Development of new tools to detect, characterize and quantify quantum entanglement in multipartite systems

Quantum entanglement is a key property of quantum information
theory, that is at the heart of numerous promising applications in fields such as
quantum cryptography, quantum computing or quantum sensing. In the past
decades, the advent of such innovative technologies has reinforced the need for
a better understanding of entanglement. The aim of this thesis is to contribute
to this effort through the development of new tools targeting the characterization
of several features of entanglement. Concerning the issue of entanglement
detection, we present an optimization of the approach that exploits the concept
of generalized concurrences to solve the separability problem for pure states. We
then reformulate the separability question of mixed states into a matrix analysis
problem, from which we obtain general separability criteria for multipartite
states of ranks two and three. We also briefly discuss some properties of separable
states. In particular, we characterize optimal separable decompositions of
symmetric (i.e. permutation invariant) states of two and three qubits with maximal
rank properties. Regarding the quantification of entanglement, we propose
a function to quantify the entanglement of symmetric multiqubit states within
classes of entangled states gathering states that are stochastically equivalent
through local operations assisted with classical communication. This function
establishes a link between the amount of entanglement of a symmetric state and
the distribution of its Majorana points on the Bloch sphere. We finally investigate
the robustness of entanglement with respect to particle loss and provide a
full description of all multiqubit states that are fragile for the loss of one of their
qubits. For symmetric states, the fragility for the loss of one qubit is shown to
be related to a particular symmetry of the Majorana points.