= Solution
Two finite-dimensional <Hermitian matrices> are related by <unitary conjugation> if and only if they have the same multiset of <eigenvalues>. Necessity follows because conjugation preserves the <characteristic polynomial>; sufficiency follows by mapping orthonormal eigenbases in the <spectral theorem>. Consequently
$$
\boxed{\rho_0\sim\rho_2,\qquad \rho_1\sim\rho_3,}
$$
and no member of the first pair is equivalent to a member of the second. For an explicit example, the <Hadamard gate> $H$ gives $H\rho_0H^\dagger=\rho_2$ and $H\rho_3H^\dagger=\rho_1$. Multiplying $H$ by a scalar phase does not change either conjugation.
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