Necessary and sufficient conditions are derived that allow one to completely characterize the nonclassicality of an arbitrary quantum state of a harmonic oscillator. Starting from a theorem by Bochner we derive an infinite hierarchy of conditions for the characteristic functions of the quantum state. Following the hierarchy step by step, the nonclassicality can be classified with respect to increasing orders. The full hierarchy of conditions is completely equivalent to the most commonly accepted criterion for nonclassicality: the failure of the Glauber–Sudarshan P -function to be a probability density. Our reformulated version of the criterion is directly related to measurable quantities that can be observed in phase-sensitive measurements such as homodyne detection. It directly generalizes a criterion for nonclassicality (Vogel 2000 Phys. Rev. Lett. 84 1849) that has already been applied in experiments (Lvovsky and Shapiro 2002 Phys. Rev. A 65 033830). Examples of phase-insensitive quantum states are considered that exhibit no first-order nonclassical effects, but that are nonclassical in second order.
Criteria for the nonclassicality of a quantum state of the harmonic oscillator are formulated in terms of normally ordered moments of quadratures. Inequalities are obtained that are equivalent to the failure of the Glauber-Sudarshan
P function to be a probability density. A possibility to measure the required moments is proposed.
A quantum state is nonclassical if its Glauber-Sudarshan
P function fails to be interpreted as a probability density. This quantity is often highly singular, so that its reconstruction is a demanding task. Here we present the experimental determination of a well-behaved
P function showing negativities for a single-photon-added thermal state. This is a direct visualization of the original definition of nonclassicality. The method can be useful under conditions for which many other signatures of nonclassicality would not persist.
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 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.