Cones of ball-ball separable elements

Let B1, B2 be balls in finite-dimensional real vector spaces E1, E2, centered around unit length vectors v1, v2 and not containing zero. An element in the tensor product space E1 ⊗ E2 is called B1 ⊗B2-separable if it is contained in the convex conic hull of elements of the form w1 ⊗w2, where w1 ∈ B1, w2 ∈ B2. We study the cone formed by the separable elements in E1⊗E2. We determine the largest faces of this cone via a description of the extreme rays of the dual cone, i.e. the cone of the corresponding positive linear maps. We compute the radius of the largest ball centered around v1⊗v2 that consists of separable elements. As an application we obtain lower bounds on the radius of the largest ball of separable unnormalized states around the identity matrix for a multi-qubit system. These bounds are approximately 12% better than the best previously known. Our results are extendible to the case where B1, B2 are solid ellipsoids.