Quantized Symmetries in Operator Algebras and Quantum Information

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For over a century it has been known that our physical universe is governed by quantum mechanics. A fundamental feature of quantum mechanics is that if you perform a measurement on a physical system, it has the effect of permanently altering the state of the system. This means that the order in which one performs measurements on certain physical systems matters. In this context, we say that measurement is "non-commutative". The inherent non-commutativity of quantum mechanics is described by the theory of operator algebras, which is the branch of mathematics that investigates how to efficiently manipulate and encode arbitrarily large matrices of numerical data. Today, operator algebras and quantum theory are both well-established and independent branches of the mathematical sciences. However, in recent years, with the advent of quantum computation and quantum information science, there has been a resurgence of fruitful interactions between the more mathematical theory of operator algebras and the more physical theory of quantum mechanics. The primary goal of this project is to explore and deepen the emerging connections between operator algebra theory and quantum information theory (QIT). The central objects of study in this project are quantized symmetries in operator algebras and QIT. Understanding the symmetries of systems has long been a powerful tool in mathematical and physical problems, and the highly non-commutative nature of both operator algebras and QIT naturally leads to the discovery of more flexible notions of symmetries, called quantum symmetries. In this project, we will use various mathematical incarnations of quantum symmetries to provide a link between operator algebra theory and QIT, and we will use this link to provide new insights into both of these important fields. This project will contribute to the development of the US workforce through the training of graduate and undergraduate students.This research project breaks up into two main directions. The first direction concerns the structural theory of some new and interesting classes of operator algebras arising from quantum isomorphism spaces. Quantum isomorphism spaces appear naturally in the study of quantum teleportation and super-dense coding schemes in QIT, and also in the representation theory of quantum permutation groups. The study of these algebras leads us to propose a quantum analogue of Lueck’s determinant conjecture from geometric group theory and investigate its potential applications to constructing new examples of strongly 1-bounded and strongly solid von Neumann algebras. The second direction of this project relates to the Principal Investigator's pioneering work on the interactions between quantum symmetries and QIT. Here, geometric and representation-theoretic tools will be used to find applications of non-local games arising in QIT to the construction of new examples of hyperlinear discrete quantum groups.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.