Consider Problem 2.10 from Ballentine (paraphrased):
Show (by constructing an example depending on a continuous parameter) that this can be done in infinitely many ways.
I'm not sure how to proceed. I know from a theorem developed in the text that I can write any nonpure state operator $\rho$ as a nontrivial convex combination of other states $\rho^{(i)}$; that is, for valid states $\rho^{(i)}$ I can write $$\rho = \sum_1^n a_i \rho^{(i)}, a_i \in [0,1]$$
where, importantly, $n \geq 2$, since the theorem's conclusion is that the convex combination must be nontrivial.
I can generate at least one other way of decomposing $\rho$: namely, taking its spectral decomposition, choosing two "pure state" projectors $|\phi_i \rangle \langle \phi_i|$ and $|\phi_j \rangle \langle \phi_j|$ with eigenvalues $p_i$ and $p_j$, and defining $|\psi_i \rangle = \sqrt{p_i} |\phi_i \rangle + \sqrt{p_j} |\phi_j \rangle$ and $|\psi_j \rangle = \sqrt{p_i} |\phi_i \rangle - \sqrt{p_j} |\phi_j \rangle$. Forming the projectors shows that I can replace the initial two projectors. It's not clear to me how to generalize this example to show that there can indeed be infinitely many such decompositions (in this case I can generate something like ${n \choose 2}$ such decompositions using this strategy).
I am looking for an elementary extension of the example I gave, or at least something closely related (which uses an example depending on a continuous parameter).
