# Construction of optimal ensemble to show quantum steerability

In Wiseman et al. (2007), in the process of deriving necessary and sufficient conditions for the steerability of some classes of states, the authors show (lemma 1, page 3) how to construct an optimal ensemble $$F^\star=\{\rho_\xi^\star\mathscr P_\xi^\star\}$$ such that, if this ensemble cannot explain Bob's state via a local hidden state, then no other ensemble can.

More precisely, the context is that of Alice trying to convince Bob that she can steer his state by means of her performing measurements on her part of the system. In other words, we assume that Alice and Bob share some state $$W$$, and that Alice performs measurements $$A\in\mathscr M$$ ($$\mathscr M$$ denoting here the measurements that Alice can choose to perform) on her part of the system. If Alice performs measurement $$A$$ and gets result $$a$$, then Bob's state becomes $$\tilde\rho_a^A=\operatorname{Tr}_\alpha[W(\Pi_a^A\otimes I)],$$ where $$\operatorname{Tr}_\alpha$$ denotes the partial trace with respect to Alice's part of the system, and the tilde is used to remember that this is an unnormalised state.

If Bob can describe this state by means of some prior local hidden state $$\rho=\sum_\xi p_\xi\rho_\xi$$ as $$\tilde\rho_a^A=\sum_\xi\mathscr P(a|A,\xi)p_\xi\rho_\xi,\tag5$$ then he is not convinced that Alice can steer his system, as it would mean that he can describe his observations by simply assuming that what he is measuring is some local state $$\rho$$ that is not affected by Alice.

My question is about the aforementioned lemma that is proved in the paper: why is the ensemble $$F^\star$$ optimal?

The lemma is as follows:

Consider a group $$G$$ with unitary representation $$U_{\alpha\beta}(g)=U_\alpha(g)\otimes U_\beta(g)$$. Suppose that for every measurement $$A\in\mathscr M$$, outcome (that is, eigenvalue) $$a\in\lambda(A)$$, and group element $$g\in G$$, we have $$U_\alpha^\dagger(g)A U_\alpha(g)\in\mathscr M$$, and moreover $$\tilde\rho_a^{U_\alpha^\dagger(g)AU_\alpha(g)}=U_\beta(g)\tilde\rho_a^A U^\dagger_\beta(g).$$ Then there exists a $$G$$-covariant optimal ensemble, that is, one such that $$\forall g\in G$$ $$\{\rho_\xi^\star\mathscr P_\xi^\star\}=\{U_\beta(g)\rho_\xi^\star U_\beta^\dagger(g)\mathscr P_\xi^\star\}.$$

They then proceed to give an ensemble that they prove (not that I understand this proof, but that is another matter) does satisfy (5). Why is this then proving that if this ensemble does not satisfy (5) then no other ensemble can? I find this quite confusing, given that we just proved that it does satisfy (5).