Entanglement Measures, Twist Fields, and Partition Functions in Quantum Field Theory

量子力学是描述原子尺度上的物理现象的理论。它定义了一组表征物理系统的数学对象，并指定需要在这些对象上执行哪些数学操作以提取有关系统的信息。在量子力学中，我们经常说“系统的状态”，意思是它的性质。在数学上，状态是一个具有某些特殊性质的向量。同样，在量子力学中，可观测的是我们可以测量的任何属性。在数学上，可观察对象用矩阵表示。这个理论的美妙之处在于，一旦我们有了向量和矩阵，我们就可以使用标准技术来执行计算（即使这些计算对于复杂的物理系统来说可能变得非常复杂）。这个研究项目的核心在于量子力学的一个特殊特征：它允许两个不同量子系统的状态纠缠在一起。这意味着，在某些情况下，有可能制备两个处于某种状态的电子，如果我们可以测量电子1的一个性质，我们将自动知道电子2的相同性质的值，而不需要进行第二次独立的测量。纠缠是一种真正的量子现象。它在经典力学（例如描述行星运动的那种物理学）中没有对应的东西，它以一种惊人的方式展示了自然界在微观尺度上的“奇怪”量子行为，引起了科学家们的极大关注。继量子力学之后，20世纪物理学最伟大的进步之一是提出了可以描述许多身体量子系统的物理学的理论。从本质上讲，这是量子力学在我们有数百个（可能无限多）基本粒子相互作用的情况下的推广。这种高度复杂的系统最好用量子力学的连续体版本来描述，它也包含了广义相对论的原理。这些理论被称为量子场论（QFTs），它们在描述许多实验结果方面已经被证明是非常成功的，比如在欧洲核子研究中心进行的实验。在这种情况下，系统的状态由希尔伯特空间中的向量描述，可测量量的值与作用于该空间的局部算子的期望值有关。在这个项目中，我们想要研究给定多体系统量子态的各种函数的数学性质，给我们在这种状态下可以存储的纠缠量的信息。所讨论的函数被称为纠缠熵（EE）和对数负性（LN），它们之前已经在特定种类的量子理论以及理论量子计算和信息论的背景下进行了研究。迄今为止已知的大多数结果适用于量子傅里叶变换的一个重要子集，即共形场论（CFTs）或临界量子傅里叶变换。CFTs具有许多特殊的特性和许多应用，包括描述多体系统中的紧急行为。多体临界系统在所有长度尺度上都显示出相关性，这意味着系统某一部分的局部小变化会迅速传播到整个系统。相比之下，另一类qft是相关长度有限的大量或间隙模型。这些模型描述了多体系统在接近临界而非临界的普遍特征，从纠缠的角度研究较少。我们的项目将通过计算大规模qft中的纠缠度量并将其推广到更高空间维度的系统来填补这一空白。在此过程中，将开发一个新的数学框架，该框架基于使用特定的局部场及其相关函数。

Quantum Mechanics is the theory that describes physical phenomena at atomic scales. It defines a set of mathematical objects which characterize a physical system and specifies which mathematical operations on those objects need to be performed in order to extract information about the system. In quantum mechanics we often speak about "the state of a system" meaning its properties. Mathematically, a state is a vector with certain special properties. Similarly, an observable in quantum mechanics is any property that we can measure. Mathematically, observables are represented by matrices. The beauty of the theory is that once we have vectors and matrices, we can use standard techniques to perform computations (even if these computations can become extremely involved for complex physical systems).At the heart of this research project lies a particular feature of quantum mechanics: it allows for the states of two different quantum systems to be entangled. This means that under certain circumstances it is possible to prepare say, two electrons in a state such that if we can measure a property of electron 1 we will automatically know the value of the same property for electron 2 without needing to perform a second, independent measurement. Entanglement is a genuine quantum phenomenon. It has no counterpart in classical mechanics (e.g. the sort of physics that describes planetary motion) and it has attracted much attention among scientists as it demonstrates in a striking way the "weird" quantum behaviour of nature at microscopic scales.Following on from quantum mechanics, one of the greatest advancements in Physics in the 20th century has been the formulation of theories which can describe the physics of many body quantum systems. This is in essence the generalisation of quantum mechanics to the situation where we have hundreds (potentially infinitely many) elementary particles in interaction. Such highly complex systems are best described by a continuum version of quantum mechanics which also incorporates the principles of general relativity. These theories are known as quantum field theories (QFTs) and they have proven incredibly successful in describing the results of many experiments such as those performed at CERN. In this setting the state of the systems is described by a vector in a Hilbert space and the values of measurable quantities are related to expectation values of local operators acting on that space. In this project we want to investigate the mathematical properties of various functions which given a quantum state of a many-body system, give us information about the amount of entanglement that can be stored in such a state. The functions in question are known as the entanglement entropy (EE) and the logarithmic negativity (LN) and they have been previously studied for particular kinds of quantum theories and also in the context of theoretical quantum computation and information theory. Most of the results hitherto known apply to an important subset of QFTs which are known as conformal field theories (CFTs) or critical QFTs. CFTs have many special features and many applications including to the description of emergent behaviours in many-body systems. Many-body critical systems display correlations at all length scales, meaning that small local changes to one part of the system quickly propagate to the whole system. In contrast, another family of QFTs are massive or gapped models where the correlation length is finite. Such models describe universal features of many-body systems near but not at criticality and have been less studied from the viewpoint of entanglement. Our project will contribute to filling this gap by computing measures of entanglement in massive QFTs and generalising these to systems in higher space dimensions. Along the way a new mathematical framework will be developed which is based on the use of a particular family of local fields and their correlation functions.