Detection and Quantification of Quantum Resources

在量子信息理论中，量子态的“有用性”有几种不同的概念，这取决于如何处理它们。例如，有时“有用”的状态是那些纠缠的状态（即具有经典力学中不存在的非局部属性），有时“有用”的状态是那些相干的状态（即叠加态）。这些不同的“有用性”概念产生了所谓的量子资源理论：我们认为“有用”的杰出量子状态的集合，以及量化量子状态有用程度的某种方法。我的研究计划旨在更好地了解量子信息理论中感兴趣的各种资源，并探索它们之间的关系。在接下来的几年里，我计划在这个整体研究计划中探索许多具体的项目：纠缠见证是一种数学装置，可以用来检测量子系统中的纠缠。我们通常没有关于量子态或纠缠见证的完整信息，所以我们会问这样的问题：“如果我们只知道一个态的特征值，我们能说什么？”我打算探讨这个问题，如果我们只知道一个证人的特征值或一个状态的特征值，我们能对纠缠说些什么。2. 同样，如果我们只知道一个状态的特征值，我们能确定它在相干资源理论中是否有用吗？我（和其他一些研究人员）有一个猜想，可以完全解决这个问题。对于如何证明这个猜想，我们也有一些想法，但每个想法都很复杂，需要大量的工作。3. 最近在量子资源理论和“完全正”矩阵集之间建立了许多联系，这已经被矩阵分析和凸优化领域的研究人员研究了几十年。然而，对于如何利用完全正矩阵的各种已知性质来告诉我们，例如，量子态是可分离的还是纠缠的，还没有任何深入的研究。我将解决这个问题，并使用完全正矩阵来开发一些新的可分性测试。4. 如果我们有一个无限多个量子态的集合，我们怎么能确定它们都是纠缠的？一般来说，这个问题是非常困难的，但我（和其他一些研究人员）已经开发了一种方法，在实践中似乎工作得很好。我们将探讨用这种方法可以做些什么，并确定它什么时候有效（什么时候不那么有效）。

In quantum information theory, there are several different notions of "usefulness" of quantum states, depending on what is to be done with them. For example, sometimes the states that are "useful" are those that are entangled (i.e., have non-local properties not present in classical mechanics), and sometimes the states that are "useful" are those that are coherent (i.e., in a superposition). These different notions of "usefulness" give rise to what is called a quantum resource theory: a collection of distinguished quantum states that we think of as "useful", and some way of quantifying how useful a quantum state is. My research program aims to develop a better understanding of the various resources that are of interest in quantum information theory, and also explore how they are related to each other. I have numerous specific projects within this overall research program that I plan to explore over the next few years: 1. An entanglement witness is a mathematical device that can be used to detect entanglement in a quantum system. We often do not have complete information about a quantum state or an entanglement witness, so we ask questions like "what can we say about a state if we only know its eigenvalues?" I intend to explore this question of what we can say about entanglement if we only know a witness's eigenvalues or a state's eigenvalues. 2. Similarly, can we determine whether or not a state is useful in the resource theory of coherence if we only know its eigenvalues? I (with a few other researchers) have a conjecture that would completely solve this problem. We also have a few ideas for how we could prove the conjecture, but each of them is quite involved and will require a significant amount of work. 3. Numerous connections have recently been made between quantum resource theories and the set of "completely positive" matrices, which have been studied for decades by researchers in the fields of matrix analysis and convex optimization. However, there has not been any thorough investigation into how the various known properties of completely positive matrices can be used to tell us, for example, if a quantum state is separable or entangled. I will tackle this question, and use completely positive matrices to develop some new tests for separability. 4. If we have a collection of infinitely many quantum states, how can we be sure that all of them are entangled? In general, this problem is very difficult, but I (with a few other researchers) have developed a method that seems to work quite well in practice. We will explore what can be done with this method, and pin down exactly when it works well (and not so well).