  We show that the manifold of quantum states is endowed with a rich and
nontrivial geometric structure. We derive the Fubini-Study metric of the
projective Hilbert space of a multi-qubit quantum system, endowing it with a
Riemannian metric structure, and investigate its deep link with the
entanglement of the states of this space. As a measure, we adopt the
Entanglement Distance E preliminary proposed in [1]. Our analysis shows that
entanglement has a geometric interpretation: E(|psi>) is the minimum value of
the sum of the squared distances between |psi> and its conjugate states, namely
the states v^mu . sigma^mu |psi>, where v^mu are unit vectors and mu runs on
the number of parties. We derive a general method to determine when two states
are not the same state up to the action of local unitary operators. We prove
that the entanglement distance, along with its convex roof expansion to mixed
states, fulfills the three conditions required for an entanglement measure:
that is i) E(|psi>) =0 iff |psi> is fully separable; ii) E is invariant under
local unitary transformations; iii) E doesn't increase under local operation
and classical communications. Two different proofs are provided for this latter
property. We also show that in the case of two qubits pure states, the
entanglement distance for a state |psi> coincides with two times the square of
the concurrence of this state. We propose a generalization of the entanglement
distance to continuous variable systems. Finally, we apply the proposed
geometric approach to the study of the entanglement magnitude and the
equivalence classes properties, of three families of states linked to the
Greenberger-Horne-Zeilinger states, the Briegel Raussendorf states and the W
states. As an example of an application for the case of a system with
continuous variables, we have considered a system of two coupled Glauber
coherent states.
