Tensile quantum-to-classical transition of macroscopic entangled states under complete coarse-grained measurements

Abstract: The macroscopic limit at which the quantum-to-classical transition occurs remains as one of the long-standing questions in the foundations of quantum theory. There are evidences that the macroscopic limit to which the quantumness of a system persists depends on the degree of interaction due to the measurement processes. For instance, with a system having a considerably large Hilbert space dimension, if the measurement is performed in such a way that the outcome of the measurement only reveals a coarse-grained version of the information about the individual level of the concerned system then the disturbance due to the measurement process can be considered to be infinitesimally small. Based on such coarse-grained measurement the dependence of Bell inequality violation on the degree of coarsening has already been investigated [Phys. Rev. Lett. 112, 010402 (2014)]. In this paper, we first capture the fact that when local-realism is taken to be the defining notion of classicality, the effect of the degree of coarsening on the downfall of quantumness of a macroscopic entangled state can be compensated by testing a Bell-inequality of a higher number of settings from a family of symmetric Bell-inequalities if the number of settings is odd. However, on the contrary, we show that such compensation can not be seen when we witness such quantum-to-classical transition using symmetric Bell inequalities having an even number of settings. Finally, complementing the above result, we show that when unsteerability is taken as the classicality, for both odd and even numbers of settings the degree of coarsening at which the quantum-to-classical transition occurs can be consistently pushed ahead by testing a linear steering inequality of a higher number of settings and observing its violation. We further extend our treatment for mixed macroscopic entangled states