We investigate the role of weak-field homodyne (WH) measurement for optical communications over a lossy bosonic channel with coherent state encoding. This kind of receiver employs photon-number resolving (PNR) detectors with finite resolution and low-intensity local oscillator. As a figure of merit, we consider the mutual information for both single and double WH detection. At first, we consider a Gaussian input modulation, showing this scheme to be suboptimal with respect to the homodyne and double homodyne capacity, respectively. Thereafter, we investigate the performance of non-Gaussian modulation, by considering a Gamma distribution of the energy of the encoded pulses, and achieve an increase in the information rate with respect to the Gaussian case in all energy regimes. In particular, WH measurement outperforms the Shannon limit for homodyne/double homodyne detection in the photon-starved limit, albeit it does not beat neither the direct detection (DD) limit achieved with Gamma modulation, nor on-off keying (OOK) strategies, e.g. OOK-DD and OOK-Dolinar receiver.

To achieve the maximum information transfer and face a possible eavesdropper, the samples transmitted in continuous-variable quantum key distribution (CV-QKD) protocols are to be drawn from a continuous Gaussian distribution. As a matter of fact, in practical implementations the transmitter has a finite (power) dynamics and the Gaussian sampling can be only approximated. This requires the quantum protocols to operate at small powers. In this paper, we show that a suitable probabilistic amplitude shaping of a finite set of symbols allows to approximate at will the optimal channel capacity also for increasing average powers. We investigate the feasibility of this approach in the framework of CV-QKD, propose a protocol employing discrete quadrature amplitude modulation assisted with probabilistic amplitude shaping, and we perform the key generation rate analysis assuming a wiretap channel and lossless homodyne detection.