From purification to unrestricted effects: compression, local tomography, and reconstruction

This is the authors' abstract. We don't add key points for this paper.

In a finite-dimensional operational theory, the allowed effects can be a proper subset of the positive functionals bounded by the deterministic effect. We study when purification prevents this restriction. Starting with causality, pure products and closed physical effect sets, we obtain exact binary steering from an informative, possibly noisy pointer, without assuming perfectly distinguishable records. Steering identifies the effect cone on a marginal support with the complementary state face and saturates the physical effect interval. These support identities yield two routes to non-restriction. Ideal compression, or suitable balanced or countable support hypotheses, forces the full dual effect cone; compression on an open set also gives homogeneity and hence, by an existing classification, Jordan state cones. Independently, purity preservation and domination of each pure bipartite state by a positive multiple of the product of its marginals force non-restriction without compression or any bound on purifying partners. Local tomography implies this domination property; real quantum theory shows that the latter is strictly weaker. This proof treats arbitrary faces and establishes injectivity of pure processes that vanish on no nonzero state. We apply the criteria to restricted balls, polytopes, quantum sectors and cones of even powers, while exhibiting limits of support-only arguments. For quantum state spaces, a single closed informative pointer suffices for every finite measurement, with an explicit approximation bound before closure. Finally, with local tomography and finite conditioned control, pure projections onto marginal supports yield a rank bound for arbitrary pure ensembles. Every pair of pure states then lies in a rank-two face, whose Euclidean geometry proves symmetry of transition probabilities. This supplies a linear self-duality and the filter hypotheses of an existing Jordan representation theorem, completing the reconstruction of complex quantum instruments on the system dimensions actually present without adding transition symmetry as an axiom. An explicit open-effect theory shows why closedness cannot simply be dropped. The reconstruction separates established ingredients from the support and effect-availability arguments developed here. Preparation note: This preprint builds on Yoann Dumont’s research programme, earlier manuscript, and research notes. ChatGPT assisted with mathematical exploration, proof development and checking, comparison with these prior materials, bibliographic searches, English drafting, and typesetting. AI-assisted checks do not constitute independent peer review. As author, Yoann Dumont takes responsibility for the claims, references, and any necessary corrections to the published text.