Quantum De Moivre-Laplace theorem for noninteracting indistinguishable particles

• Published in: arXiv.org
• Main Author: Shchesnovich, V S
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 19.06.2017
• Subjects: Asymptotic properties; Bins; Counting; Mathematics - Mathematical Physics; Physics - Atomic Physics; Physics - Mathematical Physics; Physics - Other Condensed Matter; Physics - Quantum Physics; Theorems
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1609.05007

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.