We revisit Heisenberg indeterminacy principle in the light of the Galois–Grothendieck theory for the case of finite abelian Galois extensions. In this restricted framework, the Galois–Grothendieck duality between finite K-algebras split by a Galois extension \(L\) and finite \(Gal(L{:}K)\)-sets can be reformulated as a Pontryagin duality between two abelian groups. We define a Galoisian quantum model in which the Heisenberg indeterminacy principle (formulated in terms of the notion of entropic indeterminacy) can be understood as a manifestation of a Galoisian duality: the larger the group of automorphisms \(H\subseteq G\) of the states in a G-set \({\mathcal {O}}\simeq G/H\), the smaller the “conjugate” algebra of observables that can be consistently evaluated on such states. Finally, we argue that states endowed with a group of automorphisms \(H\) can be interpreted as squeezed coherent states, i.e. as states that minimize the Heisenberg indeterminacy relations.