Continuous Variable (CV)-QKD

This chapter is devoted to the detailed description of continuous variable (CV) QKDContinuous variable (CV)-QKD schemes, in particular, with Gaussian modulation and discrete modulationCV-QKD with discrete modulation . The chapter starts with the fundamentals of Gaussian quantum information theoryGaussian quantum information theory , where the P-representation-representation is introduced and applied to represent the thermal noiseThermal noise as well as the thermal noise plus the coherent stateCoherent state signal. Then, quadrature operatorsQuadrature operators are introduced, followed by the phase-space representation. Further, Gaussian and squeezed statesSqueezed state are introduced, followed by Wigner functionWigner function definition as well as the definition of correlation matricesCorrelation matrices . The next subsection is devoted to the Gaussian transformationGaussian transformation and Gaussian channelsGaussian channels , with beam splitter operation and phase rotation operation being the representative examples. The thermal decompositionThermal decomposition of Gaussian statesGaussian states is discussed next and the von Neumann entropyVon Neumann entropy for thermal statesThermal state is derived. The focus is then moved to the nonlinear quantum opticsNonlinear quantum optics fundamentals, in particular, the three-wave mixingThree-wave mixing (TWM) and four-wave mixingFour-wave mixing (FWM) are described in detail. Further, the generation of the Gaussian states is discussed, in particular, the EPR stateEPR state . The correlation matricesCorrelation matrices for two-mode Gaussian statesTwo-mode Gaussian states are discussed next, and how to calculate the symplectic eigenvalues, relevant in von Neumann entropyVon Neumann entropy calculation. The Gaussian state measurementsGaussian state measurements and detectionGaussian state detection are discussed then, with emphasis on homodyne detectionHomodyne detection , heterodyne detectionHeterodyne detection , and partial measurementsPartial measurements . In section on CV-QKD protocols with Gaussian modulation, after the brief description ofSqueezed state squeezed state-based protocolsSqueezed state-based protocols , theCoherent state coherent state-based protocolsCoherent State-Based CV-QKD protocols are described in detail. We start the section with the description of both lossless and lossy transmission channelsLossy transmission channel , followed by the description of how to calculate the covariance matrixCovariance matrices under various transformations, including beam splitter , homodyne detectionHomodyne detection , and heterodyne detectionHeterodyne detection . The equivalence between the prepare-and-measure (PM)Prepare-and-Measure (PM) protocols and entanglement-assisted protocolsEntanglement-assisted (EA) protocols with Gaussian modulation is discussed next. The focus is then moved to the secret-key rate (SKR)Secret-key rates 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 BobHolevo information between Eve and Bob , in both cases assuming the PM protocolPrepare-and-Measure (PM) protocols and reverse reconciliation. Further, entangling cloner attackEntangling Cloner attack is described, followed by the derivation of Eve-to-Bob Holevo information. The entanglement-assisted protocolEntanglement-assisted (EA) protocols is described next as well as the corresponding Holevo information derivation. In all these derivations, both homodyne detectionHomodyne detection and heterodyne detectionHeterodyne detection are considered. Some illustrative SKRSecret-key rates results corresponding to the Gaussian modulation are provided as well. In section on CV-QKDContinuous variable (CV)-QKD with discrete modulationCV-QKD with discrete modulation , after the brief introduction, we describe the four-state and eight-state CV-QKD protocolsCoherent State-Based CV-QKD protocols . Both the PM and entanglement-assisted protocolsEntanglement-assisted (EA) 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-QKDRF-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-QKDContinuous variable (CV)-QKD schemes with discrete modulationCV-QKD with discrete modulation . We then discuss how to increase the SKRSecret-key rates through the parallelization approach. The final section in the chapter provides some relevant concluding remarks.