The nonclassicality, entanglement, and dimensionality of a noisy twin beam are determined using a characteristic function of the beam written in the Fock basis. One-to-one correspondence between the negativity quantifying entanglement and the nonclassicality depth is revealed. Twin beams, which are either entangled or nonclassical (independent of their entanglement), are observed only for the limited degrees of noise, which degrades their quantumness. The dimensionality of the twin beam quantified by the participation ratio is compared with the dimensionality of entanglement determined from the negativity. Partitioning of the degrees of freedom of the twin beam into those related to entanglement and to noise is suggested. Both single-mode and multimode twin beams are analyzed. Weak nonclassicality based on integrated-intensity quasidistributions of multimode twin beams is studied. The relation of the model to the experimental twin beams is discussed.
Gradual loss of the entanglement of a twin beam containing around 25 photon pairs with the increasing external noise is experimentally investigated. The entanglement is quantified by the nonclassicality depths and the nonclassicality counting parameters related to several nonclassicality criteria. The reduction of intensity moments of the analyzed multimode twin beams to single-mode ones allows determination of the negativity as another quantifier of the entanglement. Both the raw photocount histograms and the reconstructed photon-number distributions are analyzed in parallel.
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.
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.
Different non-classicality criteria expressed in the form of inequalities among intensity moments and elements of photon-number distributions are applied to noisy twin beams and other two-mode states obtained from a twin beam by using a beam splitter. Their performance in revealing the non-classicality is judged in comparison with the exact results provided by suitable entanglement and local non-classicality quantifiers. Whereas the non-classicality of noisy twin beams is always revealed by these criteria, not all the nonclassical states obtained at the output of the beam splitter can be identified by these experimentally easily reachable criteria.
Numerous nonclassicality criteria based on the probabilities of experimental photocount or theoretical photon-number distributions are derived using several approaches. Relations among the derived criteria are revealed and the fundamental criteria are identified. They are grouped into parametric systems that allow the analysis of the nonclassicality from different points of view (``local'' nonclassicality, pairwise character of photon correlations, etc.). Considering their structure, the criteria may be divided into groups that differ in the power to resolve the nonclassicality. Quantification of the nonclassicality using the Lee nonclassicality depth and the nonclassicality counting parameter is discussed. The number of field modes used is identified as an important parameter that may cause unexpected results. An appropriate linear transformation of a photocount (photon-number) distribution into its
s-ordered form, needed for the determination of the nonclassicality depth, is derived. For comparison, the derived criteria are applied both to an experimental photocount histogram of a twin beam and the reconstructed photon-number distributions with different levels of the noise.