The tensor as an informational resource.

Abstract

A tensor is a multidimensional array of numbers that can be used to store data, encode a computational relation, and represent quantum entanglement. In this sense, a tensor can be viewed as valuable resource whose transformation can lead to an understanding of structure in data, computational complexity, and quantum information. In order to facilitate the understanding of this resource, we propose a family of information-theoretically constructed preorders on tensors, which can be used to compare tensors with each other and to assess the existence of transformations between them. The construction places copies of a given tensor at the edges of a hypergraph and allows transformations at the vertices. A preorder is then induced by the transformations possible in a given growing sequence of hypergraphs. The new family of preorders generalizes the asymptotic restriction preorder which Strassen defined in order to study the computational complexity of matrix multiplication. We derive general properties of the preorders and their associated asymptotic notions of tensor rank and view recent results on tensor rank nonadditivity, tensor networks, and algebraic complexity in this unifying frame. We hope that this work will provide a useful vantage point for exploring tensors in applied mathematics, physics, and computer science but also from a purely mathematical point of view.