Operational Subsystem Identification from Global Quantum Processes: Similarity Obstructions, Two-Loop Rigidity, and Finite-Data Guarantees

Abstract

This preprint develops an operational identifiability theory for bipartite quantum subsystem structure without presupposing an embedded tensor-product structure. It first proves a strict boundary between simultaneous-similarity information obtainable from self-consistent global process probabilities and the multiplication, adjoint, positivity, and order structure required for physical quantum subsystems. For finite-dimensional unitary return processes, the paper establishes a sharp zero/one/two-loop hierarchy. A tree leaves the full tensor-product-structure manifold unconstrained; one compatible return process leaves a continuous ambiguity of dimension (m-1)(n-1); and two Haar-generic compatible returns identify the common factor algebra up to exchange of equal-dimensional factors. From informationally complete probability tables, the construction M_s=G^{-1}P^{(s)} yields a calibration-free common-commutant obstruction. A finite-data singular-value certificate can rule out every nontrivial bipartition of the global dimension simultaneously. In the exact generic compatible case, the ranks of the primitive commutant idempotents also determine the previously unknown factor dimensions. After an independently justified physicality lift and explicit C^*-factor tests, the framework provides local reconstruction and finite-shot error guarantees. An exact two-qubit example demonstrates that two processes may each be factorizable in some tensor-product structure while admitting no common factorization.