= Solution
The operator $D$ vanishes on the orthogonal complement of $S=\operatorname{span}\{|\alpha_0\rangle,|\alpha_1\rangle\}$, so it has rank at most two. The states are distinct rays, hence they are linearly independent and $S$ has dimension two. In an orthonormal basis of $S$, its determinant is
$$
\det(D|_S)=-p_0p_1\left(1-|\langle\alpha_0|\alpha_1\rangle|^2\right)<0.
$$
\b[Thus its two eigenvalues on $S$ are nonzero and have opposite signs. All remaining eigenvalues are zero.]
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