Boundary Geometry Turns Entanglement into Steering

Abstract: Entanglement does not in general imply Einstein-Podolsky-Rosen steering. We identify a boundary-geometric mechanism that closes this gap on product-null boundary strata of two-qubit state space, where Bob's conditional states touch the boundary of the Bloch ball. The key obstruction is local: if a projective assemblage approaches a Bloch-sphere boundary contact with a first-order tangential displacement but only a second-order inward defect, then no finite-measure local-hidden-state model can reproduce it. For two-qubit states with a product vector in the kernel, this boundary contact is exactly the tangency of Bob's steering ellipsoid to the Bloch sphere. At such a product-null tangency, a single tangential coherence controls both partial-transpose negativity and the boundary-contact scaling obstruction. The same boundary minor gives a compact experimental witness: once the product-null contact is verified or guaranteed, the tangential coherence supplies the steering signal. Consequently, every entangled two-qubit rank-two state, and every entangled rank-three state whose null space is spanned by a product vector, is two-way projectively steerable. The same boundary idea extends to arbitrary steering cuts: the Bloch-sphere contact is replaced by a rank-deficient trusted conditional state, and support-kernel first-order coherence implies both NPT entanglement and projective steering.