  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
intuitive 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. Here we introduce an information
theoretic paradigm together with an intuitive geometric picture for decoding
incompatible observables, starting from two simple ideas: Every observable can
only provide limited information and information is monotonic under data
processing. 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.
