The degree of nonclassicality of states of a field mode is analysed considering both phase-space and distance-type measures of nonclassicality. By working out some general examples, it is shown explicitly that the phase-space measure is rather sensitive to superposition of states, with finite superpositions possessing maximum nonclassical depth (the highest degree of nonclassicality) irrespective of the nature of the component states. Mixed states are also discussed and examples with nonclassical depth varying between the minimum and the maximum allowed values are exhibited. For pure Gaussian states, it is demonstrated that distance-type measures based on the Hilbert–Schmidt metric are equivalent to the phase-space measure. Analyzing some examples, it is shown that distance-type measures are efficient to quantify the degree of nonclassicality of non-Gaussian pure states.
We define the degree of nonclassicality of a one-mode Gaussian state of the quantum electromagnetic field in terms of the Bures distance between the state and the set of all classical one-mode Gaussian states. We find the closest classical Gaussian state and the degree of nonclassicality using a recently established expression for the Uhlmann fidelity of two single-mode Gaussian states. The decrease of nonclassicality under thermal mapping is carefully analyzed. Along the same lines, we finally present the evolution of nonclassicality during linear amplification.
We consider the class of the one-mode Gaussian states of the quantum radiation field. The relative entropy of such a state with respect to a similar one is derived. We analyze the entropic amount of nonclassicality of a Gaussian state defined as the minimal relative entropy of any classical Gaussian state with respect to it. A similar quantity built with the Hilbert-Schmidt distance is also calculated. Both nonclassicality measures are then compared with the Bures-metric degree of nonclassicality evaluated previously. The properties of the closest classical Gaussian state, as well as the decrease of the nonclassicality under thermal noise mappings are carefully examined in each case. For mixed states we find that only the Bures-distance amount of nonclassicality is equivalent to the nonclassical depth.
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.
The total noise of a field state is a measure of the fluctuations of the field amplitude. It is a minimum for coherent states. As the behavior of a state becomes more nonclassical, its total noise increases. This is shown first for several specific types of nonclassical states, among them squeezed and sub-Poissonian states. These results are generalized by using nonclassical distance to measure how nonclassical a field state is. A lower bound for the total noise is derived that is an increasing function of nonclassical distance. From it one can conclude that highly nonclassical states have large amplitude fluctuations.