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We consider a thermodynamic framework to quantify instrument incompatibility via a resource theory subject to thermodynamic constraints. We use the minimal thermalization time needed to erase incompatibility’s signature to measure incompatibility. Unexpectedly, this time value is equivalent to incompatibility advantage in a work extraction task. Hence, both thermalization time and extractable work can directly quantify instrument incompatibility. 

Reference: Chung-Yun Hsieh and Shin-Liang Chen, Phys. Rev. Lett. 133, 170401 (2024).