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 $ψ$ their mean values satisfy $0\leq <ψ|A|ψ>\leq <ψ|B|ψ>\leq 1$. If there exists a state $ϕ$ such that the second moments fulfill the inequality $<ϕ|A^2|ϕ> ><ϕ|B^2|ϕ>$ then the system cannot be described by the classical probabilistic scheme. An example of an optimal triple $(A,B,ϕ)$ in the case of a qubit is given.

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