Exploring quantum information theory through graphs and networks

In this manuscript, we explore different aspects of quantum information using graph theory as a principal tool. The research presented in this thesis addresses fundamental questions in quantum information theory, such as uncertainty relations and quantum entanglement, as well as practical applications such as certification of quantum systems.
First, we define and analyse the anticommutativity graph associated with a set of observables. This approach allows us to relate the expectation values of these observables to the Lovász number of their anticommutativity graph, a well-known graph invariant. More specificity, the Lovász number provides an upper bound on the sum of the squares of the expectation values. This relationship enables us to derive uncertainty relations for any set of dichotomic observables. These can be transformed into witnesses to detect entanglement, including in PPT entangled states.
Second, we address the problem of measurement scheduling for marginal state tomography, i.e., finding measurement settings that allow for the reconstruction of marginal states of a quantum system. We demonstrate that, for Pauli tomography, this problem can be mapped to a specific instance of the graph-theoretical problem of edge clique covering. Using this mapping, we show for instance that, for two-body marginal tomography of planar qubit topologies, nine Pauli settings are necessary and sufficient, regardless of the numbers of qubits in the system. Furthermore, we establish that with general local projective measurements, 3^k measurement settings are sufficient to reconstruct all k-body marginal states of a multi-qubit system. We report an experimental demonstration of the applicability of the measurement settings derived in our work.
Lastly, we develop necessary criteria for network entanglement, revealing that many graph states, as well as permutationally symmetric states, cannot be prepared in network structures without the usage of classical communication. We then propose a certification protocol for the topology of quantum networks, which has later been implemented experimentally.