Quantum information theory studies how quantum information can be represented, stored, processed and sent. On the mathematical side this often involves the study of tensor products and decompositions of positive matrices, as well as positive maps. These are also natural objects in free semialgebraic geometry, where positivity of non-commutative objects are studied from a geometric viewpoint. In this talk I will give an introduction to some important concepts in both areas, and demonstrate how results and methods from either field can be successfully employed in the other. This is joint ongoing work between the group of Gemma De las Cuevas and my own research group in Innsbruck.