## 2 Security of CV QKD
We assume that the remote trusted side (Bob) is the reference side for the error correction algorithms, thus using so-called reverse reconciliation, which was shown robust against any level of pure channel loss [30] and is applicable in the strongly attenuating links with \(\eta\ll 1\). Security of CV QKD was established against two main types of attacks, namely individual, when an eavesdropper is able to interact optimal probe states with the signal states and then individually measure the probes, and collective, when the probes are assumed to be stored in a quantum memory after the interaction and then optimally collectively measured, which increases the amount of information that can be potentially obtained by an eavesdropper [31]. Security of the coherent-state protocol with heterodyne detection against collective can be extended to security against general attacks using de Finetti reduction [32], similar extensions for the squeezed-state protocol and homodyne detection being more demanding [33; 34]. We study security of the above described protocol by evaluating the lower bound on secure key rate per channel use, which in the finite-size regime, reverse reconciliation scenario and firstly assuming collective attacks, performed by an eavesdropper, reads  
\[K=\max\Bigl{\{}0,\frac{n}{N}\Bigl{[}\beta I_{AB}-\chi_{BE}-\delta(n)\Bigr{]} \Bigr{\}}, \tag{1}\]  
where \(\beta\in[0,1]\) is postprocessing efficiency representing how close the trusted parties are able to reach the Shannon mutual information \(I_{AB}\) using realistic error correction codes,  
Figure 1: **(a)** CV QKD scheme based on a signal state preparation using a Source (laser or optical parametric oscillator for preparation of coherent or squeezed states respectively) and a quadrature Modulator (driven by the data \(x_{M},p_{M}\)) on the side of [PERSON] and on a homodyne detection on the side of [PERSON], resulting in measurement outcomes \(x_{B}\) or \(p_{B}\) after the propagation through an untrusted quantum channel with transmittance \(\eta\) and excess noise \(\epsilon\) related to the channel output; **(b)** Theoretical purification scheme of the state preparation on the side of [PERSON] based on generation of two oppositely squeezed states with variances (3) on optical parametric oscillators (squeezers) \(S_{1}\) and \(S_{2}\), coupling squeezed states on a balanced beamsplitter, and local homodyne measurement on the [PERSON]’s side resulting in \(x_{A}\) or \(p_{A}\) on one of the modes (equivalent to modulation of squeezed states; in the case of coherent-state protocol, a heterodyne detection, resulting in \(x_{A}\) and \(p_{A}\), is considered at [PERSON]’s side), while the other mode is sent to the channel towards Bob, the rest of the scheme is an in **(a)**. \(\chi_{BE}\) is the Holevo bound [35], giving the upper bound on an eavesdropper's information on the measurement results at the remote receiving side, Bob, and \(\delta(n)\approx 7\sqrt{\frac{\log_{2}\left(2/\varepsilon\right)}{n}}\) is the correction parameter related to the finite-size effects [36], where \(\tilde{\epsilon}\) is the smoothing parameter that contributes to the overall failure probability of the protocol and is set further to \(10^{-10}\). The mutual information between the trusted parties \(I_{AB}=1/2\log_{2}V_{B}/V_{B|A}\) for the Gaussian-distributed data can be expressed through variances and correlations between the measurement outcomes, i.e., through the variance of [PERSON]'s measurement \(V_{B}=\eta V+(1-\eta)+\epsilon\), where \(V\equiv V_{S}+V_{M}\), and conditional variance \(V_{B|A}=V_{B}-C_{AB}^{2}/V_{A}\), \(V_{A}\equiv V_{M}\) is the variance of [PERSON]'s data and \(C_{AB}=\sqrt{\eta}V_{M}\) is the correlation between modulation data and measurements on \(x_{B}\), for the zero-mean-distributed observables obtained as \(C_{AB}=\langle\hat{x}_{M}\hat{x}_{B}\rangle\).