Frame Phase-Retrievability and Applications to Quantum Information

框架为信号和信息提供了理想的数学表示，因此在开发解决科学和工程应用中许多具有挑战性问题的方法方面发挥了重要作用。框架与量子通信的数学理论有着天然的联系。例如，帧的相位可恢复性是一个重要的特性，它允许从帧系数测量的幅度中恢复信号。这出现在许多应用中，包括语音识别，x射线晶体学和电子显微镜。在量子信息理论中，这是量子系统区分纯态和它们的量子测量值的必要属性。该项目解决了该地区的几个基本问题。首席研究员（PI）将研究框架在涉及量子通信能力和信息传输安全问题中的应用。通过引入新的思想来解决这些实际问题，本研究将促进对量子信息理论以及应用泛函/谐波分析的科学理解。此外，这个项目将促进教学、培训和学习，因为一些研究生和本科生将直接参与这个研究项目。本研究的一个方向是建立（主要是算子值）帧相位可恢复性的量化测量，然后使用它们来解决设计（或表征）具有规定水平的零错误通信容量和信息传输安全的量子通信信道的问题。将特别注意结构化通道，这通常是由不同类型的表示框架引起的。在这里，算子值框架和投影群表示之间的相互作用将发挥关键作用。这个项目的第二个方向是在信号/状态恢复（或通道检测）领域从一个来源的不明信道。PI将与不相交框架理论和量子通道理论建立理论联系。一个关键因素是使用不相交的帧作为构建块来产生一个大的候选类，其中这个集合的任何子类都可以用作量子设置中目标信号集或状态集的未识别信道源。这种方法需要为冯·诺伊曼代数上的有界线性映射发展一个不相交框架理论（或多路复用）。第三个方向涉及量子测量。量子测度的膨胀问题不仅在上述研究领域的范围内，而且也符合PI的长期目标，即在交换和非交换条件下为算子值测度建立一个一般的膨胀理论。这样一种广义膨胀理论可能导致一种可能的量子测量分类理论，以及它们相关的量子通道。主要目标是建立一个量子测量的膨胀理论，告诉我们量子测量的某些重要信息可以通过膨胀保存在哪些时间，以及如何保存。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Frames provide desirable mathematical representations for signals and information, and consequently play an important role in developing approaches to many challenging questions in science and in engineering applications. Frames have natural connections with the mathematical theory of quantum communications. For example, phase-retrievability of a frame is an important property that allows the recovery of a signal from the magnitudes of its frame coefficient measurements. This appears in many applications including speech recognition, x-ray crystallography and electron microscopy. In quantum information theory, this is a necessary property for a quantum system to have to distinguish pure states from their quantum measurements. This project addresses several fundamental issues in the area. The principal investigator (PI) will investigate frame applications to problems involving quantum communication capability and information transmission security. By bringing a new circle of ideas to attack these practical problems, this investigation will advance scientific understanding in quantum information theory as well as in applied functional/harmonic analysis. Additionally, this project will promote teaching, training, and learning as several graduate and undergraduate students will be directly involved in this research project.One direction of this investigation is to establish quantified measurements of (mostly, operator-valued) frame phase-retrievability, and then use them to tackle problems of designing (or characterizing) quantum communication channels with prescribed levels of zero-error communication capacity and information transmission security. Special attention will be given to the structured channels, which are usually induced by different kinds of representation frames. Here, the interplay between operator-valued frames and projective group representations will play a key role. A second direction of this project is in the area of signal/state recovering (or channel detection) from a source of unidentified channels. The PI will establish its theoretical connections with the theory of disjoint frames and quantum channels. A key ingredient is to use disjoint frames as building blocks to produce a large class of candidates, where any subclass of this set can be used as a source of unidentified channels for a target set of signals or states in the quantum setting. Such an approach requires developing a disjoint frame theory (or multiplexing) for bounded linear maps on von Neumann algebras. A third direction involves quantum measures. Dilation problems for quantum measures are not only in the scope of aforementioned areas of investigation but also in line with the PI’s long-term goal of establishing a general dilation theory for operator-valued measures in both commutative and non-commutative settings. Such a general dilation theory may lead to a possible classification theory for quantum measures as well as for their associated quantum channels. The main objective is to establish a dilation theory for quantum measures that can tell us which, when, and how certain important information of a quantum measure can be preserved through dilations.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.