  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.
