In Quantum Optics, the widely accepted definition of nonclassicality is based on the P function of Glauber and Sudarshan. When it fails to be interpreted as a classical probability density, the corresponding quantum state is said to be a nonclassical one. Here we present the first reconstruction of a nonclassical P function of a single-photon added thermal state. We also consider the nonclassical properties of general spacegtime dependent correlation functions of radiation fields. For the detection of these correlation functions, a balanced homodyne correlation technique was proposed. It is shown that the measurable correlation functions also allow one to completely characterize bipartite entangled quantum states with a negative partial transposition. Finally, we present a method for identifying general bipartite entanglement for continuous variables.
The characterization of nonclassicality of quantum states in terms of measurable quadrature distributions is studied, on the basis of recently derived necessary and sufficient conditions for the failure of the Glauber–Sudarshan P -function to be a probability density. When the noise-subtracted quadrature distribution, that is the marginal of the P -function, is not a probability density, the same is true for the Glauber–Sudarshan P -function. Inversely, for a complete characterization of a nonclassical P -function one needs phase-sensitive criteria that connect quadrature distributions for different phases. Even for phase-insensitive quantum states, phase-sensitive nonclassicality conditions are needed, as is illustrated for one-photon added thermal states.
We report the experimental reconstruction of a nonclassicality quasiprobability for a single-photon-added thermal state. This quantity has significant negativities, which is necessary and sufficient for the nonclassicality of the quantum state. Our method exhibits several advantages compared to the reconstruction of the
P function, since the nonclassicality filters used in this case can regularize the quasiprobabilities as well as their statistical uncertainties. A priori assumptions about the quantum state are therefore not necessary. We also demonstrate that, in principle, our method is not limited by small quantum efficiencies.
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.
The algebraic quantification of nonclassicality, which naturally arises from the quantum superposition principle, is related to properties of regular nonclassicality quasiprobabilities. The latter are obtained by non-Gaussian filtering of the Glauber-Sudarshan
P function. They yield lower bounds for the degree of nonclassicality. We also derive bounds for convex combinations of Gaussian states for certifying quantum non-Gaussianity directly from the experimentally accessible nonclassicality quasiprobabilities. Other quantum-state representations, such as
s-parametrized quasiprobabilities, insufficiently indicate or even fail to directly uncover detailed information on the properties of quantum states. As an example, our approach is applied to multi-photon-added squeezed vacuum states.
The definition of nonclassical states in quantum optics by the nonpositivity of their Glauber–Sudarshan quasiprobability is investigated and it is shown that it hides some serious problems. It leads to a subdivision of squeezed thermal states into classical and nonclassical states which is difficult to interpret physically by some qualitatively different behaviour of the states. Nonclassical states are found in arbitrarily small neighbourhoods of every classical state that is illustrated by a very artificial modified thermal state. The observability of the criterion in comparison to that for nonclassicality of states determined by the nearest Hilbert–Schmidt distance to a class of reference states is discussed. The behaviour of the nonclassicality of states in models of phase-insensitive processes of damping and amplification is investigated and it is found that every nonclassical state eventually makes a transition to a classical state. However, this is not specific for the negativities or singularities of the Glauber–Sudarshan quasiprobability and is found in similar form for other quasiprobabilities, for example, for the Wigner quasiprobability. We discuss in quite general form some defects of the Glauber–Sudarshan quasiprobability if compared with classical distribution functions over the phase space, in particular the failure of an earlier advertised superposition formula.