  Abelian duality is realized naturally by combining differential cohomology
and locally covariant quantum field theory. This leads to a C$^*$-algebra of
observables, which encompasses the simultaneous discretization of both magnetic
and electric fluxes. We discuss the assignment of physically well-behaved
states to such algebra and the properties of the associated GNS triple. We show
that the algebra of observables factorizes as a suitable tensor product of
three C$^*$-algebras: the first factor encodes dynamical information, while the
other two capture topological data corresponding to electric and magnetic
fluxes. On the former factor we exhibit a state whose two-point correlation
function has the same singular structure of a Hadamard state. Specifying
suitable counterparts also on the topological factors we obtain a state for the
full theory, providing ultimately a unitary implementation of Abelian duality.
