2018 **Abstract:** We show how adaptive protocols of quantum and private communication through bosonic Gaussian channels can be simplifed into much easier block versions that involve resource states with finite energy. This is achieved by combining an adaptive-to-block reduction technique devised earlier, based on teleportation stretching and relative entropy of entanglement, with a recent finite-resource simulation of Gaussian channels. In this way, we derive weak converse upper bounds for the secret-key capacity of phase-insen…

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“…This step is crucial also in simulations via standard teleportation [18,26] (see also Refs. [37][38][39][40][41]). Second, we need to "stretch" the protocol [25] by replacing the various instances of the approximate channel E M with a collection of Choi matrices ρ ⊗M E and then suitably re-organizing all the remaining QOs.…”

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“…This step is crucial also in simulations via standard teleportation [18,26] (see also Refs. [37][38][39][40][41]). Second, we need to "stretch" the protocol [25] by replacing the various instances of the approximate channel E M with a collection of Choi matrices ρ ⊗M E and then suitably re-organizing all the remaining QOs.…”

“…The target state d k AB n f ( )can either be a maximally entangled state d k AB n y ( ) (see equation (13)) or a private state d k AB n g ( )(see equation (14)), depending on the task of Alice and Bob. All the details of the adaptive strategy leading to equation (20) are determined by the protocol n , that the parties are following. These details include the error threshold ò, the maximum number of rounds n, the target states d k AB n f ( ), and the set of rules that, at any round of the protocol, map the vectors of previous outcomes…”

“…a channel, led to recent intensive research for computable and simple upper bounds on channel capacities, preferably determined by a single use of the channel [9,[13][14][15][16][17][18][19][20].…”

mentioning

“…For instance, see Refs. [60,84]. Then, from the CM Vμ of Φμ Ra , it is easy to compute the CM of the output state ρ µ,μ Ra :…”