Quantum Perspectives in Banach and Metric Spaces

University Of Oklahoma Norman Campus, Norman OK

Investigators

Abstract

In many situations of interest, one has a quantitative way of expressing how far apart two objects are. A basic example is physical distance between locations in space, and a modern one is to measure how similar two words are (which is how software autocorrects our typing). The mathematical notion used to encapsulate this idea is that of a metric space. Particularly interesting kinds of such spaces are those that arise from networks, that is, those for which the measure of distance between two nodes is given by the optimal choice of a path connecting them using the links in the network: there are no direct flights between Oklahoma City and Boston, so for travel purposes what matters is not the geographical distance between the two cities but rather how to get from one to the other using the available commercial flights. Such network-type spaces play an important role in Information Theory, the mathematical framework for communications where information is encoded in a string of 0’s and 1’s as in today’s computers. The project studies the shape of the corresponding quantum network-type spaces that arise from Quantum Information Theory, which models communication systems where information is now encoded in the state of a quantum-mechanical system. Better knowledge of the shape of these quantum spaces has implications for the understanding of the capabilities of the quantum processes that will be used by the quantum computers of the future. In addition, through outreach, mentoring of postdocs, and involvement in undergraduate research, the project will contribute to the growth and diversification of the body of students and researchers in STEM fields. The project will focus on geometric questions arising from various models of quantization. It is divided into three parts. The first is about quantum graphs and other quantum metric spaces, with an emphasis on quantum expanders and the further development of a theory of the large-scale geometry of quantum metric spaces. The second part proposes to use Quantum Information Theory tools to study the nonlinear geometry of noncommutative sequence and function spaces. The focus is to study noncommutative versions of several generalized Mazur maps of interest in Banach space geometry, Operator Algebra Theory, Geometric Group Theory, and Theoretical Computer Science, as well as interpolation properties of a new generalization of weighted noncommutative Lebesgue p spaces. The third part is focused on operator spaces, which are a type of quantum Banach spaces. The study of their nonlinear geometry is a timely subject. The project intends to adapt tools coming from the linear theory of tensor products to continue the development of a nonlinear geometric theory for operator spaces, and to study asymptotic tensor products for operator spaces (whose Banach space counterparts have in recent times found applications in Quantum Information Theory). 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.