Signifying the nonlocality of NOON states using Einstein-Podolsky-Rosen steering inequalities

We construct Einstein-Podolsky-Rosen (EPR) steering signatures for the nonlocality of the entangled superposition state described by 12{|N)|0)+|0)|N)}, called the two-mode NOON state. The signatures are a violation of an EPR steering inequality based on an uncertainty relation. The violation confirms an EPR steering between the two modes and involves certification of an intermode correlation for number, as well as quadrature phase amplitude measurements. We also explain how the signatures certify an Nth order quantum coherence, so the system (for larger N) can be signified to be in a superposition of states distinct by a mesoscopic value of quantum number in each mode. Finally, we examine the limitations imposed for lossy scenarios, discussing how experimental realizations may be possible for N=2,3.