Mathematical aspects of quantum entanglement theory

在量子信息理论中，最有用的资源之一被称为“纠缠”：粒子以比经典力学中可能的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肖尔的分解算法，但是关于它的数学性质仍然存在许多问题。拟议的研究计划旨在回答其中两个问题。******我要解决的第一个问题是：“我们如何在给定的量子系统中构建所有可能的不可扩展的产品基？”不可扩展积基（UPB）是一组本身不纠缠的量子态，但在纠缠理论中已经发现了许多用途。例如，upb可以用来构建“束缚纠缠”状态：纠缠的状态，但是它们的纠缠非常弱，以至于它们对于许多量子信息处理任务（包括量子隐形传态）是无用的。尽管已经发现了许多有趣的ups特性，但到目前为止，它们的构造在很大程度上是临时的。研究人员已经在特定的案例中发现了单独的例子，但还没有通用的方法来描述或发现一般的ups。我脑海中有一个计算方法，它应该在这个问题上取得实质性进展，也许可以完全解决它（在某些维度上，取决于算法的扩展程度）。******我要解决的第二个问题是：“什么时候量子态是绝对可分离的？”绝对可分离的状态是不纠缠的状态，更进一步，我们可以通过观察它们的特征值来确定它们不纠缠。同样地，这些状态是无论施加什么量子门都无法使其纠缠的状态，因此在许多量子信息处理任务中都希望避免它们。******我们对绝对可分离状态的集合所知甚少。从本质上讲，关于这组的唯一结果实际上是关于一组称为“绝对PPT”的状态，它是作为绝对可分离状态的粗略近似引入的，更容易分析。然而，这两组似乎并没有太大的不同。我最近证明了这两个集合在特定的小维度上完全重合，并与一些研究生合作作者开发了一种通用方法，在某种意义上（有点技术性）证明了这两个集合的“紧密性”。我们开发的方法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可以抓住它，我相信有很多新的结果可以从中得出

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.**