Proposition 9 — superposition is non-unique decomposition of mixed states

Let $H$ be a complex inner product space, $\psi\in H$ a unit vector, and $(\phi_j)_{j\in J}$ a finite family of unit vectors with $\phi_j\neq\psi$ for all $j$. Then the following are equivalent:

1. $\psi\in\operatorname{span}\{\phi_j : j\in J\}$;
2. there exist $n\ge0$, unit vectors $\psi_0=\psi,\psi_1,\dots,\psi_n$, and weights $a_0,\dots,a_n>0$, $b_j>0$ with $\sum_k a_k=1$, $\sum_j b_j=1$ and

A pure state is a superposition of given pure states exactly when some mixture of those states can also be prepared as a mixture that includes the given state.

Formalization Note The paper says "complex vector space", but normalization and the operators $|\psi\rangle\langle\psi|$ require an inner product, so $H$ is a complex inner product space (not necessarily complete). The distinguished vector is indexed by $0$.