Revisit the non-locality of Majorana zero modes and teleportation: Bogoliubov-de Gennes equation based treatment.

The nonlocal nature of the Majorana zero modes implies an inherent teleportation channel and unique transport signatures for Majorana identification. In this work we make an effort to eliminate some inconsistencies between the Bogoliubov-de Gennes equation based treatment and the method using the associated regular fermion number states of vacancy and occupation within the `second quantization' framework. We first consider a rather simple `quantum dot--Majorana wire--quantum dot' system, then a more experimentally relevant setup by replacing the quantum dots with transport leads. For the latter setup, based on the dynamical evolution of electron-hole excitations, we propose a {\it single-particle-wavefunction} approach to quantum transport, whose stationary limit recovers the conventional quantum scattering theory and the steady-state nonequilibrium Green's function formalism. Further, we theoretically revisit the issue of Majorana two-probe tunneling spectroscopy and discuss the condition of the quantized conductance $2e^2/h$, together with a new prediction of {\it half} quantum conductance $\frac{1}{2}(e^2/h)$. The present work may arouse a need to reexamine some existing studies and the proposed treatment is expected to be involved in analyzing future experiments in this fast developing field.