Sliced Wasserstein Meets Quantum Optics: Provable Wavefunctions Tomography with Scarce Noisy Measurements

Quantum state tomography of pure non-Gaussian states is fundamentally limited by scarce and noisy data due to measurement-induced collapse and experimental drift. We introduce a novel reconstruction framework that parameterizes the wavefunction directly via an overparameterized neural network, trained using rank-1 factored gradient descent. We rigorously prove that this formulation's loss landscape is devoid of spurious local minima and strict saddles, guaranteeing all critical points are global minima. By utilizing the max-sliced Wasserstein-1 distance, our loss perfectly maps to the structure of homodyne measurements, structurally preventing gradient vanishing in low-shot regimes. On noisy two-mode cat states, our method achieves continuous fidelity improvements, successfully bypassing the noise-floor plateaus that limit standard iterative maximum-likelihood estimation.