Thermodynamic Approach to Quantifying Incompatible Instruments

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. Finally, we show that incompatibility signatures must vanish in non-Markovian thermalization.