Conservation relation of nonclassicality and entanglement for Gaussian states in a beam-splitter

We study the relation between single-mode nonclassicality and two-mode entanglement in a beam-splitter. We show that not all of the nonclassicality (entanglement potential) is transformed into two-mode entanglement for an incident single-mode light. Some of the entanglement potential remains as single-mode nonclassicality in the two entangled output modes. Two-mode entanglement generated in the process can be equivalently quantified as the increase in the minimum uncertainty widths (or decrease in the squeezing) of the output states compared to the input states. We use the nonclassical depth and logarithmic negativity as single-mode nonclassicality and entanglement measures, respectively. We realize that a conservation relation between the two quantities can be adopted for Gaussian states, if one works in terms of uncertainty width. This conservation relation is extended to many sets of beam-splitters.