Free Probability, Random Matrices and Quantum Information

自由概率、随机矩阵和量子信息

• 批准号: RGPIN-2015-06384
• 负责人: Collins, Benoit
• 金额: $ 2.26万
• 依托单位: University of Ottawa
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=614117
• 关键词: Free; Probability; Random; Matrices; Quantum

The main objective of my research program is to make advances in Quantum Information with the help of Random Matrix Theory, Free Probability Theory, and Quantum Groups Theory. Another objective is to pursue my ongoing program, where I study some aspects of the above-mentioned theories separately. While Quantum Information Theory (QIT) and Random Matrix Theory (RMT) are well-established and distinct fields, it has become clear that they could benefit from each other's know-how only a few years ago. Quantum Information Theory has been developed as a counterpart of classical information in the case where the physical information carriers are of quantum nature or exhibit quantum properties. The notions of entropy for classical channels has been well understood since the celebrated theorems of Shannon about optimal asymptotic transmission rates. Their quantum analogues have been widely studied but examples remain difficult to obtain, and typical behaviors are still not well understood, whereas in the meantime, quantum communication and cryptography already start to be physically implemented. There is nowadays evidence that, as in the classical case, random techniques are necessary to obtain proofs of the optimal rates of transmission. In this case, the natural objects are random matrices. Random Matrix Theory is interested in the behaviour of matrix valued probability measures as the dimension of the matrices becomes large. Their study was initiated by Wishart in the twenties and was raised to a self-standing fields of mathematics by Wigner. Since then, random matrices have developed in many directions. Free Probability Theory (FPT), a branch of Operator Algebras, was initially introduced by Voiculescu in order to study the free group factors in von Neumann algebra theory. Nowadays it has ramifications in many other branches of pure and applied mathematics. In particular, it has deep links with Random Matrix Theory. And Quantum Groups (QGT) were introduced in the late eighties by Jimbo, Drinfeld, and then Woronowicz, as deformations of classical groups. Later, a free version was introduced by Wang. Wang's free quantum groups turned out to have deep connections with free probability.  While the connection between QIT and RMT was unveiled about 10 years ago by multiple resarchers, such as  Hayden, Shor, Winter, we explored this connection systematically, and discovered links with FPT. In turn, we have now strong evidence that quantum groups will play an important role in the construction of interesting quantum channels. Our research will be primarily focused on the interplay between these fields; in the meantime we will continue to make contribution to these fields separately.

我的研究计划的主要目标是在随机矩阵理论，自由概率论和量子群理论的帮助下在量子信息方面取得进展。另一个目标是继续我正在进行的计划，在那里我分别研究上述理论的某些方面。 虽然量子信息理论（QIT）和随机矩阵理论（RMT）是成熟的和不同的领域，但很明显，它们可以在几年前从彼此的专业知识中受益。 量子信息理论是在物理信息载体具有量子性质或表现出量子特性的情况下作为经典信息的对应物而发展起来的。经典信道的熵概念自香农关于最佳渐近传输速率的著名定理以来已经得到很好的理解。它们的量子类似物已被广泛研究，但实例仍然难以获得，典型的行为仍然没有得到很好的理解，而与此同时，量子通信和密码学已经开始在物理上实现。现在有证据表明，在经典的情况下，随机技术是必要的，以获得最佳传输速率的证明。在这种情况下，自然对象是随机矩阵。 随机矩阵理论关注的是矩阵值概率测度在矩阵维数变大时的行为。他们的研究是由Wishart在20世纪20年代开始的，并由Wigner上升到一个独立的数学领域。从那时起，随机矩阵在许多方向上发展。 自由概率论（FPT）是Voiculescu为了研究von Neumann代数理论中的自由群因子而引入的算子代数的一个分支。如今，它在纯数学和应用数学的许多其他分支中产生了分支。特别是，它与随机矩阵理论有很深的联系。 量子群（QGT）是在八十年代后期由Jimbo，Drinfeld和Woronowicz引入的，作为经典群的变形。后来，一个免费的版本被王介绍。王的自由量子群被证明与自由概率有着深刻的联系。 虽然QIT和RMT之间的联系在大约10年前被Hayden，Shor，Winter等多个研究者发现，但我们系统地探索了这种联系，并发现了与FPT的联系。反过来，我们现在有强有力的证据表明，量子群将在构建有趣的量子通道中发挥重要作用。 我们的研究将主要集中在这些领域之间的相互作用;同时，我们将继续分别为这些领域做出贡献。