Coarse-graining of measurement and quantum-to-classical transition in the bipartite scenario

The connection between coarse-graining of measurement and emergence of classicality has been investigated for some time, if not well understood. Recently in (PRL $\textbf{112}$, 010402, (2014)) it was pointed out that coarse-graining measurements can lead to non-violation of Bell-type inequalities by a state which would violate it under sharp measurements. We study here the effects of coarse-grained measurements on bipartite cat states. We show that while it is true that coarse-graining does indeed lead to non-violation of a Bell-type inequality, this is not reflected at the state level. Under such measurements the post-measurement states can be non-classical (in the quantum optical sense) and in certain cases coarse-graning can lead to an increase in this non-classicality with respect to the coarse-graining parameter. While there is no universal way to quantify non-classicality, we do so using well understood notions in quantum optics such as the negativity of the Wigner function and the singular nature of the Gluaber-Sudharshan P distribution.