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Relation-First Reconstruction of Qubit Factors: Exact-Commutant Discontinuity and Soft-Complement Rigidity

Oliver Tuma

Published Sep 11, 2026• 0 citations
Publisher

Abstract

We study the reconstruction of a qubit factor from global operator relations without presupposing a tensor-product structure. Given two Hermitian operators (A,B) on an unpartitioned finite-dimensional Hilbert space, we quantify how approximate involution and anticommutation relations,[|A^2-I|,\qquad |B^2-I|,\qquad |{A,B}|,]control the existence of a nearby exact same-space qubit representation. The construction is explicit and yields quantitative upper and lower bounds relating observable relation defects to distance from an exact marked qubit factor. A central obstruction is then exhibited: arbitrarily small perturbations of an exact qubit-factor pair can generate the full ambient matrix algebra, causing the exact commutant of the raw operators to collapse discontinuously from the complementary matrix factor to the scalars. Thus exact-algebra or exact-commutant reconstruction from noisy operators is intrinsically unstable. Despite this discontinuity, the complementary factor remains quantitatively recoverable from a soft commutant. For the commutator map associated with an exact rounded pair, we determine the complete singular-value structure and use its fixed spectral separation to control the low-singular-value subspace of the original unrounded operators. If[\rho=2\sqrt{p^2+q^2}<1,]where (p) and (q) bound the same-space rounding errors, the reconstructed complementary subspace satisfies[\left|P_{\widehat{\mathcal C}}-P_{\mathcal A’}\right|_{2\to2}\le\frac{\rho}{2-\rho},]while the observed spectral gap is at least (2-2\rho). Composing the rounding bounds with the relation defects gives an explicit relation-defect-to-complement certificate. The results separate three notions that coincide only in the noiseless idealization: exact generated algebra, exact commutant, and robustly recoverable complementary factor structure. The paper also clarifies the distinction between factor existence and tensor-product-structure uniqueness and positions the result relative to stability of approximate qubit relations, almost-commuting operator algebras, finite-dimensional (C^*)-algebra stability, and numerical commutant reconstruction.

DOI
10.5281/zenodo.22708028

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