Defining quantumness via the Jordan product.

We propose alternative definitions of classical states and quantumnesswitnesses by focusing on the algebra of observables of the system. A central role is assumed by the anticommutator of the observables, namely the Jordan product. This approach turns out to be suitable for generalizations to infinite dimensional systems. We then show that the whole algebra of observables can be generated by three elements by repeated application of the Jordan product. [ABSTRACT FROM AUTHOR]