Quantum de Moivre–Laplace theorem for noninteracting indistinguishable particles in random networks

Abstract

The asymptotic form of the average probability to count N indistinguishable identical particles in a small number of binned-together output ports of a M-port Haar-random unitary network, proposed recently in Shchesnovich (2017 Sci. Rep. 7 31) in a heuristic manner with some numerical confirmation, is presented with mathematical rigour and generalised to an arbitrary (mixed) input state of N indistinguishable particles. It is shown that, both in the classical (distinguishable particles) and quantum (indistinguishable particles) cases, the average counting probability into r output bins factorises into a product of counting probabilities into two bins. This fact relates the asymptotic Gaussian law to the de Moivre–Laplace theorem in the classical case and similarly in the quantum case where an analogous theorem can be stated. The results have applications to the setups where randomness plays a key role, such as the multiphoton propagation in disordered media and the scattershot Boson sampling.