Abstract
Non-asymptotic quantum Shannon theory analyses how to transmit quantum information across a quantum channel as efficiently as possible within a specified error tolerance, given access to a finite, fixed, number of channel uses. In a recent work, we derived computable lower bounds on the non-asymptotic capacities of memoryless bosonic Gaussian channels. In this work, we extend these results to the non-Markovian bosonic Gaussian channel introduced in F. A. Mele, G. D. Palma, M. Fanizza, V. Giovannetti, and L. Lami IEEE Transactions on Information Theory 70(12), 8844-8869 (2024), which describes non-Markovian effects in optical fibres and is a non-Markovian generalisation of the pure loss channel. This allows us to determine how many uses of a non-Markovian optical fibre are sufficient in order to transmit qubits, distil ebits, or generate secret-key bits up to a given error tolerance . To perform our analysis, we prove novel properties of singular values of Toeplitz matrices, providing an error bound on the convergence rate of the celebrated Avram-Parter's theorem, which we regard as a new tool of independent interest for the field of quantum information theory and matrix analysis.