  We prove that the generator of the $L^2$ implementation of a KMS-symmetric
quantum Markov semigroup can be expressed as the square of a derivation with
values in a Hilbert bimodule, extending earlier results by Cipriani and
Sauvageot for tracially symmetric semigroups and the second-named author for
GNS-symmetric semigroups. This result hinges on the introduction of a new
completely positive map on the algebra of bounded operators on the GNS Hilbert
space. This transformation maps symmetric Markov operators to symmetric Markov
operators and is essential to obtain the required inner product on the Hilbert
bimodule.
