Quantum Information Erasure Under Sequential Weak Measurements: Empirical Manifestation of the Quantum Cramér-Rao Bound in Physics-Informed Neural Networks

Abstract Reconstructing the unobservable dynamics of quantum many-body systems from sequential weak measurements presents fundamental information-theoretic challenges due to measurement back-action. In this study, we introduce a Physics-Informed Neural Network (PINN) designed for continuous quantum state filtering on a 14-qubit Transverse Field Ising Model (TFIM). We demonstrate that while unconstrained models can exploit thermodynamic memorization, enforcing causal masking and Hamiltonian differential constraints forces the network to synthesize true unitary evolution. Under these rigorous physical constraints, we observe that the predictive accuracy for non-commuting transverse magnetization invariably reaches a strict empirical asymptote at $R^2 \approx 0.39$. We establish that this ceiling is not an architectural bottleneck, but the direct empirical manifestation of the Quantum Cramér-Rao Bound (QCRB), proving that deep learning models remain fundamentally bound by the Heisenberg uncertainty principle. Key Findings The Heisenberg Wall: We identify a fundamental predictive ceiling ($R^2 \approx 0.39$) for non-commuting observables in continuous quantum state filtering. Physics-Informed Architecture: We demonstrate that enforcing Hamiltonian consistency and causal masking eliminates "identity mapping" artifacts, allowing for true phenomenological simulation. Information Erasure: Our results map the neural network's performance bound to the residual information capacity dictated by the Quantum Cramér-Rao Bound. Keywords Physics-Informed Neural Networks; Quantum Many-Body Systems; Transverse Field Ising Model; Sequential Weak Measurements; Quantum Cramér-Rao Bound; Quantum State Filtering.