# Finite-resource teleportation stretching for continuous-variable systems

## TL;DR

This paper introduces a method to simplify adaptive quantum communication protocols over Gaussian channels into finite-energy block protocols, providing near-optimal bounds for secret-key capacities in continuous-variable systems.

## Contribution

It combines adaptive-to-block reduction with an alternative Gaussian channel simulation to derive finite-energy bounds on secret-key capacities.

## Key findings

- Derived weak converse upper bounds for secret-key capacity.
- Applicable to both point-to-point and repeater-assisted communication.
- Approximates the optimal limit for infinite energy.

## Abstract

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.