## 3 Security Analysis
For this reason, computing an upper bound on the Holevo information \(\mathsf{x}(B;E)\), that is a lower bound on \(K^{\rm asympt}_{\rm coll}\), is an important first step in security proofs for continuous-variable QKD. In Table 1, we summarize the current state-of-the-art for security proofs for CVQKD with a Gaussian modulation. Two protocols have complete security proofs in the composable security framework. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Protocol**} & \multirow{2}{*}{
\begin{tabular}{c} **(PM) State** \\ **Preparation** \\ \end{tabular} } & \multirow{2}{*}{**(PM) Modulation**} & \multicolumn{2}{c}{**Bob’s**} & \multirow{2}{*}{**Best Currently-Available Security Proofs**} \\  & & & & & \\ \hline \multirow{3}{*}{[41]} & \multirow{3}{*}{squeezed} & \multirow{3}{*}{Gaussian} & \multirow{3}{*}{homodyne} & Finite-size [38, 39] \\  & & & & \(K^{\epsilon}(N)>0\) for practical \(N\) \\  & & & & \(\lim_{N\to\infty}K^{\epsilon}(N)<K^{\rm asympt}_{\rm coll}\) \\ \hline \multirow{3}{*}{[23]} & \multirow{3}{*}{coherent} & \multirow{3}{*}{Gaussian} & \multirow{3}{*}{heterodyne} & Finite-size [24] \\  & & & & \(K^{\epsilon}_{\rm coll}(N)\approx K^{\rm asympt}_{\rm coll}\) for practical \(N\) \\  & & & & \(K^{\epsilon}(N)=0\) for practical \(N\)[37] \\ \hline \multirow{3}{*}{[22]} & coherent & Gaussian & homodyne & asymptotic collective [30, 31, 42] \\  & coherent & Gaussian 1D & homodyne & asymptotic collective [43] \\  & squeezed & Gaussian & heterodyne & asymptotic collective [45] \\  & thermal & Gaussian & homo/heterodyne & asymptotic collective [47, 48, 49] \\ \hline \multirow{3}{*}{[50]} & \multirow{3}{*}{squeezed} & Gaussian + & \multirow{3}{*}{homodyne} & asymptotic collective [50] \\  & & additional Gaussian & & \\ \hline \multirow{3}{*}{[51, 52]} & \multirow{3}{*}{coherent} & \multirow{3}{*}{Gaussian} & homo/heterodyne + & \multirow{3}{*}{asymptotic collective [51, 52]} \\  & & & Gaussian post-selection & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Current security status of the main one-way continuous-variable quantum key distribution (CVQKD) protocols. PM, prepare and measure. Their entanglement-based version consists of preparing \(N\) two-mode squeezed states and measuring each mode with either homodyne detection (of a randomly-chosen quadrature) for [41] or with heterodyne detection for [23]. The proof techniques are quite different: the security of [41] is based on an entropic uncertainty principle for continuous variables [40], while the security of [23] is obtained in two steps (security against collective attacks followed by a reduction from general attacks). Despite this success, improvements are still called for. Indeed, in the first case of [41], the security proof provides a positive secret key rate \(K^{\epsilon}(N)\), which is positive for reasonable values of \(N\), but unfortunately, the key rate does not converge to \(K^{\mathrm{asympt}}_{\mathrm{coll}}\) for \(N\rightarrow\infty\), which indicates that either the true secret key rate is overestimated by \(K^{\mathrm{asympt}}_{\mathrm{coll}}\) or, more likely, that the proof technique should be improved. On the other hand, in the case of the heterodyne protocol with coherent states [23], the quantity \(K^{\epsilon}(N)\) converges to the asymptotic value \(K^{\mathrm{asympt}}_{\mathrm{coll}}\), corresponding to a Gaussian attack, but the convergence is too slow to obtain a positive key rate for a reasonable block size \(N\). The main issue lies in the reduction from general to collective attack using either [PERSON]'s theorem [36] or the post-selection technique [37] (also known as a de Finetti reduction). We insist on the fact that these limitations might be due to insufficient proof techniques. Indeed, it is quite possible that better security proofs will be found and establish that \(K^{\epsilon}(N)\) converges to \(K^{\mathrm{asympt}}_{\mathrm{coll}}\) for reasonable block lengths. We believe that this is certainly the most pressing issue in the theoretical study of CVQKD.