Coherent states with minimum Gini uncertainty for finite quantum systems

Uncertainty relations $\mathrm{\Delta }(\rho )\ge {\eta }_d$ 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 $\mathrm{\mid }g⟩$ with minimum Gini uncertainty and displacement transformations are used to define coherent states ${\mathrm{\mid }\alpha ,\beta ⟩}_g$ (where $\alpha ,\beta \in {\mathbb{Z}}_d$) with minimum Gini uncertainty ($\mathrm{\Delta }[{\mathrm{\mid }\alpha ,\beta ⟩}_g{}_g⟨\alpha ,\beta \mathrm{\mid }]\approx {\eta }_d$). The ${\mathrm{\mid }\alpha ,\beta ⟩}_g$ resolve the identity, and therefore an arbitrary state can be expanded in terms of them. This expansion is robust in the presence of noise.