Time correlations and Leggett-Garg inequalities for probing the topological phase transition in the Kitaev chain

Topological states (TS) in the Kitaev chain have shown as robust quantum information entities with potential applications in topological quantum computation protocols. A major challenge in these new proposals is the control of both the autonomous as well as directed time evolution of TS, an issue rather unexplored up to now. We evaluate the interplay between time dependent quantum correlations and nonlocal quantum objects such as Majorana based qubits. We use two-time correlations (TTC) and Leggett-Garg inequalities (LGI) for identifying the transition between normal and the topological phase in a Kitaev chain. TTC and LGI of dichotomic quantum observables associated with fermion occupation number of both local as well as nonlocal qubit operators (formed by pairing local and non-local Majorana fermions) are analyzed for different chain lengths and chemical potentials. In order to gain further insight on the physical properties of the system's dynamics, violations of LGI are also evaluated for different string order parameter qubits. We obtain analytical results which allow us to understand the fundamental aspects of TTC in topological Kitaev chains.