  Quantum incompatibility is referred as the phenomenon that some quantum
measurements cannot be performed simultaneously, and is also used in various
quantum information tasks. However, it is still a challenge to certify whether
a given set of multiple high-dimensional measurements respects a specific
structure of incompatibility. To address this problem, we propose a modified
quantum state discrimination protocol that decomposes complex compatibility
structures into pair-wise ones and employs noise robustness to witness
incompatibility structures. Our method is capable of detecting genuine $n$-wise
incompatibility and some specific general compatibility structures, as
demonstrated by our experimental verification of incompatibility structures of
$4$ mutually unbiased bases in a qutrit system. The experimental results show
that our approach is able to witness incompatibility structures, i.e., the
hierarchy of quantum incompatibility resources.

Quantum Physics