Choi-polynomial certificates for witness-detected local-unitary orbits and maximal quantum f -divergences

Abstract We develop a Choi-polynomial method for studying witness-detected parts of local-unitary orbits through maximal quantum f -divergences. A finite family of positive maps gives Choi-matrix inequalities along the local orbit of a bipartite state; if the maps are positive but not completely positive, indecomposable, or k-positive, these inequalities certify entanglement, PPT entanglement, or Schmidt-number constraints. We prove that the witness constraints are pullbacks of Choi polynomials and hence define compact basic semialgebraic subsets of the local unitary group. For finite atomic approximants of the operator-convex representation of f , the resulting minimization models admit Schur-complement semidefinite representations and KKT conditions with explicit Choipolynomial multipliers. Applications give positive divergence gaps for separable targets, PPT-entangled regions and Schmidt-number regions, and certified locality gaps between full-unitary, local-unitary and witness-detected local optima.