Correcting Measurement-Induced Endogeneity in Quantum Systems via Physics-Informed Latent Constraints

We present a novel inference framework to address measurement-induced endogeneity in quantum systems, a phenomenon where detector efficiency is coupled to the underlying quantum state. We establish that such state-dependent detection failures are formally equivalent to the statistical mechanism of Missing Not At Random (MNAR), which fundamentally obscures genuine quantum correlations and prevents the empirical verification of Bell-type inequalities (S > 2) under high-dimensional noise. To counteract this, we develop a physics-informed architecture that integrates a Conditional Variational Autoencoder (CVAE) with Penalized Empirical Likelihood (PEL). By embedding the Hilbert-Schmidt manifold directly into the latent space, our model strictly enforces the physical axioms of positive semidefiniteness (ˆρ ⪰ 0) and unit trace (Tr(ˆρ) = 1). The core of our methodology lies in its ability to decouple state-dependent selection bias from the intrinsic quantum signal. We demonstrate that this framework successfully restores hidden correlations, enabling the observation of Bell violations in regimes where conventional error mitigation techniques fail. Furthermore, we provide rigorous theoretical guarantees, including the asymptotic normality of the estimator and an oracle inequality for high-dimensional state reconstruction. Our results offer a robust pathway for causal directionality inference even in the presence of non- Markovian decoherence, bridging the gap between quantum information theory and semi-parametric statistical inference.