EPSRC Reference: 
EP/P006108/1 
Title: 
Entanglement Measures, Twist Fields, and Partition Functions in Quantum Field Theory 
Principal Investigator: 
CastroAlvaredo, Dr OA 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Sch of Engineering and Mathematical Sci 
Organisation: 
City, University of London 
Scheme: 
Standard Research 
Starts: 
02 December 2016 
Ends: 
01 December 2019 
Value (£): 
229,185

EPSRC Research Topic Classifications: 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
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 manybody 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 manybody systems. Manybody 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 manybody 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.

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