Passive Observer Fibers in Two-Level Quantum Dynamics: Local Rigidity, Global Ambiguity, and Controlled Active Completion

Abstract

We study identifiability of finite-dimensional Hermitian dynamics from complete site-to-site transition probabilities. On the two-level spectral stratum $H = \alpha I + \beta P$, passive probabilities factor through a simple projector-modulus observer. We give the exact static/visible classification of this observer and then exhibit two sharply different forms of nonidentifiability. For a rank-three harmonic projector on seven sites, one passive law has at least four pairwise diagonal-gauge-inequivalent preimages, while an exact cyclotomic tangent calculation proves local modulus rigidity at each of the four constructed branches. In contrast, a standard one-parameter qutrit SIC family gives a continuous family of gauge-inequivalent zero-diagonal Hamiltonians with identical complete passive probabilities. We formulate the fixed-diagonal tangent map and an equivalent Bargmann-weighted triangle-curl system, making the local/global contrast explicit. Finally, under a calibrated isolated edge-phase intervention model, two cubic-response settings recover a complex rooted triangle invariant and at most $2(n-2)$ settings reconstruct the full diagonal-gauge class on complete support. The results separate passive nonidentifiability, local differential rigidity, and active identifiability without requiring a generic rigidity theorem for projector-modulus fibers.