Coherent states with minimum Gini uncertainty for finite quantum systems

Abstract

Uncertainty relations ∆(ρ) ≥ ηdin terms of the Gini index are studied. The ‘Gini uncertainty constant’ ηdis estimated numerically and compared to an upper bound η̃d ≥ ηd. It is shown that for large d we get η̃d ≈ ηd. States |g〉 with minimum Gini uncertainty and displacement transformations are used to define coherent states |α, β〉g (where α, β ∈ ℤd) with minimum Gini uncertainty (∆[|α, β〉g g〈α, β|] ≈ ηd). The |α, β〉g resolve the identity, and therefore an arbitrary state can be expanded in terms of them.