Algebraic methods in quantum information

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Quantum information science (QIS) is the study of information and computation, starting from a quantum-mechanical description of the physical world. On the theoretical side, we explore these concepts using mathematical models. While the questions we ask about these models come from an information science perspective, the natural mathematical language for QIS lies in the subjects of operator algebras and Lie theory. Combining these two perspectives poses unique mathematical challenges. Since QIS is still a developing field, in many cases we don't currently have a good understanding of which problems or problem instances are tractable (i.e. can be solved mathematically), and which are intractable. Broadly speaking, my interest in the mathematics of QIS lies in exploring the boundaries between tractable and intractable problems/problem instances. For instance, this might be accomplished by finding natural problems in QIS which are intractable; or identifying classes of problems or problem instances where, despite initially appearing intractable, some geometric or algebraic structure makes the problem tractable. ******Non-local games provide a fertile area to explore these boundaries. In a non-local game, two players cooperate to win a simple game. Although they know the rules in advance, they are unable to communicate while the game is in progress, so they may not be able to win with probability one. Bell's famous theorem states that the players can achieve a higher winning probability than expected classically if they share an entangled quantum state. Thus, non-local games can be regarded as simple distributed tasks which can have a quantum advantage. Since Bell's discovery, non-local games have been implemented (usually under the name of Bell tests) in many experiments, and now form one of the cornerstones of the evidence for quantum mechanics. On the theory side, non-local games have been heavily studied (often under the name of Bell inequalities or Bell scenarios) in physics, mathematics, and computer science, and have many potential applications in quantum information science. ******Despite being heavily studied, the fundamental mathematical questions about non-local games remain unanswered. These include:******1. How do we model a quantum strategy, i.e. a strategy with an entangled quantum state?******2. Given a game, can we compute the optimal winning probability over quantum strategies?******3. Given a game, how much entanglement is required to play optimally or near-optimally?******Recently we have been able to make some of the first progress on questions 1-3 by making a connection between non-local games and the theory of finitely-presented groups. In particular, we have been able to show that it is impossible to determine the exact optimal winning probability over quantum strategies. The goal of this research is to continue to develop this connection to get further insight into these questions. **