PIRSA:18030118

Hyperlinear profile and entanglement

Slofstra, W. (2018). Hyperlinear profile and entanglement. Perimeter Institute. https://pirsa.org/18030118

Slofstra, William. Hyperlinear profile and entanglement. Perimeter Institute, Mar. 19, 2018, https://pirsa.org/18030118 

          @misc{ pirsa_PIRSA:18030118,
            doi = {10.48660/18030118},
            url = {https://pirsa.org/18030118},
            author = {Slofstra, William},
            keywords = {Mathematical physics},
            language = {en},
            title = {Hyperlinear profile and entanglement},
            publisher = {Perimeter Institute},
            year = {2018},
            month = {mar},
            note = {PIRSA:18030118 see, \url{https://pirsa.org}}
          }

William Slofstra Institute for Quantum Computing (IQC)

DOI 10.48660/18030118
Collection
Talk Type Scientific Series
Subject

Abstract

An approximate representation of a finitely-presented group is an assignment of unitary matrices to the generators, such that the defining relations are close to the identity in the normalized Hilbert-Schmidt norm. A group is said to be hyperlinear if every non-trivial element can be bounded away from the identity in approximate representations of the group. Determining whether all groups are hyperlinear is a major open problem, as a non-hyperlinear group would provide a counterexample to the famous Connes embedding problem.

Given the difficulty of the Connes embedding problem, it makes sense to look at an easier problem: how fast does the dimension of approximate representations grow (as a function of how close the defining relations are to the identity) when we require a given set of elements to be bounded away from the identity. These growth rates are called the hyperlinear profile of the group.

In this talk, I will explain our best lower bounds on hyperlinear profile, as well as the connection to entanglement requirements for non-local games (joint work with Thomas Vidick). Time permitting,  I will also mention some other approaches to looking for non-hyperlinear groups, including the recent work of De Chiffre, Glebsky, Lubotzky, and Thom on a group which is not approximable in the unnormalized Hilbert-Schmidt norm.