Achievable rates in non-asymptotic bosonic quantum communication

TL;DR

This paper develops new tools to analyze the non-asymptotic capacities of bosonic Gaussian channels, providing bounds and algorithms for finite-use quantum communication scenarios.

Contribution

It introduces easily computable lower bounds on non-asymptotic capacities and new methods for bounding photon number probabilities and computing trace distances.

Findings

Bound on probability of Gaussian state having more than N photons decreases exponentially with N.

First algorithm for computing trace distance between Gaussian states with fixed precision.

Provides practical bounds for finite-use quantum communication in bosonic channels.

Abstract

Bosonic quantum communication has extensively been analysed in the asymptotic setting, assuming infinite channel uses and vanishing communication errors. Comparatively fewer detailed analyses are available in the non-asymptotic setting, which addresses a more precise, quantitative evaluation of the optimal communication rate: how many uses of a bosonic Gaussian channel are required to transmit $k$ qubits, distil $k$ Bell pairs, or generate $k$ secret-key bits, within a given error tolerance $\varepsilon$? In this work, we address this question by finding easily computable lower bounds on the non-asymptotic capacities of Gaussian channels. To derive our results, we develop new tools of independent interest. In particular, we find a stringent bound on the probability $P_{>N}$ that a Gaussian state has more than $N$ photons, demonstrating that $P_{>N}$ decreases exponentially with $N$. Furthermore, we design the first algorithm capable of computing the trace distance between two Gaussian states up to a fixed precision.