Mathematical aspects of quantum entanglement theory

量子纠缠理论的数学方面

• 批准号: RGPIN-2016-04003
• 负责人: Johnston, Nathaniel
• 金额: $ 1.58万
• 依托单位: Mount Allison University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=614774
• 关键词: Mathematical; aspects; quantum; entanglement; theory

In quantum information theory, one of the most useful resources is called "entanglement": the ability for particles to be correlated with each other in much stronger ways than those possible in classical mechanics. Entanglement is arguably the main ingredient in most interesting quantum algorithms and protocols, such as quantum teleportation and Shor's factoring algorithm, but many questions remain about its mathematical properties. The proposed program of research aims to answer two of these questions. The first question that I will tackle is: "How can we construct all possible unextendible product bases in a given quantum system?" An unextendible product basis (UPB) is a set of quantum states that are themselves not entangled, but nonetheless have found numerous uses throughout entanglement theory. For example, UPBs can be used to construct "bound entangled" states: states that are entangled, but their entanglement is so weak that they are useless for many quantum information processing tasks (including quantum teleportation). Despite the many interesting properties of UPBs that have been found, their construction until now has largely been ad hoc. Researchers have found solitary examples in specific cases, but no general-purpose method for characterizing or finding UPBs in general yet exists. I have a computational method in mind that should make substantial progress on this problem, and perhaps solve it completely (up to certain dimensions, depending on how well the algorithm scales). The second question that I will tackle is: "When is a quantum state absolutely separable?" Absolutely separable states are states that are not entangled, and furthermore it is possible to determine that they are not entangled just by looking at their eigenvalues. Equivalently, these are the states that cannot be made entangled no matter what quantum gate is applied to them, so it is desirable to avoid them in many quantum information processing tasks. Very little is known about the set of absolutely separable states. Essentially the only results about this set are actually about a set called the "absolutely PPT" states, which was introduced as a coarse approximation of the absolutely separable states that is easier to analyze. However, it seems that these two sets might not be so different after all. I recently proved that these two sets exactly coincide in certain small dimensions, and with some graduate student co-authors we developed a general method for proving "closeness" of these two sets in a certain (somewhat technical) sense. The method that we developed is quite computational in nature, so I believe that it has a natural foothold that an undergraduate student researcher would be able to latch onto, and I believe that there are many new results that can be derived from it.

在量子信息理论中，最有用的资源之一被称为“纠缠”：粒子以比经典力学更强的方式相互关联的能力。纠缠可以说是大多数有趣的量子算法和协议的主要成分，例如量子隐形传态和Shor的因子分解算法，但关于它的数学性质仍然存在许多问题。研究计划旨在回答其中两个问题。 我将解决的第一个问题是：“我们如何在给定的量子系统中构造所有可能的不可扩展的乘积基？不可扩展的乘积基（UPB）是一组量子态，它们本身并不纠缠，但在整个纠缠理论中有许多用途。例如，UPB可以用来构造“束缚纠缠”态：纠缠的状态，但它们的纠缠是如此之弱，以至于它们对许多量子信息处理任务（包括量子隐形传态）毫无用处。尽管已经发现了UPB的许多有趣的特性，但到目前为止，它们的构造在很大程度上是临时的。研究人员在特定情况下发现了孤立的例子，但还没有通用的方法来表征或发现一般的UPB。我有一个计算方法，应该在这个问题上取得实质性的进展，也许可以完全解决它（直到某些维度，取决于算法的扩展程度）。 我将要解决的第二个问题是：“什么时候量子态是绝对可分的？“绝对可分的状态是没有纠缠的状态，而且可以通过观察它们的本征值来确定它们没有纠缠。等价地，这些是无论应用什么量子门都不能使它们纠缠的状态，因此在许多量子信息处理任务中希望避免它们。 关于绝对可分态的集合，我们所知甚少。本质上，关于这个集合的唯一结果实际上是关于一个被称为“绝对PPT”状态的集合，它被引入作为更容易分析的绝对可分状态的粗略近似。不过，这两套似乎也没什么区别。我最近证明了这两个集合在某些小维度上完全重合，并且与一些研究生合著者一起开发了一种通用方法来证明这两个集合在某种（有点技术）意义上的“接近性”。我们开发的方法在本质上是相当计算的，所以我相信它有一个自然的立足点，一个本科生研究人员将能够抓住，我相信有许多新的结果，可以从中得出。