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.
Categories: quant-ph