Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 60 / 1 / c

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. Consequentlyand no member of the first pair is equivalent to a member of the second. For an explicit example, the Hadamard gate $H$ gives $H{\rho }_0{H}^{†}={\rho }_2$ and $H{\rho }_3{H}^{†}={\rho }_1$. Multiplying $H$ by a scalar phase does not change either conjugation.