  We investigate the Kondo effect in an open quantum system, motivated by
recent experiments with ultracold alkaline-earth(-like) atoms. Because of
inelastic collisions and the associated atom losses, this system is described
by a complex-valued Kondo interaction and provides a non-Hermitian extension of
the Kondo problem. We show that the non-Hermiticity induces anomalous reversion
of renormalization-group flows which violate the $g$-theorem due to
non-unitarity and produce a quantum phase transition unique to non-Hermiticity.
Furthermore, we exactly solve the non-Hermitian Kondo Hamiltonian using a
generalized Bethe ansatz method and find the critical line consistent with the
renormalization-group flow.
