Let’s forget everything again. We have a reference device characterized by some bistochastic . We suppose that is a Born matrix, satisfying , where is the matrix of all 1’s. As we’ve learned this implies that for , If we want , we must have . We observe that and , which projects on . For any observable , we assume a lower bound on the variance which is linear in and quadratic in . We thus have some three-index tensor such that or Let . We want to be symmetric in the first two indices so that is symmetric. Since on , and projects onto that subspace, we have iff is a valid state. Just as we assumed a particular simple form for , we assume a particular simple form for , namely Let us work out the matrix elements of . First, so that and since , this simplifies to or Let , since . We obtain at last In the case of quantum theory according to a 3-design, , , , . Then , and so or which is precisely the matrix which we’ve derived elsewhere, and in particular , so that .