Coherent states with minimum Gini uncertainty for finite quantum systems

TL;DR

This paper introduces a new class of coherent states for finite quantum systems that minimize Gini uncertainty, providing a robust basis for state expansion and analysis.

Contribution

It defines coherent states with minimal Gini uncertainty in finite quantum systems and demonstrates their properties and robustness.

Findings

Numerical estimation of Gini uncertainty constant $\eta_d$ and its upper bound.

Construction of coherent states $\ket{\alpha, eta ext{ extunderscore}g$ with minimal Gini uncertainty.

These states resolve the identity and enable noise-robust state expansion.

Abstract

Uncertainty relations $\Delta(\rho)\ge \eta_d$ in terms of the Gini index are studied. The `Gini uncertainty constant' $\eta_d$ is estimated numerically and compared to an upper bound $\tilde \eta_d\ge \eta_d$. It is shown that for large $d$ we get $\tilde \eta_d\approx \eta_d$. States $\ket{g}$ with minimum Gini uncertainty and displacement transformations are used to define coherent states $\ket{\alpha, \beta}_g$ (where $\alpha, \beta \in {\mathbb Z}_d$) with minimum Gini uncertainty ($\Delta[\ket{\alpha, \beta}_g\;_g\bra{\alpha, \beta}]\approx \eta_d$). The $\ket{\alpha, \beta}_g$ resolve the identity, and therefore an arbitrary state can be expanded in terms of them. This expansion is robust in the presence of noise.