What are the necessary and sufficient conditions for a linear superoperator to be Unitary or Self-adjoint with respect to the Hilbert-Schmidt inner product $\left(\hat{A},\hat{B}\right)=Tr\left(\hat{A}^{\dagger}\hat{B}\right)$?

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    $\begingroup$ It might be a stupid question, but in what they should this differ from "ordinary Hilbert spaces"? In other words, AFAIK, the Hilbert-Schmidt space for a given Hilbert space $H$ is itself a Hilbert space (with the said inner product). So why would the notion of unitarity and self-adjointness change? $\endgroup$ Mar 6 at 14:12
  • $\begingroup$ The notion doesn't change, of course. What I mean is that superoperators generally involve operators, like the Liouville-von Neumann $\mathcal{L}=[\hat{H},]$ that gives time evolution to density operators. It is easy to see that $\exp[i\mathcal{L}]$ is unitary, but are all unitary superoperators of this form? $\endgroup$
    – AndresB
    Mar 6 at 15:15
  • $\begingroup$ Background. $\endgroup$ Mar 6 at 21:00


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