Non-commutative analysis for quantum computing

Research Project, 2024 – 2027

The future technological advancements very much rely on the development of quantum computing. Understanding its capabilities is one of the most important tasks which requires novel tools anchored across different disciplines. The project will concentrate on theoretical questions central to quantum information processing and use machinery of operator algebras and systems.  In particular, we will investigate quantum no-signalling correlations with focus on the quantisation of synchronous correlations and their application to non-local game theory and quantum entanglement.  Our results will allow to study non-commutative versions of the graph homomorphism  and capture the colouring problem for quantum graphs and related capacities of quantum channels. We will exploit quantum non-local games and study their universal C*-algebras as well as their connection to the Connes Embedding Problem. We will establish expressions for values of quantum games, i.e. maximum success probabilities with possible applications to quantum cryptography. Another goal is to investigate different capacities  of communication channels, in particular, the zero capacity for channels with memory that captures the classical zero-error capacity of a memoryless channel as a one-shot parameter. We will also study non-local games with memory and define their winning rate that extends the winning rate of a memoryless game, providing a one-shot view on this limit with high relevance in computer science  and QIT.