Algebraic methods in quantum information

量子信息科学(QIS)是对信息和计算的研究，从对物理世界的量子力学描述开始。在理论方面，我们使用数学模型来探索这些概念。虽然我们对这些模型提出的问题来自信息科学的观点，但QIS的自然数学语言存在于算子代数和李理论的主题中。将这两种观点结合起来会带来独特的数学挑战。由于QIS仍然是一个发展中的领域，在许多情况下，我们目前还不能很好地理解哪些问题或问题实例是可以处理的(即可以用数学方法解决的)，哪些是难以处理的。总的来说，我对QIS数学的兴趣在于探索易处理和难处理的问题/问题实例之间的界限。例如，这可能是通过在QIS中找到难以处理的自然问题来实现的；或者识别问题的类别或问题实例，尽管最初看起来很难处理，但某些几何或代数结构使问题容易处理。*外地游戏为探索这些边界提供了肥沃的领域。在外地游戏中，两个玩家合作才能赢得一场简单的游戏。虽然他们提前知道了规则，但他们无法在比赛进行时进行交流，因此他们可能无法以概率1取胜。贝尔著名的定理指出，如果玩家共享一个纠缠的量子态，他们可以获得比经典预期更高的获胜概率。因此，非本地游戏可以被视为简单的分布式任务，可以具有量子优势。自从贝尔的发现以来，非局域游戏已经在许多实验中实现(通常以贝尔测试的名义)，现在构成了量子力学证据的基石之一。在理论方面，非局域博弈在物理、数学和计算机科学中得到了广泛的研究(通常以Bell不等式或Bell情景的名义)，并在量子信息科学中有许多潜在的应用。*尽管进行了大量的研究，但关于外地游戏的基本数学问题仍然没有答案。这些问题包括：*1.我们如何建模量子策略，即具有纠缠量子态的策略？*2.给定一个游戏，我们能否计算出比量子策略更优的获胜概率？*3.给定一个游戏，需要多少纠缠才能发挥最优或接近最优？*最近，我们通过将非局部博弈与有限群理论联系起来，在问题1-3上取得了一些初步进展。特别是，我们已经能够证明，不可能确定相对于量子策略的确切最优获胜概率。这项研究的目标是继续发展这种联系，以进一步深入了解这些问题。**

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. **