# Workshop on Entanglement Entropy in Many Body Quantum Systems

Lead Research Organisation:
King's College London

Department Name: Mathematics

### Abstract

Quantum mechanics has very intriguing features. One of them is that we just cannot know all physical properties of any system for sure - there is always some fundamental uncertainty. Wherefore with quantum mechanics, we only predict probabilities of measurements. Another intriguing feature is that what we thought were particles sometimes behave like waves, and what we thought were waves sometimes behave like particles. Electrons will show interference patterns, as if there was a "probability wave".

But perhaps the most intriguing, and arguable the most "quantum" of all features, is quantum entanglement. This is the single property that displays most clearly the dichotomy between the description as particles with probabilities, and as waves with interference. Entanglement was famously referred to as "spooky action at a distance" by Einstein, and led to what is still known as the Einstein-Podolsky-Rosen "paradox". In simple terms, it says that, according to the rules of quantum mechanics, if two particles (or two quantum systems of any kind) are entangled, then a measurement on one particle will instantaneously affect the actual physical state of the other particle. This is spooky because entanglement can in principle exist between particles that are as far as we want from each other: for instance, pairs of particles spontaneously created at some point and traveling in opposite directions. Something happening here on one of these particles can affect instantaneously the state of the other while it's on the other side of the galaxy!

Entanglement has led to many interpretative issues in quantum mechanics, especially with respect to the principle of locality that was so dear to Einstein (no information can travel faster than the speed of light), and work as been done beyond the Copenhagen interpretation we implicitly referred to here. Perhaps most interestingly, however, as Feynman envisioned, quantum mechanics, and in particular quantum entanglement, led to a revolution in information and computing science. Quantum entanglement, this very quantum correlation between particles, is nowadays perceived as a resource, and gives rise to algorithm that are exponentially faster than their classical counterpart, like Shor's algorithm for prime factorization of large numbers; this may have very deep technological implications.

In recent years, the quantum information viewpoint led to an unexpected direction: quantum entanglement, it turns out, is also at the basis of many phenomena of theoretical physics that occur when many particles interact with each other. These many-body "emergent" phenomena are some of the most interesting and complex in theoretical physics, and have been known and studied for a long time; one of the most well-known being the Kondo effect, by which magnetic impurities in metals drastically affect its conductivity at very small temperatures. In the quantum entanglement viewpoint, this is simply because of the strong entanglement between metallic electrons and the magnetic impurities. Studying entanglement in many-body systems has led to surprising realizations, has challenged what we thought we understood about many-body systems, and has led to new methods, new theoretical frameworks and even new classes of many-body behaviours. This is a very active research area, with theoretical, numerical and potentially technological implications.

This workshop will bring together the leading researchers worldwide in the area of quantum entanglement in many-body systems, with an emphasis on, but not restricted to, the entanglement entropy, a mathematical characterization of entanglement which has found deep underpinning in many-body systems. The workshop will provide the most recent research in the area, and will be a platform for determining and disseminating the important problems and ideas to be developed in the near future.

But perhaps the most intriguing, and arguable the most "quantum" of all features, is quantum entanglement. This is the single property that displays most clearly the dichotomy between the description as particles with probabilities, and as waves with interference. Entanglement was famously referred to as "spooky action at a distance" by Einstein, and led to what is still known as the Einstein-Podolsky-Rosen "paradox". In simple terms, it says that, according to the rules of quantum mechanics, if two particles (or two quantum systems of any kind) are entangled, then a measurement on one particle will instantaneously affect the actual physical state of the other particle. This is spooky because entanglement can in principle exist between particles that are as far as we want from each other: for instance, pairs of particles spontaneously created at some point and traveling in opposite directions. Something happening here on one of these particles can affect instantaneously the state of the other while it's on the other side of the galaxy!

Entanglement has led to many interpretative issues in quantum mechanics, especially with respect to the principle of locality that was so dear to Einstein (no information can travel faster than the speed of light), and work as been done beyond the Copenhagen interpretation we implicitly referred to here. Perhaps most interestingly, however, as Feynman envisioned, quantum mechanics, and in particular quantum entanglement, led to a revolution in information and computing science. Quantum entanglement, this very quantum correlation between particles, is nowadays perceived as a resource, and gives rise to algorithm that are exponentially faster than their classical counterpart, like Shor's algorithm for prime factorization of large numbers; this may have very deep technological implications.

In recent years, the quantum information viewpoint led to an unexpected direction: quantum entanglement, it turns out, is also at the basis of many phenomena of theoretical physics that occur when many particles interact with each other. These many-body "emergent" phenomena are some of the most interesting and complex in theoretical physics, and have been known and studied for a long time; one of the most well-known being the Kondo effect, by which magnetic impurities in metals drastically affect its conductivity at very small temperatures. In the quantum entanglement viewpoint, this is simply because of the strong entanglement between metallic electrons and the magnetic impurities. Studying entanglement in many-body systems has led to surprising realizations, has challenged what we thought we understood about many-body systems, and has led to new methods, new theoretical frameworks and even new classes of many-body behaviours. This is a very active research area, with theoretical, numerical and potentially technological implications.

This workshop will bring together the leading researchers worldwide in the area of quantum entanglement in many-body systems, with an emphasis on, but not restricted to, the entanglement entropy, a mathematical characterization of entanglement which has found deep underpinning in many-body systems. The workshop will provide the most recent research in the area, and will be a platform for determining and disseminating the important problems and ideas to be developed in the near future.

### Planned Impact

Quantum entanglement is perhaps the currently known physical phenomenon with the most far-reaching but yet unexploited potential for technological impact. There are obvious ways in which entanglement can deeply change modern technologies: these are actual implementations of theoretical quantum computing and quantum information elements. Many laboratories worldwide are attempting to develop such technologies both in the academics and in the private sector, and there is no doubt, for instance, that quantum computers would affect the internet security landscape in an important fashion. Since quantum entanglement occurs at the level of the smallest particles like nuclei and electrons, where quantum mechanics dominates, and since many physical systems contain a very large number of such smallest particles, it is very natural to expect that the understanding of entanglement in many-body systems will be helpful for technological implementations. For instance, one possible implementation of quantum information elements is through nitrogen-vacancy diamond defects, where many-body physics is likely to play an important role.

But there are also more subtle ways in which a better understanding of quantum entanglement may have an important impact. Many-body physics is at the basis of most truly nontrivial behaviours of condensed matter; for instance, the Kondo effect of magnetic impurities in metals, or the special aspects of graphene which have potential for exciting new material properties, are effects of many electrons in interaction. The understanding of how entanglement affects or control many-body behaviours, in particular through the study of the entanglement entropy and related measures of entanglement, may shed light on such material properties, with potential technological implications.

The fact that this workshop be held in the United Kingdom also makes any potential impact from new research results and ideas more likely to occur in the U.K. More generally, it puts the U.K. at the center of research in entanglement in many-body systems, which, hopefully, will help research groups in this area to grow and potentially to generate important technological and economic impact.

Therefore, in summary, although we do not expect specific and immediate technological or economic impact from the workshop, the subjects dealt with are of a great importance in current research and have a large potential for future impact. The study of these subjects and the dissemination of the new research results and ideas in the U.K. is then of paramount importance.

But there are also more subtle ways in which a better understanding of quantum entanglement may have an important impact. Many-body physics is at the basis of most truly nontrivial behaviours of condensed matter; for instance, the Kondo effect of magnetic impurities in metals, or the special aspects of graphene which have potential for exciting new material properties, are effects of many electrons in interaction. The understanding of how entanglement affects or control many-body behaviours, in particular through the study of the entanglement entropy and related measures of entanglement, may shed light on such material properties, with potential technological implications.

The fact that this workshop be held in the United Kingdom also makes any potential impact from new research results and ideas more likely to occur in the U.K. More generally, it puts the U.K. at the center of research in entanglement in many-body systems, which, hopefully, will help research groups in this area to grow and potentially to generate important technological and economic impact.

Therefore, in summary, although we do not expect specific and immediate technological or economic impact from the workshop, the subjects dealt with are of a great importance in current research and have a large potential for future impact. The study of these subjects and the dissemination of the new research results and ideas in the U.K. is then of paramount importance.

### Publications

Hoogeveen M
(2015)

*Entanglement negativity and entropy in non-equilibrium conformal field theory*in Nuclear Physics B
Bianchini D
(2015)

*Entanglement entropy of non-unitary conformal field theory*in Journal of Physics A: Mathematical and Theoretical
Blondeau-Fournier O
(2016)

*Universal scaling of the logarithmic negativity in massive quantum field theory*in Journal of Physics A: Mathematical and TheoreticalDescription | Quantum mechanics is the theory of physics that describes the very small particles. One of the most important phenomena that arise from quantum mechanics is quantum entanglement: this is the fact that particles or degrees of freedom that are separated from each other by very large distances can influence each other immediately. Observing one can affect the physical state of the other without any delay, even if light itself would take many years to go from one to the other! Quantum entanglement is at the basis of many of the modern quantum-information algorithms that show that quantum computers, when they are built, will be much more powerful than classical ones. In this workshop, modern works on this property of quantum mechanics were presented, in the context of applications to the very many interacting electrons and other particles present in day-to-day matter. There are very special phenomena that emerge due to entanglement and the interaction of many particles, and the talks in this workshop related recent works on this. |

Exploitation Route | Many researchers attended this workshop, and the new ideas discussed helped them developing new research directions and solving problems that will lead to important new results, and new understanding of quantum entanglement in many-electron systems. |

Sectors | Digital/Communication/Information Technologies (including Software),Other |

URL | http://eemanybodyquantumsystems.weebly.com/ |

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 |

Description | conference EE june 2014 |

Form Of Engagement Activity | Participation in an activity, workshop or similar |

Part Of Official Scheme? | No |

Geographic Reach | International |

Primary Audience | Professional Practitioners |

Results and Impact | 3-day workshop on Entanglement entropy in many body quantum systems, co-organized with City University and hosted at King's and City. 50-60 participants, EPSRC and IoP funded. |

Year(s) Of Engagement Activity | 2014 |

URL | http://eemanybodyquantumsystems.weebly.com/ |