We calculate numerically the capacity of a lossy photon channel assuming photon number resolving detection at the output. We consider scenarios of input Fock and coherent states ensembles and show that the latter always exhibits worse performance than the former. We obtain capacity of a discrete-time Poisson channel as a limiting behavior of the Fock states ensemble capacity. We show also that in the regime of a moderate number of photons and low losses the Fock states ensemble with direct detection is beneficial with respect to capacity limits achievable with quadrature detection.

Quantum fingerprinting reduces communication complexity of determination whether two n-bit long inputs are equal or different in the simultaneous message passing model. Here we quantify the advantage of quantum fingerprinting over classical protocols when communication is carried out using optical signals with limited power and unrestricted bandwidth propagating over additive white Gaussian noise (AWGN) channels with power spectral density (PSD) much less than one photon per unit time and unit bandwidth. We identify a noise parameter whose order of magnitude separates near-noiseless quantum fingerprinting, with signal duration effectively independent of n, from a regime where the impact of AWGN is significant. In the latter case the signal duration is found to scale as O(√n), analogously to classical fingerprinting. However, the dependence of the signal duration on the AWGN PSD is starkly distinct, leading to quantum advantage in the form of a reduced multiplicative factor in O(√n) scaling.