Operator theory with applications to quantum information theory

量子信息论是一门研究量子属性的学科，这些属性可用于以高效、准确和安全的方式存储、传输和处理信息。我的方法是通过算符理论和矩阵代数技术为量子力学中的物理实现建立数学基础，最终目标是推动量子信息理论背后的数学。 量子纠错用于从通过量子通道发送量子信息时出现的噪声引入的错误中恢复信息。另一方面，量子密码术用于在通过量子通道发送量子信息时隐藏信息，以便第三方无法恢复原始消息。耐人寻味的是，量子纠错和量子密码学之间存在着一座代数桥梁，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具有明确的现实应用，因此为如何做到这一点开发一个明确的数学框架是很重要的。在给定特定的量子信道的情况下，希望有一种简单的程序来确定哪些状态(如果有的话)可以通过该信道私下发送。我的工作旨在通过发展一种重要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一种强大的工具。量子信息论的一个主要研究领域是纠缠变换问题：即，人们能否通过局域操作和经典通信来操纵复合系统的纯态，并使其转变为另一种特定的态？最近，这个问题被用优化理论给出了解答，从而使优化在量子信息论中占有重要的地位。 无限维纯态的纠缠熵可以是无限的，这意味着给无限维态分配一个纠缠值并不是对有限维设置的直接概括。我计划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，特别是量子态上的优胜化和王牌的偏序，有助于我们理解纠缠，这最终将导致充分利用纠缠作为一种资源。 如果希望通过测量设备测量处于特定状态的系统，可以首先通过可以被认为是噪声源的量子通道作用于该系统，然后使用不同的测量设备来测量所得到的系统。通过量子通道的预处理，可以得到量子概率度量的偏序“干净”，以及量子概率度量具有“干净”这一理想(最佳)性质的概念。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一种新的使用来探讨这一主题。我希望通过测量空间的结构方面，为最优性这个主题提供新的见解。我想回答的许多结构性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.