A geometric measure of non-classicality

This paper aims to stress the role of the Cahill-Glauber quasi-probability densities in defining, detecting, and quantifying the non-classicality of field states in quantum optics. The distance between a given pure state and the set of all pure classical states is called here a geometric degree of non-classicality. As such, we investigate non-classicality of a pure single-mode state of the radiation field by using the coherent states as a reference set of pure classical states. It turns out that any such distance is expressed in terms of the maximal value of the Husimi $Q$ function. As an insightful application we consider the de-Gaussification process produced when preparing a quantum state by adding $p$ photons to a pure Gaussian one. For a coherent-state input, we get an analytic degree of non-classicality which compares interestingly with the previously evaluated entanglement potential. Then we show that addition of a single photon to a squeezed vacuum state causes a considerable enhancement of non-classicality, especially at weak and moderate squeezing of the original state. By contrast, addition of further photons is less effective.