Emergent Causality and Robust Estimation in Open Quantum-Compatible Systems under Non-Unitary Selection

We propose a novel framework for causal inference inspired by the process matrix formalism, where causal structure is not fixed but emerges through the act of observation. Unlike classical Directed Acyclic Graphs (DAGs), our approach treats the measurement process as a source of causal directionality, necessitating a symmetric, observer-aware inference architecture. We reinterpret Missing Not At Random (MNAR) data not as a nuisance, but as a \textit{non-unitary operation} within a causal network, reflecting back-action from the measurement apparatus. By mapping high-dimensional, low-sample-size (HDLSS) challenges to pseudo-density matrix reconstruction, we develop a selection-aware doubly robust estimator. This estimator integrates variational autoencoders (VAE) for latent state tomography and penalized empirical likelihood to achieve sparsity. We demonstrate that the fundamental limits of causal estimation under observer-induced perturbation are mathematically equivalent to decoherence-induced information loss, aligning our framework with the Quantum Cram\'er-Rao bound.