Entanglement transitions in a boundary-driven open quantum many-body system

We introduce a numerical framework for integrating Markovian dynamics on tree tensor operator ansatz states. This framework enables the simulation of both transient and steady-state regimes of systems governed by the Lindblad master equation, while preserving positivity of the density matrix and providing direct access to entanglement monotones. We demonstrate its capability to probe entanglement in open quantum many-body systems and to distinguish it from other correlations by studying a boundary-driven XXZ spin chain. Our analysis uncovers entanglement transitions driven by both the coupling to the environment and the anisotropy, revealing a striking connection between spatial entanglement scaling and spin current.