Continuous Variable (CV)-QKD

This chapter is devoted to the detailed description of continuous variable (CV) QKD schemes, in particular, with Gaussian modulation and discrete modulation. The chapter starts with the fundamentals of Gaussian quantum information theory, where the P-representation is introduced and applied to represent the thermal noise as well as the thermal noise plus the coherent state signal. Then, quadrature operators are introduced, followed by the phase-space representation. Further, Gaussian and squeezed states are introduced, followed by Wigner function definition as well as the definition of correlation matrices. The next subsection is devoted to the Gaussian transformation and Gaussian channels, with beam splitter operation and phase rotation operation being the representative examples. The thermal decomposition of Gaussian states is discussed next and the von Neumann entropy for thermal states is derived. The focus is then moved to the nonlinear quantum optics fundamentals, in particular, the three-wave mixing and four-wave mixing are described in detail. Further, the generation of the Gaussian states is discussed, in particular, the EPR state. The correlation matrices for two-mode Gaussian states are discussed next, and how to calculate the symplectic eigenvalues, relevant in von Neumann entropy calculation. The Gaussian state measurements and detection are discussed then, with emphasis on homodyne detection, heterodyne detection, and partial measurements. In section on CV-QKD protocols with Gaussian modulation, after the brief description of squeezed state-based protocols, the coherent state-based protocols are described in detail. We start the section with the description of both lossless and lossy transmission channels, followed by the description of how to calculate the covariance matrix under various transformations, including beam splitter, homodyne detection, and heterodyne detection. The equivalence between the prepare-and-measure (PM) and entanglement-assisted protocols with Gaussian modulation is discussed next. The focus is then moved to the secret-key rate (SKR) calculation under collective attacks. The calculation of mutual information between Alice and Bob is discussed first, followed by the calculation of Holevo information between Eve and Bob, in both cases assuming the PM protocol and reverse reconciliation. Further, entangling cloner attack is described, followed by the derivation of Eve-to-Bob Holevo information. The entanglement-assisted protocol is described next as well as the corresponding Holevo information derivation. In all these derivations, both homodyne detection and heterodyne detection are considered. Some illustrative SKR results corresponding to the Gaussian modulation are provided as well. In section on CV-QKD with discrete modulation, after the brief introduction, we describe the four-state and eight-state CV-QKD protocols. Both the PM and entanglement-assisted protocols are discussed. The SKR calculation for discrete modulation is discussed next, with illustrative numerical results being provided. We also identify conditions under which the discrete modulation can outperform the Gaussian modulation. In section on RF-assisted CV-QKD scheme, we describe a generic RF-assisted scheme applicable to arbitrary two-dimensional modulation schemes, including M-ary PSK, and M-ary QAM. This scheme exhibits better tolerance to laser phase noise and frequency offset fluctuations compared to conventional CV-QKD schemes with discrete modulation. We then discuss how to increase the SKR through the parallelization approach. The final section in the chapter provides some relevant concluding remarks.