Tight bounds for private communication over bosonic Gaussian channels based on teleportation simulation with optimal finite resources

TL;DR

This paper introduces a novel method using finite-energy resource states for teleportation simulation to derive tighter upper bounds on the secret key capacity of bosonic Gaussian channels, improving upon previous asymptotic approaches.

Contribution

It proposes a new approach employing finite-energy resource states for teleportation simulation, enabling tighter bounds on secret key capacity of Gaussian channels.

Findings

Finite-energy resource states closely approximate ultimate bounds at high energy.

Optimization over resource states bounds maximum secret key rates in finite channel uses.

Method improves bounds compared to asymptotic resource state approaches.

Abstract

Upper bounds for private communication over quantum channels can be derived by adopting channel simulation, protocol stretching, and relative entropy of entanglement. All these ingredients have led to single-letter upper bounds to the secret key capacity which can be directly computed over suitable resource states. For bosonic Gaussian channels, the tightest upper bounds have been derived by employing teleportation simulation over asymptotic resource states, namely the asymptotic Choi matrices of these channels. In this work, we adopt a different approach. We show that teleporting over an analytical class of finite-energy resource states allows us to closely approximate the ultimate bounds for increasing energy, so as to provide increasingly tight upper bounds to the secret-key capacity of one-mode phase-insensitive Gaussian channels. We then show that an optimization over the same class of resource states can be used to bound the maximum secret key rates that are achievable in a finite number of channel uses.