  A general theory of quantum stochastic processes was formulated by Accardi,
Frigerio and Lewis in 1982 within the operator-algebraic framework of quantum
probability theory, as a non-commutative extension of the Kolmogorovian
classical stochastic processes. More recently, studies on non-Markovian quantum
processes have led to the discrete-time process tensor formalism in the
Schr\"{o}dinger picture to describe the outcomes of sequential interventions on
open quantum systems. However, there has been no treatment of the relationship
of the process tensor formalism to the quantum probabilistic theory of quantum
stochastic processes. This paper gives an exposition of quantum stochastic
processes and the process tensor and the relationship between them. In
particular, it is shown how the latter emerges from the former via extended
correlation kernels incorporating ancillas.
