  Among several tasks in Machine Learning, a specially important one is that of
inferring the latent variables of a system and their causal relations with the
observed behavior. Learning a Hidden Markov Model of given stochastic process
is a textbook example, known as the positive realization problem (PRP). The PRP
and its solutions have far-reaching consequences in many areas of systems and
control theory, and positive systems theory.
  We consider the scenario where the latent variables are quantum states, and
the system dynamics is constrained only by physical transformations on the
quantum system. The observable dynamics is then described by a quantum
instrument, and the task is to determine which quantum instrument --if any--
yields the process at hand by iterative application.
  We take as starting point the theory of quasi-realizations, whence a
description of the dynamics of the process is given in terms of linear maps on
state vectors and probabilities are given by linear functionals on the state
vectors. This description, despite its remarkable resemblance with the Hidden
Markov Model, or the iterated quantum instrument, is nevertheless devoid of any
stochastic or quantum mechanical interpretation, as said maps fail to satisfy
any positivity conditions. The Completely-Positive realization problem then
consists in determining whether an equivalent quantum mechanical description of
the same process exists.
  We generalize some key results of stochastic realization theory, and show
that the problem has deep connections with operator systems theory, yielding
possible insight to the lifting problem in quotient operator systems. Our
results have potential applications in quantum machine learning,
device-independent characterization and reverse-engineering of stochastic
processes and quantum processors, and dynamical processes with quantum memory.
