Free Probability, Random Matrices and Quantum Information

我的研究计划的主要目标是借助随机矩阵理论、自由概率理论和量子群理论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项目，在那里我分别学习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)是成熟而截然不同的领域，但很明显，就在几年前，它们还可以从彼此的技术诀窍中受益。 在物理信息载体具有量子性质或表现出量子性质的情况下，量子信息理论已经发展成为经典信息的对应物。自从香农关于最佳渐近传输速率的著名定理以来，经典信道的熵的概念就已经被很好地理解了。它们的量子类似物已经被广泛研究，但例子仍然很难获得，典型的行为仍然没有被很好地理解，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，就像在经典情况下一样，随机技术对于获得最佳传输速率的证明是必要的。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当矩阵的维度变大时，矩阵值概率度量的行为。他们的研究是由Wishart在20世纪20年代发起的，并由Wigner提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得到了发展。 自由概率论(FPT)是算子代数的一个分支，它最初是由Vocerescu引入的，目的是研究von Neumann代数理论中的自由群因子。如今，它已经在理论数学和应用数学的许多其他分支中产生了影响。特别是，它与随机矩阵理论有着深刻的联系。 量子群(QGT)作为经典群的变形，在80年代末由Jim、Drinfeld和Woronowicz引入。后来，王健林推出了一个免费版本。事实证明，王的自由量子群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年前由多个研究人员揭示，如海登、肖尔、温特，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。

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.