Incompatibility of quantum devices is a useful resource in various quantum information theoretical tasks, and it is at the heart of some fundamental features of quantum theory. In the work [A. Mitra and M. Farkas, Phys. Rev. A 105, 052202 (2022)], we revise a notion of instrument compatibility that was already introduced in the literature, and we call this notion as traditional compatibility. Then, we discuss the notion of parallel compatibility and show that these two notions are inequivalent. We then argue that the notion of traditional compatibility has conceptual drawbacks and prove that parallel compatibility does not have such drawbacks. Hence, we propose to adopt parallel compatibility as the definition of the compatibility of quantum instruments. On other hand, in the work [A. Mitra, M. Farkas, arXiv:2209.02621 [quantph] (accepted in Phys. Rev. A)], we try to characterize and quantify parallel compatibility of quantum instruments.
