Minimal positive realizations: A survey

Minimal resources for exact simulation of quantum walks

“…In this work, we develop a general framework and foundational tools for a theory of CPTP model reduction for quantum dynamics. While finding minimal linear models can be done by relying on linear system theory, as soon as positivity constraints are imposed the problem becomes challenging, and still open even for classical linear systems [7]. In quantum engineering, effective reduced models have been proposed for control and filtering purposes, but their properties are hard to characterize [49], [50].…”

“…make the model physically admissible or satisfy some locality constraints. This problem, in general, proves difficult to solve: for example, model reduction for linear dynamical systems with positivity constraints is still a partially open issue [6], [7].…”

Externally Positive Linear Systems From Transfer Function Properties

“…To fully exploit these simplifying results, a general characterisation of the class of externally positive systems is required. Several seminal results have been obtained for positive systems, especially on the positive-realisation problem, whereby a state-space realisation with a positivity preserving input-state and state-output map is sought [4]. A complete solution to this problem was given in [14] which built upon the earlier results of [1] and [28] where observable and reachable cones were introduced.…”