“Commutator formalism” for pairs correlated through Schmidt decomposition as used in Quantum Information

Institut des NanoSciences de Paris, CNRS, Université Pierre et Marie Curie - 4 place Jussieu, Paris, France, EU

^a monique.combescot@insp.jussieu.fr

Received: 15 June 2011
Accepted: 24 October 2011

Abstract

To easily calculate statistical properties of pairs correlated through Schmidt decomposition, as commonly used in Quantum Information, we propose a “commutator formalism” for these single-index pairs, somewhat simpler than the one we previously developed for double-index Wannier excitons. We use it here to get the pair number threshold for bosonic behavior of N pairs through the requirement that their number operator mean value must stay close to N. While the main term of this mean value is controlled by the second moment of the Schmidt distribution, so that to increase this threshold, we must increase the Schmidt number, higher momenta appearing at higher orders lead to choosing a distribution as flat as possible.