Benjamin McKenna, Georgia Tech

Title: The injective norm of random tensors

Abstract: The injective norm is a natural generalization to tensors of the operator norm of a matrix. In quantum information, the injective norm is one important measure of genuine multipartite entanglement of quantum states, where it is known as the geometric entanglement. We present two methods to give upper bounds on the injective norm of real and complex random tensors, corresponding to lower bounds on the geometric entanglement of random quantum states. The first method is based on spin-glass methods, the Kac—Rice formula, and recent progress coming from random matrices. The second method, more recent, is a kind of moment method for random tensors. Joint work with Stéphane Dartois.