A simple test of quantumness for a single system

We propose a simple test of quantumness which can decide whether for the given set of accessible experimental data the classical model is insufficient. Take two observables $A,B$ such that for any state $\psi$ their mean values satisfy $0\le <\psi \mathrm{\mid }A\mathrm{\mid }\psi >\le <\psi \mathrm{\mid }B\mathrm{\mid }\psi >\le 1$. If there exists a state $\varphi$ such that the second moments fulfill the inequality $<\phi|A^2|\phi>

<\phi|B^2|\phi>$ then the system cannot be described by the classical probabilistic scheme. An example of an optimal triple $(A,B,\varphi )$ in the case of a qubit is given.