Operator theory with applications to quantum information theory

算子理论及其在量子信息论中的应用

• 批准号: RGPIN-2014-06457
• 负责人: Plosker, Sarah
• 金额: $ 1.09万
• 依托单位: Brandon University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635288
• 关键词: Operator; theory; applications; quantum; information

Quantum information theory is the study of quantum properties that can be used to store, transmit, and process information in an efficient, accurate, and secure way. My approach is to build up the mathematical foundations for physical realizations in quantum mechanics through operator theory and matrix algebra techniques with the end goal of advancing the mathematics behind quantum information theory. Quantum error correction is used to recover information from errors introduced by noise that occurs when sending quantum information through a quantum channel. Quantum cryptography, on the other hand, is used to hide information when sending quantum information through a quantum channel so that the original message cannot be recovered by a third party. Intriguingly, there is an algebraic bridge linking quantum error correction with quantum cryptography, and the two fields can be thought of as two sides of a coin. At the same time, my research has shown that this connection breaks down in a certain general setting; I am presently exploring this in further detail.Being able to protect information against possibly malicious eavesdroppers has clear real-world applications, and it is therefore important to develop a clear mathematical framework for how to do so. Given a particular quantum channel, it would be desirable to have a straightforward procedure for determining which states (if any) can be sent privately over the channel. My work aims to address this issue through the development of an overarching private quantum code theory. Entanglement is a key to quantum information theory; being able to manipulate entanglement makes quantum information a powerful tool. A major area of research in quantum information theory is the problem of entanglement transformations: that is, can one manipulate a pure state of a composite system via local operations and classical communication and have it transform into another particular state? Recently, this question has been answered using majorization theory, thus giving majorization an important role in quantum information theory. The entropy of entanglement of an infinite-dimensional pure state can be infinite, meaning that assigning a value of entanglement to an infinite-dimensional state is not a straightforward generalization of the finite-dimensional setting. I plan to focus on the infinite-dimensional setting, since quantum mechanics is inherently infinite-dimensional. Understanding entanglement transformations, and in particular the partial orders of majorization and trumping on quantum states, informs our understanding of entanglement, which will ultimately lead to the full use of entanglement as a resource.If one wishes to measure a system that is in a particular state via a measurement apparatus, one can first act upon the system by a quantum channel, which can be thought of as a noise source, and then measure the resulting system using a different measurement apparatus. Preprocessing by a quantum channel leads to the partial order “cleaner than” on quantum probability measures and the notion of a quantum probability measure having the desirable (optimal) quality of being “clean”. Some work has been done to formalize the very general issue of optimality in quantum measurements; I plan to pursue this topic through a novel use of measurement spaces. I hope to provide new insights into the subject of optimality via structural aspects of the measurement spaces. Many of the structural questions I wish to answer are of independent interest in operator theory.

量子信息理论是对可用于以有效，准确且安全的方式存储，传输和处理信息的量子特性的研究。我的方法是通过操作者理论和矩阵代数技术在量子力学中为物理现实建立数学基础，以最终目标促进量子信息理论背后的数学。量子误差校正用于从通过量子通道发送量子信息时发生的噪声引入的错误中恢复信息。另一方面，量子密码学通过量子通道发送量子信息时，用于隐藏信息，以便第三方无法恢复原始消息。有趣的是，有一个代数桥将量子误差校正与量子密码学连接起来，并且两个场可以被认为是硬币的两个侧面。同时，我的研究表明，这种联系在某个一般环境中崩溃了。我目前正在进一步详细地探索这一点。可以保护信息免受可能的恶意窃听器具有清晰的现实应用程序，因此，为如何制定一个清晰的数学框架很重要。给定特定的量子通道，希望有一个直接的程序来确定可以通过频道私下发送（如果有）的状态。我的工作旨在通过开发总体私人量子代码理论来解决这个问题。纠缠是量子信息理论的关键。能够操纵纠缠使量子信息成为强大的工具。量子信息理论研究的主要研究领域是纠缠转换的问题：也就是说，一个人可以通过本地操作和经典交流来操纵复合系统的纯状态，并可以转化为另一个特定状态吗？最近，该问题已使用大化理论回答，因此在量子信息理论中赋予了多数化的重要作用。无限维纯状态的纠缠的熵可以是无限的，这意味着将纠缠的值分配给无限维状态并不是对有限维设置的直接概括。我计划专注于无限维度的设置，因为量子力学本质上是无限维度的。理解纠缠转变，尤其是在量子状态下进行大规模化和胜过的部分顺序，可以告知我们对纠缠的理解，这最终将导致完全使用纠缠作为一种资源。在量子概率测量上导致部分阶“更清洁”，以及具有“清洁”的理想（最佳）质量的量子概率测量的概念。已经完成了一些工作，以正式化量子测量中的最优性问题。我计划通过新颖的测量空间来购买此主题。我希望通过测量空间的结构方面对最优主题提供新的见解。我希望回答的许多结构性问题对操作者理论具有独立的兴趣。