  Accurate modeling of quantum systems interacting with environments requires
addressing non-unitary dynamics, which significantly complicates computational
approaches. In this work, we enhance an open quantum system (OQS) theory using
density-matrix purification, enabling a unitary description of dynamics by
entangling the system with an environment of equal dimension. We first
establish the connection between density-matrix purification and conventional
OQS methods. We then demonstrate the standalone applicability of purification
theory by deriving system-environment interactions from fundamental design
principles. Using model systems, we show that the purification approach extends
beyond the complete positivity condition and effectively models both Markovian
and non-Markovian dynamics. Finally, we implement density-matrix purification
on a quantum simulator, illustrating its capability to map non-unitary OQS
dynamics onto a unitary framework suitable for quantum computers.
