Tensor decompositions: invariance and computational complexity

Tensors are combinations of several vectors such that a bigger vector space, also calledthe tensor space, emerges. The tensor space has a richer structure that that of theseparate vector spaces. Tensors are universally used in quantum mechanics to modelmutipartite systems. Tensor decompositions aim to write these tensors as combina-tions of elements of the original vector paces. and have been used historically toease computations and to gain insight in the properties of tensors. More recently,these decompositions have been used to study pure states with high success, butmixed states have presented more challenges due to their positivity structure. In thiswork we present a general framework to encode physical structure and symmetriesfor tensor decompositions. The main result shows that every tensor that fulfils thesymmetry conditions imposed by the physical structure can be finitely decomposedin this framework. To acheive this we introduce three separate decompositions, twoof which explicitly exhibit convex properties relevant to physics: positivity and sepa-rability. We study these decompositions, both separately and their relations. We alsorelate particular cases of these decompositons to previously studied decompositionsof nonnegative matrices, showing that this is a generalisation. Moreover, we studythe computational complexity of related problems. We review complexity classes thathave been useful to solve related problems and how these classes can relate to ourproblems.Arbeit an der Bibliothek noch nicht eingelangt - Daten nicht geprüftInnsbruck, Univ., Masterarb., 2020(VLID)475789