Free Probability, Random Matrices and Quantum Information

我的主要研究目标是利用随机矩阵理论、自由概率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，我将分别研究上述理论的某些方面。

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.