Quantum De Moivre-Laplace theorem for noninteracting indistinguishable particles

TL;DR

This paper rigorously derives a quantum analogue of the De Moivre-Laplace theorem for indistinguishable particles in Haar-random networks, revealing factorization properties of counting probabilities with applications in quantum optics and Boson Sampling.

Contribution

It provides a rigorous mathematical proof and generalization of the asymptotic counting probability distribution for indistinguishable particles in random unitary networks.

Findings

Average counting probability factorizes into products of two-bin probabilities.

Quantum case exhibits an analogous Gaussian law to the classical De Moivre-Laplace theorem.

Results applicable to multiphoton propagation and Boson Sampling scenarios.

Abstract

The asymptotic form of the average probability to count $N$ indistinguishable identical particles in a small number $r \ll N$ of binned-together output ports of a $M$-port Haar-random unitary network, proposed recently in \textit{Scientific Reports} \textbf{7}, 31 (2017) in a heuristic manner with some numerical confirmation, is presented with the mathematical rigor and generalized 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 factorizes into a product of $r-1$ 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.