Measuring Nonclassicality of Bosonic Field Quantum States via Operator Ordering Sensitivity

We introduce a new distance-based measure for the nonclassicality of the states of a bosonic field, which outperforms the existing such measures in several ways. We define for that purpose the operator ordering sensitivity of the state which evaluates the sensitivity to operator ordering of the Renyi entropy of its quasiprobabilities and which measures the oscillations in its Wigner function. Through a sharp control on the operator ordering sensitivity of classical states we obtain a precise geometric image of their location in the density matrix space allowing us to introduce a distance-based measure of nonclassicality. We analyze the link between this nonclassicality measure and a recently introduced quantum macroscopicity measure, showing how the two notions are distinct.

Figure 1

Plots of the Wigner functions of ${\rho }_{2M}$ (solid line) and of ${\rho }_{\mathrm{even},M}$ (dashed line) as a function of $|\alpha |$ for $M=10$. Both states have comparable photon number: $\mathrm{Tr}({\rho }_{2M}{a}^{†}a)=M+\frac{1}{2}$, $\mathrm{Tr}({\rho }_{\mathrm{even},M}{a}^{†}a)=M+1$. The oscillations in the Wigner function of one are visibly much more pronounced than in the other, as detailed in the text.