Interactive Quantum Information Theory: Fundamentals and Applications

克劳德·香农的信息论的发展是20世纪科学的基石之一。在一篇论文中，香农为通信技术革命奠定了数学基础。他的两个经典结果，无噪声和有噪声编码定理，永远改变了我们处理压缩和纠错码的方式。这些定理构成了当前许多数字技术的基础，从移动电话到文件压缩，再到互联网的通信层。传统上，通信是在单向设置中研究的，在单向设置中，发送者可以等待任意长的时间来编码长消息；相反，对于交互式通信，如在对话中，必须传输在数据通信速率上不是最佳的较短消息。通信复杂性是研究这种交互过程的数学理论，也是理论计算机科学研究的核心计算模型之一。它在具体的计算模型中有多种实际应用，如流算法和数据结构。近年来，经典交互信息理论中出现了一系列结果，为Shannon的结果提供了对这种交互通信环境的扩展。在过去的几十年里，这种想法的延伸已经在量子信息处理领域得到了研究，为此，我们试图在亚原子水平上利用违反直觉的自然法则，以便以与我们日常生活中所经历的经典信息处理完全不同的方式来处理信息。量子信息的独特性质使其本质上比经典信息更难处理，更容易受到噪声的影响。提出的研究计划的主要目标是为交互式量子通信协议发展一种信息论，并将所开发的工具应用于研究量子通信复杂性中的计算问题的复杂性以及其他具体的量子计算模型。我提出的研究计划可以分为两大研究主题：开发与香农压缩和纠错码相类似的交互式量子编码。此外，我们开发了为这种交互式量子信息理论创建的工具在量子通信复杂性和其他量子计算的具体模型中的相关问题的应用。研究量子信息处理的主要原因之一是，它导致了比经典信息处理在质量和数量上的优势，而加拿大在发展这种量子技术方面处于世界领先地位。这是一个非常有前途的经济领域，我的项目也将有助于在这一领域培养有才华的科学家，这是加拿大一个新兴的经济部门，有许多年轻的有前途的公司。

One of the cornerstones of 20th century science was the development of information theory by Claude Shannon. In a single paper, Shannon laid the mathematical foundations for a revolution in communication technology. His two quintessential results, the noiseless and noisy coding theorems, changed forever the way we approach compression and error correction codes. These theorems form the backbone for much of current digital technology, from mobile telephony to file compression and the communication layer of the Internet. Traditionally, communication is studied in a unidirectional setting, where a sender can wait for an arbitrarily long time to encode a long message; in contrast, with interactive communication, as in a conversation, it is necessary to transmit shorter messages that are not optimal in rate of data communication. Communication complexity is the mathematical theory of such interactive processes, and is one of the core models of computation studied in theoretical computer science. It has multiple practical applications to concrete models of computation such as streaming algorithms and data structures. Recent years have seen a flurry of results in classical interactive information theory, providing extensions of Shannon's results to this interactive communication setting. In the last decades, extensions of such ideas have been studied in the realm of quantum information processing, for which we try to harness the counter-intuitive laws of nature at the subatomic level in order to process information in fundamentally different ways than what is offered by classical information processing as experienced in our day-to-day lives. The unique properties of quantum information make it intrinsically harder to handle and more susceptible to noise than classical information. The main goal of the proposed research program is to develop a theory of information for interactive quantum communication protocols, and apply the developed tools to the study of the complexity of computational problems in quantum communication complexity and other concrete models of quantum computation. My proposed research program can be split into two broad research themes: the development of interactive quantum analogues to Shannon's compression and error correcting codes. Moreover, we develop applications of the tools created for such an interactive quantum information theory to relevant problems in quantum communication complexity and other concrete models of quantum computing. One of the main reasons to study quantum information processing is that it leads to qualitative and quantitative advantages over classical information processing, and Canada is a world leader in developing such quantum technology. This is a very promising economical area and my program will also serve towards the formation of talented scientists in this field which is a burgeoning economical sector in Canada, with many young promising companies.