= Solution
Each operator has <trace> one, so its <characteristic polynomial> is $\lambda^2-\lambda+\det\rho$. The first and last are diagonal; for the others, the symmetric and antisymmetric vectors diagonalize the matrix. Their <eigenvalues> are
$$
\boxed{\operatorname{spec}(\rho_0)=\operatorname{spec}(\rho_2)=\{3/4,1/4\},\qquad \operatorname{spec}(\rho_1)=\operatorname{spec}(\rho_3)=\{1,0\}.}
$$
For $\rho_1$, $(1,-1)^T/\sqrt2$ has <eigenvalue> one and $(1,1)^T/\sqrt2$ has <eigenvalue> zero. For $\rho_2$, those <eigenvalues> are $1/4$ and $3/4$, respectively. All four <density operators> are positive semidefinite, as these spectra confirm.
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