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Cipriani and Sauvageot have shown that for any $L^2$-generator $L^{(2)}$ of atracially symmetric quantum Markov semigroup on a C*-algebra $\mathcal{A}$there exists a densely defined derivation $\delta$ from $\mathcal{A}$ to aHilbert bimodule $H$ such that $L^{(2)}=\delta^*\circ \overline{\delta}$. Herewe show that this construction of a derivation can in general not begeneralised to quantum Markov semigroups that are symmetric with respect to anon-tracial state. In particular we show that all derivations to Hilbertbimodules can be assumed to have a concrete form, and then we use this form toshow that in the finite-dimensional case the existence of such a derivation isequivalent to the existence of a positive matrix solution of a system of linearequations. We solve this system of linear equations for concrete examples usingMathematica to complete the proof.

On the existence of derivations as square roots of generators of state-symmetric quantum Markov semigroups