If the transmissivity of an optical fibre falls below a critical value, its use as a reliable quantum channel is known to be drastically compromised. 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 work 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.

We study the problem of transmitting classical information on a quantum channel in the absence of a shared phase reference. This problem is relevant for long-distance communications in free space and optical fiber, where phase noise is typically considered as a limiting factor. Previous analyses considered phase noise that acts independently on each communication mode, thus completely decohering successive signals and making it impossible to establish a phase reference. In the present work we analyze instead the realistic case in which the phase reference is lost only after m uses of the transmission line, due to a finite decoherence time. In this setting, focusing on the simplest case m = 2, we analyze two communication strategies using coherent states of the electromagnetic field and show that it is not beneficial to employ the total energy to establish a reference frame but rather to spread out the energy on all the modes and directly transmit information using their relative degrees of freedom.

We consider the problem of correctly classifying a given quantum two-level system (qubit) which is known to be in one of two equally probable quantum states. We assume that this task should be performed by a quantum machine which does not have at its disposal a complete classical description of the two template states, but can only have partial prior information about their level of purity and mutual overlap. Moreover, similarly to the classical supervised learning paradigm, we assume that the machine can be trained by n qubits prepared in the first template state and by n more qubits prepared in the second template state. In this situation, we are interested in the optimal process which correctly classifies the input qubit with the largest probability allowed by quantum mechanics. The problem is studied in its full generality for a number of different prior information scenarios and for an arbitrary size n of the training data. Finite size corrections around the asymptotic limit n → ∞ are derived. When the states are assumed to be pure, with known overlap, the problem is also solved in the case of d-level systems.