Almost Complete Coherent State Subsystems and Partial Reconstruction of Wave Functions in the Fock-Bargmann Phase-Number Representation

TL;DR

This paper develops partial reconstruction formulas for wave functions in the Fock-Bargmann phase-number representation using finite phase samples, demonstrating near-complete coherence subsystems and analyzing approximation accuracy.

Contribution

It introduces partial reconstruction formulas and studies the completeness of coherent state subsystems for finite and infinite particle numbers.

Findings

Reconstruction formulas are exact for finite particle numbers.

Subsystems are almost complete (pseudo-frames) for unbounded particles.

Approximation accuracy improves when mean number p is less than N or as N approaches infinity.

Abstract

We provide (partial) reconstruction formulas and discrete Fourier transforms for wave functions in standard Fock-Bargmann (holomorphic) phase-number representation from a finite number $N$ of phase samples $\{\theta_k=2\pi k/N\}_{k=0}^{N-1}$ for a given mean number $p$ of particles. The resulting Coherent State (CS) subsystem ${\cal S}=\{|z_k=p^{1/2}e^{i\theta_k}>\}$ is complete (a frame) for truncated Hilbert spaces (finite number of particles) and reconstruction formulas are exact. For an unbounded number of particles, ${\cal S}$ is "almost complete" (a \textit{pseudo-frame}) and partial reconstruction formulas are provided along with an study of the accuracy of the approximation, which tends to be exact when $p<N$ and/or $N\to\infty$.