  Localization and entanglement properties of edge states of HgTe quantum wells
in a finite strip geometry of width $L$ are studied under quantum information
concepts such as: 1) inverse participation ratio (IPR), which measures
localization, and 2) entropies of the reduced density matrix (RDM) for the spin
sector, which measures quantum correlations due to the spin-orbit coupling
(SOC). Qualitative and quantitative information on the edge states energies and
wavefunctions is extracted from analytic and numerical Hamiltonian
diagonalization approaches. The previously observed exponential decay of the
energy gap with $L$ and its modulations is confirmed and nontrivial
consequences of the strip width and Rashba terms on the charge conductance are
also reviewed. Analysis of the structure of the edge-state wave functions in
terms of spin, momentum $k_x$ in the $x$-direction and position $y$, evidences
the spin polarization structure of edge states at the boundaries. An IPR
analysis reveals that the valence edge states show maximum localization on the
boundaries for certain values of the momenta $k_x$ in the vicinity of the
$\Gamma$ point. The edge-state wave packets participate of less and less
momenta as we approach to the boundaries $y=0,L$ (and also the center $y=L/2$,
for some of them) of the strip. A study of the RDM to the spin sector of edge
states sheds complementary information on the structure of spin probabilities
in $(k_x,y)$ space, giving clear location of extremal values. The purity and
entropies of the RDM inform on the regions $(k_x,y)$ where the spin sector is
more and less entangled with the rest of the system, due to SOC.
