Quantum hypergraph states and the theory of multiparticle entanglement

This thesis is devoted to learning different aspects of quantum entanglement theory.
More precisely, it concerns a characterization of certain classes of pure multipartite
entangled states, their nonlocal and entanglement properties, comparisons with the
other well-studied classes of states and, finally, their utilization in certain quantum
information processing tasks.

The most extensive part of the thesis explores an interesting class of pure multipartite
entangled states, quantum hypergraph states. These states are generalizations of the
renowned class of graph states. Here we cover their nonlocal properties in various
scenarios, derive graphical rules for unitary transformations and Pauli bases measurements.
Using these rules, we characterize entanglement classes of hypergraph states
under local operations, obtain tight entanglement witnesses, and calculate entanglement
measures for hypergraph states. Finally, we apply all the aforementioned analysis
to endorse hypergraph states as powerful resource states for measurement-based
quantum computation and quantum error-correction.

The rest of the thesis is devoted to three disjoint problems, but all of them are still in
the scope of entanglement theory. First, using mathematical structure of linear matrix
pencils, we coarse grain entanglement in tripartite pure states of local dimensions
2 x m x n under the most general local transformations. In addition, we identify the
structure of generic states for every m and n and see that for certain dimensions there
is a resemblance between bipartite and tripartite entanglement. Second, we consider
the following question: Can entanglement detection be improved, if in addition to
the expectation value of the measured witness, we have knowledge of the expectation
value of another observable? For low dimensions we give necessary and sufficient
criterion that such two product observables must satisfy in order to be able to detect
entanglement. Finally, we derive a general statement that any genuine N-partite entangled
state can always be projected on any of its k-partite subsystems in a way that
the new state in genuine k-partite entangled.