Operator theory with applications to quantum information theory

量子信息理论是对量子特性的研究，这些特性可用于以有效，准确和安全的方式存储，传输和处理信息。我的方法是通过算子理论和矩阵代数技术建立量子力学物理实现的数学基础，最终目标是推进量子信息理论背后的数学。量子纠错用于从通过量子信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，量子密码学用于在通过量子信道发送量子信息时隐藏信息，使得第三方无法恢复原始消息。有趣的是，量子纠错和量子密码学之间有一座代数桥梁，这两个领域可以被认为是一枚硬币的两面。与此同时，我的研究表明，这种联系在某种一般情况下会破裂;我目前正在进一步详细探讨这一点。能够保护信息免受可能的恶意窃听者的攻击具有明确的现实应用，因此开发一个明确的数学框架来实现这一点非常重要。给定一个特定的量子信道，希望有一个简单的过程来确定哪些状态（如果有的话）可以通过信道私下发送。我的工作旨在通过发展一个总体的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。量子信息理论的一个主要研究领域是纠缠变换问题：也就是说，可以通过局部操作和经典通信来操纵复合系统的纯态，并使其变换为另一种特定的状态吗？最近，这个问题已经回答了使用优控制理论，从而使优控制在量子信息理论中的重要作用。无限维纯态的纠缠熵可以是无穷大，这意味着给无限维态分配一个纠缠值并不是有限维设置的简单推广。我打算把重点放在无限维的设置上，因为量子力学本质上是无限维的。理解纠缠变换，特别是量子态的优序和劣序，有助于我们理解纠缠，最终将导致纠缠作为一种资源的充分利用。如果希望通过测量设备测量处于特定状态的系统，可以首先通过量子信道作用于系统，可以将其视为噪声源，然后通过量子信道测量系统。然后使用不同的测量装置测量所得到的系统。通过量子信道的预处理导致量子概率测度上的偏序“比”更干净，以及量子概率测度具有“干净”的期望（最佳）质量的概念。已经做了一些工作来形式化量子测量中非常普遍的最优性问题;我计划通过测量空间的新用途来追求这个主题。我希望通过测量空间的结构方面为最优性主题提供新的见解。我想回答的许多结构问题都与算子理论有关。

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.