Engineered dynamical maps that combine not only coherent, but also unital and dissipative transformations of quantum states, have demonstrated a number of technological applications, and promise to be a beneficial tool also in quantum thermodynamic processes. Here, we exploit control of a spin qutrit to investigate energy exchange fluctuations of an open quantum system. The qutrit engineer dynamics can be understood as an autonomous feedback process, where random measurement events condition the subsequent dissipative evolution. To analyze this dynamical process, we introduce a generalization of the Sagawa-Ueda-Tasaki relation for dissipative dynamics and verify it experimentally. Not only we characterize the efficacy of the autonomous feedback protocol, but also find that the characteristic function of energy variations $G(eta)$ becomes insensitive to the process details at a single specific value of its argument. This allows us to demonstrate that a fluctuation theorem of the Jarzynski type holds for this general dissipative feedback dynamics, while previous relations were limited to unital dynamics. Moreover, in addition to the feedback efficacy, we find a witness of unitality associated with the fixed point of the dynamics.
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