Revising the quantum work fluctuation framework to encompass energy conservation

Work is a process-based quantity, and its measurement typically requires interaction with a measuring device multiple times. While classical systems allow for non-invasive and accurate measurements, quantum systems present unique challenges due to the influence of the measuring device on the final value of work. As recent studies have shown, among these challenges is the impossibility of formulating a universal definition of work that respects energy conservation for coherent quantum systems and is compatible with the Jarzynski equality - a fluctuation relation linking the equilibrium free energy difference to the non-equilibrium work. Here we overcome this challenge by introducing a genuinely quantum, positive correction to the Jarzynski equality stemming from imposing energy conservation. When sufficiently large, this correction forces quantum work to violate the second law more often. Moreover, we construct modified two-point measurement (TPM) schemes for work that ensure energy conservation for coherent quantum states and align with our quantum-corrected fluctuation relation. We further underscore the practicality and effectiveness of these schemes by providing detailed circuit implementations for them.