  Positivity or the stronger notion of complete positivity, and contextuality
are central properties of quantum dynamics. In this work, we demonstrate that a
physical unitary-universe dilation model could be employed to characterize the
completely positive map, regardless of the initial correlation condition.
Particularly, the problem of initial correlation can be resolved by a swap
operation. Furthermore, we discuss the physical essence of completely positive
map and highlights its limitations. Then we develop the quantum
measurement-chain formula beyond the framework of completely positive map in
order to describe much broader quantum dynamics, and therein the property of
contextuality could be captured via measurement transfer matrix.
