  We study the ground state properties of the one-dimensional extended Hubbard
model at half-filling from the perspective of its particle reduced density
matrix. We focus on the reduced density matrix of $2$ fermions and perform an
analysis of its quantum correlations and coherence along the different phases
of the model. Specifically, we study its (i) entanglement entropy, (ii)
$\ell_1$ norm of coherence, (iii) irreducible two-body cumulant matrix and (iv)
entanglement spectrum. Our results show that these different properties are
complementary to each other depending on the phase of the system, exhibiting
peculiar behaviors such as discontinuities, maximum or minimum values at the
quantum phase transitions, thus providing a qualitative view of the phase
diagram of the model. In particular, in the superconducting region, we obtain
that the entanglement spectrum signals a transition between a dominant singlet
(SS) to triplet (TS) pairing ordering in the system. Moreover, from the
analysis of the dominant eigenvector in the reduced state, we can relate the
SS-TS transition to the spatial separation between the fermion pairs in the two
different pairing orderings. The entanglement gap is also able to highlight a
transition - at a few-body level - in the groundstate wavefunction, not
discussed previously in the literature. While other quantifiers are less
sensitive to few-body defects in the wavefunction, the entanglement gap can
work as a magnifying glass for these, capturing such small fluctuations.
