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This study derives a non-destructive quantum R\'enyi-Jarzynski equality that links nonequilibrium work to equilibrium free energy changes without destroying quantum coherence. By employing a resource-theoretic framework, the authors quantify a finite bath's drift from equilibrium under non-adiabatic conditions using the R\'enyi $k$-divergence, revealing insights into quantum optimal control. The findings highlight that minimizing bath drift in quantum systems necessitates the generation of system-bath entanglement, particularly when drive parameters vary across bath energy levels.
Minimizing bath drift in quantum systems may require generating entanglement, challenging conventional approaches to quantum control.
The Jarzynski equality provides a strict link between nonequilibrium work and equilibrium free energy changes. Its typical quantum formulations, however, rely on measurement protocols that destroy coherence. In this Letter, we use the resource-theoretic approach to derive a non-destructive quantum Jarzynski equality conditioned on the outcomes of an arbitrary bath observable. This yields the R\'enyi-Jarzynski equality, which quantifies a finite bath's drift from equilibrium under a non-adiabatic drive via the R\'enyi $k$-divergence. We further demonstrate that the R\'enyi-Jarzynski equality provides a tunable cost function for quantum optimal control problems where minimizing bath drift is desired, such as state preparation and gate design, enabling the minimization of cross-talk in finite quantum systems. Our toy model exhibits a transition between competing minima for some critical value of $k$, illustrating how the R\'enyi order tunes sensitivity to different regions of a bath distribution. Strikingly, when drive parameters vary across bath energy levels, minimizing bath drift requires generating system-bath entanglement.
