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

TL;DR

This paper introduces a simplified commutator formalism for pairs correlated via Schmidt decomposition in Quantum Information, enabling easier calculation of their statistical properties and bosonic behavior thresholds.

Contribution

It develops a new, simpler commutator formalism for single-index pairs in quantum systems, improving analysis of their statistical and bosonic properties.

Findings

The formalism allows calculation of the pair number threshold for bosonic behavior.

The second moment of the Schmidt distribution controls the main term of the mean value.

A flatter Schmidt distribution increases the bosonic threshold.

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 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.