Let for . Let Then and where . Let and . Moreover, Suppose we demand commutativity: . We have from which we conclude that . Let us further assume self-duality for our reference states and effects so that invoking Bayes’ rule, we have so that . If we assume self-duality in general, then If , then In 3-design quantum theory, , , and , which gives .
Let us now consider , or If we further demand that , or , then and since , we have or , which fixes So by assuming , that is, , we can fix . By assuming , and for that and , we can fix and then . (Notice we haven’t said anything about the extremal states as defined by the variance restriction alone.) Indeed, now We can go further using by substituting in the expression for . Thus so that In 3-design quantum theory, and , and so . Can we keep going? No! For example, we also have so that which for 3-design quantum theory becomes , which is true! But the pure probability-assignments don’t live on -norm spheres. (Does the 2-norm and 3-norm sphere condition imply that ?)
It would be nice to fix in terms of . The next thing, however, to consider is the Jordan identity itself, . On the one hand, On the other hand, and so that and then And then it is a matter of calculation.