Boundary Geometry Turns Entanglement into Steering

Abstract

Quantum entanglement does not necessarily imply Einstein-Podolsky-Rosen steering. We identify a \emph{boundary mechanism} that closes this gap when an entangled state meets the boundary of the trusted state space in a nondegenerate way. The mechanism is local: a projective assemblage that approaches a boundary contact with first-order tangential coherence but only second-order inward defect cannot be reproduced by any finite-measure local-hidden-state model. In finite dimensions this yields a support-kernel criterion for an arbitrary fixed steering cut: if a trusted conditional state is rank-deficient and its first-order block couples its support to its kernel, then the state is both NPT (negative partial transpose) and projectively steerable. The two-qubit product-null sector provides the minimal transparent realization. A product vector in the kernel gives the pure boundary contact, and a single coherence entry simultaneously serves as the tangential displacement, the NPT minor, and the steering obstruction. Hence, on this boundary, entanglement collapses to two-way projective steering -- including the genuinely mixed rank-three branch and the rank-two case as a low-dimensional instance. We also provide a filtered standard form, an explicit rank-three Cholesky parametrization, and a compact boundary certificate.