A simple test of quantumness for a single system

Abstract

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 $0\leq <\psi|A|\psi>\leq <\psi|B|\psi>\leq 1$. If there exists a state $\phi$ 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 $(A,B,\phi)$ in the case of a qubit is given.