Information complementarity: A new paradigm for decoding quantum incompatibility

The existence of observables that are incompatible or not jointly measurable is a characteristic feature of quantum mechanics, which lies at the root of a number of nonclassical phenomena, such as uncertainty relations, wave-particle dual behavior, Bell-inequality violation, and contextuality.

However,  no simple  criterion  is available for determining  the compatibility of even two (generalized) observables, despite the overarching importance of this problem  and intensive efforts of many researchers over more than 80 years.

Starting from  the simple idea that every  observable can only provide limited  information and  that information is monotonic  under data processing, we introduce a powerful information theoretic  paradigm together with an intuitive geometric picture for decoding incompatible observables. By virtue of quantum estimation theory, we introduce a family of universal criteria for detecting incompatible observables and a natural measure of incompatibility, which are applicable to arbitrary number of arbitrary observables. Based on this framework, we derive a family of universal  measurement uncertainty relations, provide a simple information theoretic explanation of quantitative wave-particle duality, and offer new perspectives for understanding Bell nonlocality, contextuality, and quantum precision limit.