Quantum Information Theory

这个项目的主要重点是研究量子计算和信息论中的各种数学问题。 尽管构建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计算复杂性超出经典数字计算机范围的问题。 该提议的某些方面，如纠错和多位门的探索，与量子计算直接相关。 然而，主要的重点是量子通信相关领域的问题，特别是信道容量，最佳编码和解码方案以及纠缠分类。 私家侦探过去在强磁场中多粒子系统的分析、量子力学熵的性质以及算子代数上的映射方面的工作为开发处理这些具有挑战性的问题所需的新数学工具提供了一个强有力的起点。 使用量子计算机有效分解大量数字的可能性，是对现有加密协议安全性的潜在威胁。 幸运的是，使用量子粒子进行通信，这在实验上更加可行，为分发密钥以及编码消息和传输信息提供了新的机制。 这些包括基于量子粒子传输的编码和发送信息的程序;以及基于称为纠缠的量子现象的创新新协议，涉及通过一些经典通信增强的遥远站点的粒子。与经典通信一样，必须准备好处理噪声信道，并且该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最大限度地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.