reducing the secret key rate. According to the numerical results, the protocol combing post-selection and reverse reconciliation can endure more loss and excess noise than others. It is also known that the impact the finite-key effect has on the secret key rate is related to the number of exchanged signals as well as the transmission.(3) The finite-key effects for two-state protocol and four-state protocol belonging to discrete-modulated protocols are studied. In two-state protocol and four-state protocol, the EC has high efficiency even at low signal-to-noise ratio(SNR) which means that these protocols don’t need post-selection to improve the reconciliation efficiency, nevertheless, the post-selection is still employed by them to compare with no-switching protocol under the same conditions. The detailed derivation of the finite-key rate for four-state protocol combing post-selection and reverse reconciliation is analyzed. In comparison with the no-switching protocol, the similarity is that the impact the finite-key effect has on two-state protocol and four-state protocol is also related to the number of exchanged signals and the transmission. The distinction is that the non-Gaussian entanglement of the two-state protocol and the four-state protocol makes the differences of secret key rates between asymptotic and nonasymptotic regimes larger. What’s more, the advantage two-state protocol and four-state protocol have in EC phase makes them achieve longer transmission distances. Meanwhile, it is found that the more coherent states the discrete-modulated protocol uses, the more loss it can endure, and the less the number of signals is required.