We describe a method to probe the quantum phase transition between the
short-range topological phase and the long-range topological phase in the
superconducting Kitaev chain with long-range pairing, both exhibiting subgap
modes localized at the edges. The method relies on the effects of the finite
mass of the subgap edge modes in the long-range regime (which survives in the
thermodynamic limit) on the single-particle scattering coefficients through the
chain connected to two normal leads. Specifically, we show that, when the leads
are biased at a voltage V with respect to the superconducting chain, the Fano
factor is either zero (in the short-range correlated phase) or 2e (in the
long-range correlated phase). As a result, we find that the Fano factor works
as a directly measurable quantity to probe the quantum phase transition between
the two phases. In addition, we note a remarkable "critical fractionalization
effect" in the Fano factor, which is exactly equal to e along the quantum
critical line. Finally, we note that a dual implementation of our proposed
device makes it suitable as a generator of large-distance entangled
two-particle states.Comment: 24 pages, 8 .eps figures Published versio