  Non-classical probability (along with its underlying logic) is a defining
feature of quantum mechanics. A formulation that incorporates them, inherently
and directly, would promise a unified description of seemingly different
prescriptions of non-classicality of states that have been proposed so far.
This paper sets up such a formalism. It is based on elementary considerations,
free of ad-hoc definitions, and is completely operational. It permits a
systematic construction of non-classicality conditions on states and also to
quantify the non-classicality, at the same time. This quantification, as shown
for the example of two level systems, can serve as a measure of coherence and
can be furthermore, harnessed to obtain a measure for pure state entanglement
for coupled two level systems.
