Identifiability limits and deep-learning-assisted reconstruction of rotational density matrices for symmetric-top molecules

Abstract

Recovering the rotational density matrix of a molecular ensemble from time-resolved angular distributions is central to understanding ultrafast rotational dynamics, yet the inverse problem is severely underdetermined. We analyze the forward operator that maps the density matrix of laser-aligned symmetric-top molecules to the angular distribution retrieved in pump–probe experiments and demonstrate through singular value decomposition that 74%–88% of the real density matrix unknowns lie in the null space for maximum angular momentum quantum numbers Jmax = 2–5. This rank deficiency is intrinsic to the measurement geometry and imposes a linear lower bound on reconstruction error: the minimum-norm least-squares (pseudoinverse) solution sets all null-space components to zero, establishing the best achievable error for any linear, unbiased estimator. Nonlinear constraints—positive semidefiniteness, trace conservation, and block symmetries—partially recover null-space information, but the residual error grows with Jmax, reaching 15%–44% for Jmax = 5. We present a two-stage pipeline in which a convolutional neural network trained on simulated data provides a warm start for the fast iterative shrinkage-thresholding algorithm. For both CF3I and CH3Cl across Jmax = 2–6, this approach reduces the Frobenius reconstruction error by 80%–99% relative to optimization from thermal equilibrium, maintaining stable errors of 0.5%–1.1% as Jmax increases. The pipeline is robust to data noise down to a 20 dB signal-to-noise ratio and operates ten times faster than the baseline and outperforms the maximum-entropy approach by a factor of 15–39×. The classical iterative quantum tomography algorithm becomes numerically unstable for Jmax ≥ 4, whereas the proposed method converges reliably at all tested truncation levels.