Quantum Information Theory

本项目的主要重点是研究量子计算和信息理论中的各种数学问题。尽管建造量子计算机仍然是一个艰巨的实验挑战，但它们有潜力有效地处理那些计算复杂性超出经典数字计算机范围的问题。该方案的某些方面，如纠错和多比特门的探索，与量子计算直接相关。然而，主要的重点是量子通信的相关领域的问题，特别是信道容量，最优编码和解码方案和纠缠分类。侦探他过去在强磁场中对多粒子系统的分析、量子力学熵的性质以及算子代数上的映射等方面的工作为开发新的数学工具提供了一个强有力的起点，这些工具需要处理这些具有挑战性的问题。利用量子计算机高效分解大数的可能性，对现有加密协议的安全性构成了潜在威胁。幸运的是，使用量子粒子进行通信，这在实验上更为可行，为分发加密密钥、编码消息和传输信息提供了新的机制。这包括基于量子粒子传输的编码和发送信息的过程；创新的新协议，基于一种被称为纠缠的量子现象，涉及远距离的粒子，通过一些经典的通信来增强。与经典通信一样，必须准备好处理噪声信道，本建议的大部分内容涉及噪声信道的数学。其中一些工作对实验设计有直接的影响，因为知道如何最好地分配资源和选择编码方案以尽量减少噪声的影响是很重要的。

ABSTRACT The primary focus of this project is the study of a variety of mathematical problems in quantum computation and information theory. Although building quantum computers remains a formidable experimental challenge, they have the potential to efficiently handle problems whose computational complexity puts them beyond the scope of classical digital computers. Some aspects of this proposal, such as error correction and the exploration of multi-bit gates, are directly related to quantum computation. However, the main emphasis is on problems in the related area of quantum communication, particularly channel capacity, optimal encoding and decoding schemes and classification of entanglement. The P.I.'s past work on the analysis of multi-particle systems in strong magnetic fields, the properties of quantum-mechanical entropy, and maps on operator algebras provides a strong starting point for developing new mathematical tools needed to deal with these challenging issues. The possibility of using quantum computers to efficiently factoring large numbers, is a potential threat to the security of existing cryptographic protocols. Fortunately, using quantum particles for communication, which is more feasible experimentally, offers new mechanisms for distributing cryptographic keys as well as encoding messages and transmitting information. These include both procedures based on the transmission of quantum particles for encoding and sending information; and innovative new protocols, based on a quantum phenomenon known as entanglement, involving particles at distant sites augmented by some classical communication. As with classical communication, one must be prepared to deal with noisy channels and much of this proposal deals with the mathematics of noisy channels. Some of this work has direct implications for experimental design, since it is important to know how to best allocate resources and choose coding schemes to minimize the effects of noise.