Optical fibres with memory effects and their quantum communication capacities

The development of quantum repeaters poses significant challenges in terms of cost and maintenance, prompting the exploration of alternative approaches for achieving long-distance quantum communication. In the absence of quantum repeaters and under the memoryless (iid) approximation, it has been established that some fundamental quantum communication tasks are impossible if the transmissivity of an optical fibre falls below a known critical value, resulting in a severe constraint on the achievable distance for quantum communication. However, if the memoryless assumption does not hold -- e.g. when input signals are separated by a sufficiently short time interval -- the validity of this limitation is put into question. In this paper we introduce a model of optical fibre that can describe memory effects for long transmission lines. We then solve its quantum capacity, two-way quantum capacity, and secret-key capacity exactly. By doing so, we show that -- due to the memory cross-talk between the transmitted signals -- reliable quantum communication is attainable even for highly noisy regimes where it was previously considered impossible. As part of our solution, we find the critical time interval between subsequent signals below which quantum communication, two-way entanglement distribution, and quantum key distribution become achievable.