# nLab entanglement entropy

Contents

### Context

#### Measure and probability theory

measure theory

probability theory

# Contents

## Idea

entropy induced by entanglement in quantum physics, essentially a synonym for subsystem entropy.

If $\rho$ is a quantum state (density matrix) of some quantum systems and $A$ is a subsystem with complementary subsystem $\overline{A}$, then its entanglement entropy is

$S_A \;\coloneqq\; - Tr_{A}\big( \rho_A \ln \rho_A \big) \,, \;\;\; \text{where} \;\;\; \rho_A \;\coloneqq\; Tr_{\bar A}(\rho) \,.$

Special aspects:

A constant contribution to $S_A$ (i.e. independent of the choice of $A$, in a suitable sense) is called a topological entanglement entropy indicating long-range entanglement and “topological order”, being proportional to the total quantum dimension of anyon-excitations. (See the references below.)

On the other extreme, for short range entanglement the entanglement entropy is thought to scale with the surface area of the subsystem $A$ (to the extent that this makes sense, say in say in lattice models), a behaviour reminiscent of Bekenstein-Hawking entropy of black holes. For more on this see at holographic entanglement entropy.

## References

### General

• Matthew Headrick, Lectures on entanglement entropy in field theory and holography (arXiv:1907.08126)

In AQFT:

### Topological entanglement entropy

#### General

Identification of a contribution to entanglement entropy at absolute zero which is independent of the subsystem‘s size (“topological entanglement entropy”, “long-range entanglement”), reflecting topological order and proportional to the total quantum dimension of anyon excitations:

Review:

In terms of Renyi entropy (it’s independent of the Renyi entropy parameter):

• Ulrich Schollwöck, (Almost) 25 Years of DMRG - What Is It About? (pdf)

Discussion in the dimer model:

• Shunsuke Furukawa, Gregoire Misguich, Topological Entanglement Entropy in the Quantum Dimer Model on the Triangular Lattice, Phys. Rev. B 75, 214407 (2007) (arXiv:cond-mat/0612227)

Discussion via holographic entanglement entropy:

• Ari Pakman, Andrei Parnachev, Topological Entanglement Entropy and Holography, JHEP 0807: 097 (2008) (arXiv:0805.1891)

• Andrei Parnachev, Napat Poovuttikul, Topological Entanglement Entropy, Ground State Degeneracy and Holography, Journal of High Energy Physics volume 2015, Article number: 92 (2015) (arXiv:1504.08244)

• Tatsuma Nishioka, Tadashi Takayanagi, Yusuke Taki, Topological pseudo entropy (arXiv:2107.01797)

#### Relation to strong interaction

Relation of long-range entanglement to strong interaction:

• Jan Zaanen, Yan Liu, Ya-Wen Sun, Koenraad Schalm, Holographic Duality in Condensed Matter Physics, Cambridge University Press 2015 $[$doi:10.1017/CBO9781139942492$]$

In a way it appears obvious that the strongly interacting bosonic quantum critical state is subject to long-range entanglement. Nonetheless, the status of this claim is conjectural.

It is at present impossible to arrive at more solid conclusions that are based on rigorous mathematical procedures. It does illustrate emphatically the central challenge in the pursuit of field-theoretical quantum information: there are as yet not general measures available to precisely enumerate the meaning of long- range entanglement in such seriously quantum field-theoretical systems. $[$p. 527$]$

• Tsung-Cheng Lu, Sagar Vijay, Characterizing Long-Range Entanglement in a Mixed State Through an Emergent Order on the Entangling Surface $[$arXiv:2201.07792$]$

strongly interacting quantum phases of matter at zero temperature can exhibit universal patterns of long-range entanglement

#### Simulation and experiment

Experimental observation:

• A. Hamma, W. Zhang, S. Haas, and D. A. Lidar, Entanglement, fidelity, and topological entropy in a quantum phase transition to topological order, Phys. Rev. B 77, 155111 (2008) (doi:10.1103/PhysRevB.77.155111, arXiv:0705.0026)

• Hong-Chen Jiang, Zhenghan Wang, Leon Balents, Identifying Topological Order by Entanglement Entropy, Nature Physics 8 902-905 (2012) $[$arXiv:1205.4289$]$

Detection of long-range entanglement entropy in quantum simulations on quantum computers:

• Realizing topologically ordered states on a quantum processor, Science 374 6572 (2021) 1237-1241 $[$doi:10.1126/science.abi8378$]$

• Probing topological spin liquids on a programmable quantum simulator, Science 374 6572 (2021) 1242-1247 $[$doi:10.1126/science.abi8794$]$

exposition in:

Last revised on May 26, 2022 at 10:12:35. See the history of this page for a list of all contributions to it.