Einstein-Podolsky-Rosen steering and the steering ellipsoid

The question of which two-qubit states are steerable (i.e. permit a demonstration of EPR-steering) remains open. Here, a strong necessary condition is obtained for the steerability of two-qubit states having maximally-mixed reduced states, via the construction of local hidden state models. It is conjectured that this condition is in fact sufficient. Two provably sufficient conditions are also obtained, via asymmetric EPR-steering inequalities. Our work uses ideas from the quantum steering ellipsoid formalism, and explicitly evaluates the integral of $\boldsymbol n/(\boldsymbol n^\intercal A\boldsymbol n)^2$ over arbitrary unit hemispheres for any positive matrix $A$.