Single- and two-mode quantumness at a beam splitter

Abstract

In the context of bipartite bosonic systems, two notions of classicality of correlations can be defined: $P$ classicality, based on the properties of the Glauber-Sudarshan $P$ function; and $C$ classicality, based on the entropic quantum discord. It has been shown that these two notions are maximally inequivalent in a static (metric) sense, as they coincide only on a set of states of zero measure. We extend and reinforce quantitatively this inequivalence by addressing the dynamical relation between these types of nonclassicality in a paradigmatic quantum-optical setting: the linear mixing at a beam splitter of a single-mode Gaussian state with a thermal reference state. Specifically, we show that almost all $P$-classical input states generate outputs that are not $C$ classical. Indeed, for the case of zero thermal reference photons, the more $P$-classical resources at the input the less $C$ classicality at the output. In addition, we show that the $P$ classicality at the input---as quantified by the nonclassical depth---does instead determine quantitatively the potential of generating output entanglement. This endows the nonclassical depth with a new operational interpretation: it gives the maximum number of thermal reference photons that can be mixed at a beam splitter without destroying the output entanglement.