QLunch: Freek Witteveen

Speaker: Freek Witteveen

Title: The minimal canonical form for a tensor network

Abstract: Tensor networks have a gauge degree of freedom on the virtual degrees of freedom that are contracted. A canonical form is a choice of fixing this degree of freedom. For matrix product states, choosing a canonical form is a powerful tool. For tensor networks in dimension two or greater there is only limited understanding of the gauge symmetry. In this talk I will discuss a new canonical form, the minimal canonical form, which applies to translation invariant tensor networks in any dimension. Already for matrix product states this gives a new canonical form, while in higher dimensions it is the first rigorous definition of a canonical form valid for any choice of tensor. The construction is based on fundamental results in geometric invariant theory, which is a framework for studying group actions. We prove a weak fundamental theorem: two tensors have a common minimal canonical form if and only if they are gauge equivalent up to taking limits; moreover, this is the case if and only if they give the same quantum state for any geometry. This talk is based on arXiv:2209.14358, which is joint work with Arturo Acuaviva, Visu Makam, Harold Nieuwboer, David Pérez-García, Friedrich Sittner and Michael Walter.