Nonclassicality tests by classical bounds on the statistics of multiple outcomes

We derive simple practical tests revealing the quantum nature of states by the violation of classical upper bounds on the statistics of multiple outcomes of an observable. These criteria can be expressed in terms of the Kullback-Leibler divergence (or relative entropy). Nonclassicality tests for multiple outcomes can be satisfied by states that do not fulfill the corresponding single-outcome criteria.

Figure 1

Plot of $P(\mathit{n}|{\Phi }_{t,c})$ for a mixed thermal state (solid line) and for pure coherent states (dashed line) as functions of the mean number of photons $\mathrm{n̄}$ for the statistics of photon-number measurement in squeezed vacuum with $r=1$.

Figure 2

Kullback-Leibler divergence $K(\rho ||{\Phi }_n)$ comparing the actual observed statistics $P(\mathit{n}|\rho )$ and the closest statistics compatible with classical physics $P(\mathit{n}|{\Phi }_n)$ given by Eqs. (29) and (30), as a function of $p(1|\rho )$ showing nonclassicality $K(\rho ||{\Phi }_n)>0$ for all $p(1|\rho )\ne 1$.