  We show how adaptive protocols of quantum and private communication through
bosonic Gaussian channels can be simplified into much easier block versions
that involve resource states with finite energy. This is achieved by combining
the adaptive-to-block reduction technique devised earlier [S. Pirandola et al.,
Nat. Commun. 8, 15043 (2017)], based on teleportation stretching and relative
entropy of entanglement, with an alternative simulation of Gaussian channels
recently introduced by Liuzzo-Scorpo et al. [Phys. Rev. Lett. 119, 120503
(2017)]. In this way, we derive weak converse upper bounds for the secret-key
capacity of phase-insensitive Gaussian channels, which approximate the optimal
limit for infinite energy. Our results apply to both point-to-point and
repeater-assisted private communications.
