In spite of its fundamental importance in quantum science and technology, the experimental certification of nonclassicality is still a challenging task, especially in realistic scenarios where losses and noise imbue the system. Here, we present the first experimental implementation of the recently introduced phase-space inequalities for nonclassicality certification, which conceptually unite phase-space representations with correlation conditions. We demonstrate the practicality and sensitivity of this approach by studying nonclassicality of a family of noisy and lossy quantum states of light. To this end, we experimentally generate single-photon-added thermal states with various thermal mean photon numbers and detect them at different loss levels. Based on the reconstructed Wigner and Husimi Q functions, the inequality conditions detect nonclassicality despite the fact that the involved distributions are nonnegative, which includes cases of high losses (93%) and cases where other established methods do not reveal nonclassicality. We show the advantages of the implemented approach and discuss possible extensions that assure a wide applicability for quantum science and technologies.
If the phase space-based Glauber-Sudarshan distribution,
Pϱ, has negative values the quantum state,
ϱ, it describes is nonclassical. Due to
P's singular behavior, this simple criterion is impractical to use. Recent work has presented a general, sensitive, and noise-tolerant certification functional for the detection of nonclassical behavior of quantum states
ϱ. There, it was shown that when this functional takes on negative values somewhere in phase space,
ξP(x,p)<0, this is sufficient to certify the nonclassicality of a state. Here we give examples where this certification fails. We investigate states which are known to be nonclassical but the certification function is nonnegative,
ξP(x,p)≥0, everywhere in phase space. We generalize
ξ, giving it an appealing form,
S, which allows for slight improvements in certification, but
S also fails for mixed very weakly nonclassical states. More important than a slight improvement in sensitivity of
S over
ξ is that we showed how very sensitive
ξ and
S are, and more important still is our simple derivation of
S allowing us to generalize
ξ and
S to multiple modes.
We derive a family of inequalities involving different phase-space distributions of a quantum state which have to be fulfilled by any classical state. The violation of these inequalities is a clear signature of nonclassicality. Our approach combines the characterization of nonclassical effects via negativities in phase-space distributions with inequality conditions usually being formulated for moments of physical observables. Importantly, the obtained criteria certify nonclassicality even when the involved phase-space distributions are non-negative. Moreover, we show how these inequalities are related to correlation measurements. The strength of the derived conditions is demonstrated by different examples, including squeezed states, lossy single-photon states, and even coherent states.
We devise a method to certify nonclassical features via correlations of phase-space distributions by unifying the notions of quasiprobabilities and matrices of correlation functions. Our approach complements and extends recent results that were based on Chebyshev's integral inequality \cite{BA19}. The method developed here correlates arbitrary phase-space functions at arbitrary points in phase space, including multimode scenarios and higher-order correlations. Furthermore, our approach provides necessary and sufficient nonclassicality criteria, applies to phase-space functions beyond s -parametrized ones, and is accessible in experiments. To demonstrate the power of our technique, the quantum characteristics of discrete- and continuous-variable, single- and multimode, as well as pure and mixed states are certified only employing second-order correlations and Husimi functions, which always resemble a classical probability distribution. Moreover, nonlinear generalizations of our approach are studied. Therefore, a versatile and broadly applicable framework is devised to uncover quantum properties in terms of matrices of phase-space distributions.