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

量子场论中的纠缠测度、扭曲场和配分函数

基本信息

批准号：
EP/P006108/1
负责人：
Olalla Castro-Alvaredo
金额：
$ 29.2万
依托单位：
City, University of London
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2016
资助国家：
英国
起止时间：
2016 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FP006108%2F1
关键词：
Entanglement Measures Twist Fields Partition

项目摘要

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.

量子力学是描述原子尺度物理现象的理论。它定义了一组数学对象，这些对象描述了物理系统的特征，并指定了需要对这些对象执行哪些数学运算，以提取有关系统的信息。在量子力学中，我们经常谈论“系统的状态”，意思是它的属性。在数学上，状态是具有某些特殊性质的向量。类似地，量子力学中的可观测量是我们可以测量的任何性质。在数学上，可观测量由矩阵表示。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会变得非常复杂）。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，有可能准备两个处于某种状态的电子，如果我们可以测量电子1的属性，我们将自动知道电子2的相同属性的值，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。它在经典力学中没有对应物（例如，描述行星运动的那种物理学），它以惊人的方式展示了微观尺度上自然界的“怪异”量子行为，引起了科学家的极大关注。继量子力学之后，世纪物理学最大的进步之一是建立了描述多体量子物理的理论系统.这本质上是量子力学的推广，我们有数百个（可能是无限多个）基本粒子相互作用的情况。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，它也结合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（如在CERN进行的实验）方面取得了令人难以置信的成功。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，我们想研究多体系统的量子态的各种函数的数学性质，为我们提供关于可以存储在这种状态中的纠缠量的信息。所讨论的函数被称为纠缠熵（EE）和对数负性（LN），它们以前已经在特定类型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。迄今已知的大多数结果适用于被称为共形场论（CFTs）或临界QFTs的QFTs的重要子集。CFTs有许多特殊的功能和许多应用，包括在多体系统的涌现行为的描述。多体临界系统在所有长度尺度上都显示出相关性，这意味着系统一部分的微小局部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规模或有隙模型。这类模型描述了多体系统在临界附近但不处于临界的普遍特征，但从纠缠的角度研究较少。我们的项目将通过计算大规模QFT中的纠缠度并将其推广到更高空间维度的系统来填补这一空白。沿着的方式将开发一个新的数学框架，这是基于使用一个特定的家庭的本地字段和它们的相关函数。