Nonclassicality cannot be a single-observable property, since the statistics of any quantum observable is compatible with classical physics. We develop a general procedure to reveal nonclassical behavior of light states from the joint statistics arising in the practical measurement of multiple observables. Beside embracing previous approaches, this protocol can disclose nonclassical features for standard examples of classical-like behavior, such as SU(2) and Glauber coherent states. When combined with other criteria, this would imply that every light state is nonclassical.
A bosonic state is commonly considered nonclassical (or quantum) if its Glauber-Sudarshan
P function is not a classical probability density, which implies that only coherent states and their statistical mixtures are classical. We quantify the nonclassicality of a single qubit, defined by the vacuum and single-photon states, by applying the following four well-known measures of nonclassicality: (1) the nonclassical depth, $\ensuremath{\tau}$, related to the minimal amount of Gaussian noise which changes a nonpositive
P function into a positive one; (2) the nonclassical distance
D, defined as the Bures distance of a given state to the closest classical state, which is the vacuum for the single-qubit Hilbert space; together with (3) the negativity potential (NP), and (4) concurrence potential, which are the nonclassicality measures corresponding to the entanglement measures (i.e., the negativity and concurrence, respectively) for the state generated by mixing a single-qubit state with the vacuum on a balanced beam splitter. We show that complete statistical mixtures of the vacuum and single-photon states are the most nonclassical single-qubit states regarding the distance
D for a fixed value of both the depth $\ensuremath{\tau}$ and NP in the whole range
[0,1] of their values, as well as the NP for a given value of $\ensuremath{\tau}$ such that $\ensuremath{\tau}>0.315\phantom{\rule{0.16em}{0ex}}4$. Conversely, pure states are the most nonclassical single-qubit states with respect to $\ensuremath{\tau}$ for a given
D, NP versus
D, and $\ensuremath{\tau}$ versus NP. We also show the ``relativity'' of these nonclassicality measures by comparing pairs of single-qubit states: if a state is less nonclassical than another state according to some measure then it might be more nonclassical according to another measure. Moreover, we find that the concurrence potential is equal to the nonclassical distance for single-qubit states. This implies an operational interpretation of the nonclassical distance as the potential for the entanglement of formation.
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.