  We employ the quantum jump trajectory approach to construct a systematic
framework to study the thermodynamics at the trajectory level in a
nonequilibrium open quantum system under discrete feedback control. Within this
framework, we derive quantum versions of the generalized Jarzynski equalities,
which are demonstrated in an isolated pseudospin system and a coherently driven
two-level open quantum system. Due to quantum coherence and measurement
backaction, a fundamental distinction from the classical generalized Jarzynski
equalities emerges in the quantum versions, which is characterized by a large
negative information gain reflecting genuinely quantum rare events. A possible
experimental scheme to test our findings in superconducting qubits is
discussed.
