A Geometric Approach to the Projective Tensor Norm
Sandra Lang
The main focus of this thesis is on the projective norm on finite-dimensional real or complex tensor products. There are various mathematical subjects with relations to the projective norm. For instance, it appears in the context of operator algebras or in quantum physics.
The projective norm on multipartite tensor products is considered to be less accessible. So we use a method from convex algebraic geometry to approximate the projective unit ball by convex supersets, so-called theta bodies. For real multipartite tensor products we obtain theta bodies which are close to the projective unit ball, leading to a generalisation of the Schmidt decomposition. In a second step the method is applied to complex tensor products, in a third step to separable states.
In a more general context, the projective norm can be related to binomial ideals, especially to so-called Hibi relations. In this respect, we also focus on a generalisation of the projective unit ball, here called Hibi body, and its theta bodies. It turns out that many statements also hold in this general context.