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Let $A$ and $B$ $2^n \times 2^n$ Hermitian matrices. What are sufficient and necessary conditions that they are equal up to some unitary, i.e. there exists $U$ such that $A = U B U^\dagger$?

The first immediate condition is that both share the same eigenspectrum as $U$ doesn't change eigenvalues. This naturally states that operator norms of the two operators are the same.

What are the other conditions if there's any?

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It holds precisely if they have the same spectrum: You already argue for necessity. For sufficiency: $A=VDV^\dagger$, $B=WDW^\dagger$, then $D=W^\dagger BW$ and thus $A=VW^\dagger DV^\dagger W$. Thus, $U=VW^\dagger$.

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