A quantum information approach to the Hyperlinearity Conjecture (25rit036)

The Banff International Research Station will host the "A quantum information approach to the Hyperlinearity Conjecture" workshop in Banff from September 1 - 8, 2025.

The question about the genuine distinction between classical and quantum mechanics, notably raised by Einstein in the 1930's, was definitively settled in the 1960's through the celebrated Bell Theorem. Since then, other models of quantum physics, utilising seemingly larger resource types, have appeared, with the prime example the commuting operator model. Its genuinely greater power, when compared to the more conventional tensor product model, has been proved only recently,
simultaneously settling an over 40 year old open problem in pure mathematics, known as the Connes Embedding Problem.
Nevertheless, a special case of the latter problem, known as the Hyperlinearity Conjecture, and relating to the complexity of the structure of mathematical objects based on underlying symmetries (groups), has so far resisted any resolution efforts. In the present project, we propose a new route towards this conjecture, linking it to functions defined on groups that constitute
non-commutative versions of Fourier transforms. As a result of our research, we expect to strengthen the existing links between operator algebra theory and quantum computing, leading to new pure mathematical directions, and developing more mathematical tools for use in theoretical physics.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), and Alberta's Advanced Education and Technology.