  Topological insulators and topological superconductors display various
topological phases that are characterized by different Chern numbers or by
gapless edge states. In this work we show that various quantum information
methods such as the von Neumann entropy, entanglement spectrum, fidelity, and
fidelity spectrum may be used to detect and distinguish topological phases and
their transitions. As an example we consider a two-dimensional $p$-wave
superconductor, with Rashba spin-orbit coupling and a Zeeman term. The nature
of the phases and their changes are clarified by the eigenvectors of the
$k$-space reduced density matrix. We show that in the topologically nontrivial
phases the highest weight eigenvector is fully aligned with the triplet pairing
state. A signature of the various phase transitions between two points on the
parameter space is encoded in the $k$-space fidelity operator.
