# PhD Defense Vincent Steffan

Title: Tensor Decompositions Theory and Applications in Quantum Information

Abstract:

In this thesis, we shed light from various angles on ways of decomposing tensors. Our investigation consists of four parts. In the first part, we study entanglement structures, which are a natural generalization of tensor networks: While tensor network states can be seen as locally transformed versions of a tensor “built” from two-party tensors laid out according to the geometry of a graph, entanglement structures are constructed from tensors put on the hyperedges of a hypergraph. We find constructions and obstructions for the conversion between entanglement structures in the sense of restriction and degeneration, and calculate the tensor rank of specific entanglement structures. We then go on to study tensor networks in more depth. More precisely, we study the quantum max-flow, which quantifies the amount of entanglement between two regions of a tensor network. We relate the quantum max-flow in the so-called bridge graph to the theory of prehomogeneous tensor spaces and the representation theory of quivers, a connection that enables us to calculate the quantum max-flow in this graph in a large number of cases. After that, we define and study partial degeneration, an intermediate version of restriction and degeneration, which are well-known preorders for tensors. By constructing various examples and showing obstructions, we demonstrate that partial degeneration is inequivalent to both restriction and degeneration. We also relate this concept to the notion of aided rank, a generalization of tensor rank. Here, we again highlight differences between the concepts of degeneration and partial degeneration. Finally, we analyze stabilizer rank decompositions, which are relevant in the theory of simulating quantum circuits. In particular, we present a technique to lower bound stabilizer rank and approximate stabilizer rank. This technique yields, together with other interesting consequences, a strong lower bound on the stabilizer rank of tensor powers of the so-called T-state – a quantity gauging the efficiency of the simulation of quantum circuits built from Clifford+T gates using the Gottesman-Knill theorem.