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I know that phase shift gates like $S$ and $T$ are not hermitian operators. But are the $S^\dagger$ and $T^\dagger$ gates non-hermitian too?

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A matrix $A$ is a Hermitian if and only if

$$ A = A^\dagger$$

So if $A^\dagger$ is Hermitian then that means $A^\dagger = (A^\dagger)^\dagger = A$ and so of course $A$ is hermitian.

By the way, by this definition you can see that the diagonal elements of Hermitian matrix must be real.

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