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

Lead Research Organisation:
King's College London

Department Name: Mathematics

### Abstract

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.

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.

## People |
## ORCID iD |

Benjamin Doyon (Principal Investigator) |

### Publications

Castro-Alvaredo O
(2017)

*Irreversibility of the renormalization group flow in non-unitary quantum field theory*in Journal of Physics A: Mathematical and Theoretical
Blondeau-Fournier O
(2017)

*Expectation values of twist fields and universal entanglement saturation of the free massive boson*in Journal of Physics A: Mathematical and Theoretical
Castro-Alvaredo O
(2018)

*Conical twist fields and null polygonal Wilson loops*in Nuclear Physics B
Castro-Alvaredo O
(2018)

*Entanglement content of quantum particle excitations. Part I. Free field theory*in Journal of High Energy Physics
Castro-Alvaredo OA
(2018)

*Entanglement Content of Quasiparticle Excitations.*in Physical review letters
Doyon B
(2019)

*Fluctuations in Ballistic Transport from Euler Hydrodynamics*in Annales Henri PoincarĂ©
Castro-Alvaredo O
(2019)

*Entanglement content of quantum particle excitations. III. Graph partition functions*in Journal of Mathematical Physics
Castro-Alvaredo O
(2019)

*Entanglement content of quantum particle excitations. Part II. Disconnected regions and logarithmic negativity*in Journal of High Energy Physics
Doyon B
(2020)

*Lecture notes on Generalised Hydrodynamics*in SciPost Physics Lecture NotesDescription | The behaviour of physical systems composed of many particles in interactions is extremely difficult to predict and to understand. One of the deepest idea which has allowed us to gain a large understanding is that of the relation between the small and large scales, and how information or energy flows between scales. The emergence of large scale structures from small scales fluctuations is a very important phenomenon which is starting to be better understood. An important old result is that information can only be lost as we go to larger scales (sometimes referred to as the "c-theorem"). One key finding of our research is the first proof, from tools of theoretical physics, that this result holds in a larger class of quantum systems in one dimension (the "PT-symmetric systems"), than what in which the result was initially understood. This was a long-standing problem, and for the first time we have provided an understanding. This result was in fact very unexpected in the present research project, and would have seemed very unlikely had we explicitly proposed it as an objective. Another key finding is directly related to the stated objectives of the grant, and relates to the behaviour of quantum entanglement in systems with very many particles in interaction. Quantum entanglement is a very non-classical effect that is used by quantum computers in order to gain efficiency. One question is to understand how much entanglement there is in systems with a large number of particles in interaction. There has been a lot of advances in recent years. As a key finding coming out of this grant, we understood how the presence of few "excitations" above the fundamental state of a many-body system changes its entanglement. The picture is extremely elegant: these excitations are certain kind of quantum waves, often themselves seen as quantum particles, or quasiparticles, in accordance with the wave-particle duality of quantum mechanics. We have found that the entanglement due to such quasiparticles has a very simple expression, which can be interpreted with a simple probability argument, and that this expression is extremely universal, largely independent of the details of the underlying many-body system. This is a beautiful, universal result, allowing us to "see" the quasiparticles using quantum entanglement. A number of papers have been published in excellent journals detailing the ideas and its application in a variety of situations. A final key finding, made towards the end of this grant, has been the connection between certain entanglement measures, and certain measures of fluctuations, thus connecting two apparently distinct physical phenomena. This finding is relatively new and further research is required in order to fully develop the idea and its consequences. |

Exploitation Route | Concerning the first key finding, many researchers in quantum field theory (a theory for emergent behaviours in many-body quantum systems) might make use of the first key finding result, especially in the context of new states of matter where non-unitary PT-symmetric quantum mechanics may play a role. Concerning the second key finding, it might be used within the context of quantum information, for building states with specific entanglement properties, or in condensed matter physics for characterising excitations as quasiparticles. |

Sectors | Other |

Description | Collaboration with Davide Fioravanti |

Organisation | University of Bologna |

Department | Department of Physics and Astronomy |

Country | Italy |

Sector | Academic/University |

PI Contribution | This was a collaboration on one paper. Me and my team provided the main calculations and the idea of the conical twist field. |

Collaborator Contribution | Davide Fioravanti provided the knowledge about the relation with polyognal Wilson loops. |

Impact | Olalla A. Castro-Alvaredo, B. Doyon, Davide Fioravanti, Conical twist fields and null polygonal Wilson loops, Nucl. Phys. B 931 (2018) 146--178, preprint arXiv: 1709.05980 (33 pages). DOI 10.1016/j.nuclphysb.2018.04.002 |

Start Year | 2017 |

Description | Collaboration with Olivier Blondeau-Fournier |

Organisation | King's College London |

Country | United Kingdom |

Sector | Academic/University |

PI Contribution | I wrote a paper with Olivier supported by this grant. I proposed the project and we performed the calculations together. In particular, I developed the aspects relating to angular quantization and the general discussions of twist fields. |

Collaborator Contribution | Olivier performed the more technical calculations related to conformal field theory, as well as various computations necessary to related the general formulae to the angular quantization results. |

Impact | Two papers were written, including that linked above relevant for the award. |

Start Year | 2015 |

Description | Collaboration with Prof. Francesco Ravanini |

Organisation | University of Bologna |

Department | Department of Physics and Astronomy |

Country | Italy |

Sector | Academic/University |

PI Contribution | This is a scientific collaboration on the subject of non-unitary quantum field theory. I have helped find a proof of the ceff-theorem. |

Collaborator Contribution | The partner mentioned proposed the study of the ceff-theorem in non-unitary CFT. |

Impact | no output yet. |

Start Year | 2016 |

Description | OCA |

Organisation | City, University of London |

Country | United Kingdom |

Sector | Academic/University |

PI Contribution | This is a scientific collaboration with O. Castro Alvaredo, mainly on the subject of entanglement entropy in extended quantum systems, but also on other subjects within integrable quantum field theory. We have both provided equally to this collaboration, in ideas, calculations and in the writing of papers. |

Collaborator Contribution | This is a scientific collaboration with O. Castro Alvaredo, mainly on the subject of entanglement entropy in extended quantum systems, but also on other subjects within integrable quantum field theory. We have both provided equally to this collaboration, in ideas, calculations and in the writing of papers. |

Impact | 10.1088/1751-8113/49/12/125401, 10.1016/j.nuclphysb.2015.06.021, 10.1016/j.nuclphysb.2015.05.013, 10.1088/1751-8113/48/4/04FT01, 10.1088/1742-5468/2014/03/P03011, 10.1103/PhysRevB.88.094439, 10.1088/1742-5468/2013/02/P02016, 10.1103/PhysRevLett.108.120401, 10.1088/1751-8113/44/49/492003, 10.1088/1742-5468/2011/02/P02001, 10.1088/1751-8113/42/50/504006, 10.1007/s10955-008-9664-2, 10.1103/PhysRevLett.102.031602, 10.1088/1751-8113/41/27/275203, 10.1007/s10955-007-9422-x |

Start Year | 2006 |