Bridging Classical and Quantum Worlds: Maps, States, and Evolutions

In this work, we present several aspects of the interplay between classical and quantum theories. After reviewing the equivalence between positivity and complete positivity in the commutative setting, we introduce and analyze intermediate notions that interpolate between these two properties for linear maps on the space of operators on a Hilbert space, highlighting their natural connections to quantum dynamics and entanglement theory. Finally, we discuss possible constructions that allow one to derive quantum states and evolutions from classical counterparts, and vice versa.