Necessary and Sufficient Condition for Quantum State-Independent Contextuality

We solve the problem of whether a set of quantum tests reveals state-independent contextuality and use this result to identify the simplest set of the minimal dimension. We also show that identifying state-independent contextuality graphs R. Ramanathan and P. Horodecki, Phys. Rev. Lett. 112, 040404 (2014) is not sufficient for revealing state-independent contextuality.