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# Coherent states with minimum Gini uncertainty for finite quantum systems

Dec 2022

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

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