## 3 CV QKD over Satellite Channels
### Quantum Channel and Protocol Parameters  
The protocols are essentially influenced by the excess noise on the channel _output_, \(\epsilon\), further fixed to \(10^{-4}\) shot-noise units (SNU), which are the vacuum quadrature fluctuations. This complies with the experiment in the 100-km optical fiber with the total attenuation of \(-20\) dB, where the noise on the channel _input_ was estimated as 3% SNU [43] and with the recent experiment in the 300 km long low-loss fiber with the total attenuation of \(-32\) dB, where the noise at the channel _input_ was estimated in the worst case as 3.8% SNU [44]. Note that the excess noise is mainly concerned with imperfect parameter estimation from the homodyne data on the receiving side of the protocol (even if the channel noise is physically absent, the pessimistic assumption on the level of noise, related to noise estimation error, results in effectively nonzero level of channel noise in order to comply with the required probability of failure of channel estimation procedures [29]), which substantially depends on the stability of the set-up. It is essential that we fix the noise at the channel output contrary to the standard approach in CV QKD, when noise was fixed as relates to the channel input and then scaled by the channel attenuation, making the assessment of the channel noise in long-distance channels too optimistic. Alternatively and taking into account the fact that the excess noise atop of the calibrated electronic noise of the detector appears most likely due to the imperfect estimation of a noiseless quantum channel, one may assume passive eavesdropping such that no untrusted channel excess noise is present. In the case of satellite-based links, where line of sight between the sender and the receiver suggests the absence of equipment capable of active eavesdropping, this is a particularly sensible assumption and it was applied recently for feasibility study of DV QKD over satellite links [45]. We also take this assumption into account in CV QKD by assuming the excess noise to be trusted (i.e., being out of control by an eavesdropper) and including it in the state purification using the scheme similar to the entangling cloner with a strongly unbalanced coupling to the signal prior to detection [37]. For the data ensemble size, we rely on the typical passage time of 300 s observed for the Micius quantum satellite [7]. Assuming repetition rate of a CV QKD system to be of order of GHz, which is challenging but feasible with the current technology [46], we may expect \(N=10^{11}\) data points acquired during a satellite passage, half of which will then contribute to the key. Lastly, postprocessing efficiency is taken \(\beta=0.95\), complying with the currently available algorithms [47]. The maximum tolerable channel attenuation gives the idea of the protocols applicability independently of the aperture setting and random atmospheric disturbances, i.e., based only on the link optical budget. The results are summarized in Table 1 for given security assumptions and protocol parameters (signal states and clock rates). Evidently, in the assumption of passive eavesdropping there is no substantial difference between collective and individual attacks. On the other hand, when the channel excess noise is assumed to be untrusted, relaxing the assumptions on the possible attacks from collective to individual ones can substantially extend the tolerable loss. Note that the use of squeezed states typically increases the tolerable channel attenuation by 4-6 dB depending on the attack assumption (the better improvement being observed upon more strict collective attacks). Furthermore, with the high repetition rate the squeezed-state protocol can tolerate from 30 to 44 dB of channel attenuation depending on the attack assumption, making it potentially feasible in the geostationary scenario.