  Quantum detailed balance conditions and quantum fluctuation relations are two
important concepts in the dynamics of open quantum systems: both concern how
such systems behave when they thermalize because of interaction with an
environment. We prove that for thermalizing quantum dynamics the quantum
detailed balance conditions yield the validity of a quantum fluctuation
relation (where only forward-time dynamics is considered). This implies that to
have such a quantum fluctuation relation (which in turn enables a precise
formulation of the second law of thermodynamics for quantum systems) it
suffices to fulfill the quantum detailed balance conditions. We, however, show
that the converse is not necessarily true; indeed, there are cases of
thermalizing dynamics which feature the quantum fluctuation relation without
satisfying detailed balance. We illustrate our results with three examples.
