## 2 Security of CV QKD
The resulting expression for the mutual information then reads  
\[I_{AB}=\frac{1}{2}\log_{2}\left[1+\frac{\eta V_{M}}{\eta V_{S}+1-\eta+\epsilon}\right] \tag{2}\]  
and is essentially determined by the signal modulation variance \(V_{M}\), signal state variance \(V_{S}\), channel transmittance \(\eta\), and channel noise \(\epsilon\) related to the channel output. Evaluation of the Holevo bound on the other hand is more involved. It is based on the assumption that [PERSON] holds purification of the noise added in the channel (due to losses and excess noise) and relies on the evaluation of von Neumann entropies of the state shared between [PERSON] and [PERSON] [37]. This is performed in the equivalent entanglement-based representation, when state preparation is purified using two-mode entangled state. In the case of coherent-state protocol a symmetrical two-mode squeezed vacuum (TMSV) [38] state with variance \(V=1+V_{M}\) is used for purification, such that [PERSON] is measuring one of the modes using a heterodyne (balanced homodyne) detector [37]. For a general state preparation in the squeezed-state protocol, assuming independent levels of signal squeezing and modulation variance (contrary to the standard symmetrically modulated protocol, where squeezing and modulation are essentially related as \(V_{M}=1/V_{S}-V_{S}\)[39]), we use the generalized entanglement-based scheme using an asymmetrical entangled state instead of TMSV and a homodyne detection on the local mode [17], as shown in Figure 1b. The preparation in this case is equivalent to the prepare-and-measure scheme provided the asymmetrical state is constructed of the oppositely squeezed states with variances in \(x\)-quadratures being  
\[V_{1}=V_{S}+V_{M}-\sqrt{V_{M}(V_{S}+V_{M})},\hskip 28.452756 ptV_{2}=1/\left[V_{S }+V_{M}+\sqrt{V_{M}(V_{S}+V_{M})}\right], \tag{3}\]  
while having the opposite variances (\(1/V_{1}\) and \(1/V_{2}\) respectively) in the \(p\)-quadratures. In the purification-based scenario, the Holevo bound is evaluated as \(\chi_{BE}=S(AB)-S(A|B)\), where \(S(\cdot)\) denotes the von Neumann (quantum) entropy of a state, \(S(AB)\) is the quantum entropy of a (generally noisy) state shared between the trusted parties and \(S(A|B)=S(A|x_{B})\) is the von Neumann entropy of the state of the trusted parties, conditioned by [PERSON]'s measurement results in \(x\)-quadrature. We obtain the relevant von Neumann entropies from bosonic entropic functions of symplectic eigenvalues of respective covariance matrices of the states, shared between [PERSON] and [PERSON] (see [37] for details of security analysis techniques in CV QKD). In the case of individual attacks and reverse reconciliation scenario, the upper bound on the information leakage is reduced to the classical (Shannon) information between [PERSON] and [PERSON], which, similarly to \(I_{AB}\) (2) described above, reads \(I_{BE}=(1/2)\log_{2}\left(V_{B}/V_{B|E}\right)\). The evaluation of \(V_{B|E}\) is done in the assumption that [PERSON] is able to purify the channel noise. The optimal individual attack in this case is the entangling cloner attack [40], which is a TMSV state of variance \(V_{N}=1+\frac{\epsilon}{1-\eta}\) set so to emulate the channel loss \(\eta\) and noise \(\epsilon\). One of the modes of the cloner interacts with the signal with a linear coupling \(\eta\), corresponding to the channel loss and resulting in \(V_{B}=\eta V+(1-\eta)V_{N}\) (which gives exactly \(V_{B}\) as described above and as expected by the trusted parties), while the other mode is measured by [PERSON] in order to reduce her uncertainty on the quantum noise added in the channel. The mutual information between [PERSON] and [PERSON] then reads [40; 41]  
\[I_{BE}=\frac{1}{2}\log_{2}\biggl{\{}\frac{1}{V}[1+\epsilon+\eta(V-1)][\eta(V-1)+ V(\epsilon+1)]\biggr{\}}, \tag{4}\]  
which gives the bounds on the secure key rate in the case of individual eavesdropping attacks, similarly to (1).