  While the fluctuation theorem in classical systems has been thoroughly
generalized under various feedback control setups, an intriguing situation in
quantum systems, namely under continuous feedback, remains to be investigated.
In this work, we derive the generalized fluctuation theorem under continuous
quantum measurement and feedback. The essence for the derivation is to newly
introduce the operationally meaningful information, which we call
quantum-classical-transfer (QC-transfer) entropy. QC-transfer entropy can be
naturally interpreted as the quantum counterpart of transfer entropy that is
commonly used in classical time series analysis. We also verify our theoretical
results by numerical simulation and propose an experiment-numerics hybrid
verification method. Our work reveals a fundamental connection between quantum
thermodynamics and quantum information, which can be experimentally tested with
artificial quantum systems such as circuit quantum electrodynamics.
