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 nonclassicality of single-mode quantum states is studied in relation to the entanglement created by a beam splitter. It is shown that properly defined quantifications---based on the quantum superposition principle---of the amounts of nonclassicality and entanglement are strictly related to each other. This can be generalized to the amount of genuine multipartite entanglement, created from a nonclassical state by an
N splitter. As a consequence, a single-mode state of a given amount of nonclassicality is fully equivalent, as a resource, to exactly the same amount of entanglement. This relation is also considered in the context of multipartite entanglement and multimode nonclassicality.
Two basic ingredients in the quantum formalism are the concepts of states and observables, and apart from the average value (which includes the Born probability as a paramountly important case), the variance is the most fundamental and prominent quantity arising from the coupling of a state and an observable. Given the simplicity and ubiquity of the variance, it seems surprising that we can still exploit the variance to gain conceptually different insight into the quantum substratum. The purpose of the present work is to advocate the following idea: By inputting a general quantum state as the observable in the variance and incorporating a resolution of identity induced by coherent states, we obtain a mathematically simple and physically intuitive method for quantifying nonclassicality. We reveal its fundamental properties and highlight its connection with phase-space quantification of nonclassicality. The idea further suggests a whole family of appealing quantifiers of nonclassicality involving deep mathematical subtlety and an intriguing conjecture with physical significance. Some prototypical examples are worked out to illustrate the concept. This approach of regarding states as observables presents an opportunity for investigating quantum features from an alternative perspective of uncertainty.
The characterization of physical systems requires a comprehensive understanding of quantum effects. One aspect is a proper quantification of the strength of such quantum phenomena. Here, a general convex ordering of quantum states will be introduced which is based on the algebraic definition of classical states. This definition resolves the ambiguity of the quantumness quantification using topological distance measures. Classical operations on quantum states will be considered to further generalize the ordering prescription. Our technique can be used for a natural and unambiguous quantification of general quantum properties whose classical reference has a convex structure. We apply this method to typical scenarios in quantum optics and quantum information theory to study measures which are based on the fundamental quantum superposition principle.
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.
Quantification of nonclassicality and entanglement in a quantum state is crucial for quantum advantage in information processing and computation. Robustness is one of the tractable measures for quantifying quantum resources. Gaussian states are important in continuous variable quantum information for their theoretically simple and experimentally easily accessible. We provide the method of how to calculate the robustness of nonclassicality and enatnglement for Gaussian states. The robustness of nonclassicality or entanglement is demonstrated analytically for one-mode, two-mode Gaussain states and multimode symmetric Gaussian states, the result shows a clear physical meaning for the origin of nonclassicality and entanglement. For squeezed thermal states, the nonclassicality is equal to the entanglement for the two-mode case, while they are far apart for multimode cases.