Quantum-information theory of a Dirichlet ring with Aharonov-Bohm field

TL;DR

This paper analyzes quantum information measures such as Shannon entropy, Fisher information, and Rényi entropy for a particle in a two-dimensional Dirichlet annulus with Aharonov-Bohm flux, revealing asymptotic behaviors and topological effects.

Contribution

It provides a comprehensive calculation of various quantum information measures in a Dirichlet ring with AB flux, highlighting asymptotic behaviors and the impact of topology and magnetic flux.

Findings

Position Shannon entropy grows logarithmically with inner radius

Momentum Shannon entropy approaches a common asymptote for large radii

Fisher information in momentum space grows exponentially with radius

Abstract

Shannon quantum information entropies $S_{\rho,\gamma}$, Fisher informations $I_{\rho,\gamma}$, Onicescu energies $O_{\rho,\gamma}$ and R\'{e}nyi entropies $R_{\rho,\gamma}(\alpha)$ are calculated both in the position (subscript $\rho$) and momentum ($\gamma$) spaces as functions of the inner radius $r_0$ for the two-dimensional Dirichlet unit-width annulus threaded by the Aharonov-Bohm (AB) flux $\phi_{AB}$. Discussion is based on the analysis of the corresponding position and momentum waveforms. Position Shannon entropy (Onicescu energy) grows logarithmically (decreases as $1/r_0$) with large $r_0$ tending to the same asymptote $S_\rho^{asym}=\ln(4\pi r_0)-1$ [$O_\rho^{asym}=3/(4\pi r_0)$] for all orbitals whereas their Fisher counterpart $I_{\rho_{nm}}(\phi_{AB},r_0$) approaches in the same regime the $m$-independent limit mimicking in this way the energy spectrum variation with $r_0$, which for the thin structures exhibits quadratic dependence on the principal index. Frequency of the fading oscillations of the radial parts of the wave vector functions increases with the inner radius what results in the identical $r_0\gg1$ asymptote for all momentum Shannon entropies $S_{\gamma_{nm}}(\phi_{AB};r_0)$ with the alike $n$ and different $m$. The same limit causes the Fisher momentum components $I_\gamma(\phi_{AB},r_0)$ to grow exponentially with $r_0$. It is proved that the lower limit $\alpha_{TH}$ of the semi-infinite range of the dimensionless coefficient $\alpha$, where the momentum component of this one-parameter entropy exists, is \textit{not} influenced by the radius; in particular, the change of the topology from the simply, $r_0=0$, to the doubly, $r_0>0$, connected domain is \textit{un}able to change $\alpha_{TH}=2/5$. AB field influence on the measures is calculated too.