Consider the set
$$\mathcal{D} = \left\{ \mathbf{A} \in \mathbb{C}^{2 \times 2} \mid \mbox{rank} (\mathbf{A}) = 1, \|\mathbf{A}\|_{F} = 1 \right\}$$
of all $2 \times2$ rank-$1$ matrices with unit norm in Frobenius sense. Is a rank-$2$ matrix $\mathbf{M} = \mathbf{X}_1 + \mathbf{X_2},$ with $\mathbf{X}_1, \mathbf{X}_2 \in \mathcal{D}$, constructed in a unique way?
I know that if we consider any linear combination of general two rank-one matrices, a rank-$2$ matrix can be constructed in infinitely many ways. But $\mathcal{D}$ is a subset of all $2 \times 2$ rank-one matrices.
Let $u,v$ be mutually orthogonal unit vectors in $\Bbb C^2$. Then $$ uu^* + vv^* = \pmatrix{u & v}\pmatrix{u^*\\v^*} = I $$ Note that $uu^*,vv^* \in \mathcal D$ however $u,v$ are chosen. So, $I$ is a matrix that can be constructed in infinitely many ways.