A quantum information theoretic quantity sensitive to the neutrino mass-hierarchy

In this work, we derive a quantum information theoretic quantity similar to the Leggett-Garg inequality, which can be defined in terms of neutrino transition probabilities. For the case of $ u_mu to u_e/bar{ u}_mutobar{ u}_e$ transitions, this quantity is sensitive to CP violating effects as well as the neutrino mass-hierarchy, namely which neutrino mass eigenstate is heavier than the other ones. The violation of the inequality for this quantity shows an interesting dependence on mass-hierarchy. For normal (inverted) mass-hierarchy, it is significant for $ u_mu to u_e$ ($bar{ u}_mu to bar{ u}_e$) transitions. This is applied to the two ongoing accelerator experiments T$2$K and NO$ u$A as well as the future experiment DUNE.