nLab entanglement entropy

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.

Related concepts

• tensor network

• holographic entanglement entropy

• black hole entropy via AdS-CFT duality

quantum probability theory – observables and states

• states

  • classical state

  • quantum state

    • space of states (in geometric quantization)

    • state on a star-algebra, quasi-state

    • qbit, Bell state

      dimer, tensor network state

    • quantum state preparation

    • probability amplitude, quantum fluctuation

    • pure state

      • wave function

      • bra-ket

      • Bell state

    • quantum superposition, quantum interference

    • quantum entanglement

    • quantum measurement

      wave function collapse

      Born rule

      deferred measurement principle

      quantum reader monad

      measurement problem

    • superselection sector

    • mixed state, density matrix

      entanglement entropy

      holographic entanglement entropy

    • coherent quantum state

  • ground state, excited state

  • quasi-free state

  • Fock space, second quantization

  • vacuum, vacuum state

    • Hadamard state

    • vacuum diagram

    • vacuum expectation value, vacuum amplitude, vacuum fluctuation

    • vacuum energy

    • vacuum polarization

    • interacting vacuum

    • thermal vacuum, KMS state

    • vacuum stability

      • false vacuum, tachyon, Coleman-De Luccia instanton

    • theta vacuum

    • perturbative string theory vacuum

      • non-geometric string theory vacuum

      • landscape of string theory vacua

  • entangled state

  • tensor network state

    • matrix product state

    • tree tensor network state

• observables

  • quantum observable, beable

  • algebra of observables, star-algebra

  • Bohr topos

  • quantum operator (in geometric quantization)

  • quantum operation, quantum effect, effect algebra

  • in quantum field theory

    • local observable

    • polynomial observable

      • linear observable

        • field observable

    • regular observable

    • microcausal observable

    • normal-ordered product, time-ordered products, retarded product

    • Wick algebra

    • scattering amplitude

    • interacting field algebra of observables, Bogoliubov's formula

• GNS construction

• theorems

  • order-theoretic structure in quantum mechanics

    • Gleason's theorem

    • Alfsen-Shultz theorem

    • Harding-Döring-Hamhalter theorem

    • Kochen-Specker theorem

    • Nuiten's lemma

  • Wigner's theorem

  • no-cloning theorem

  • Bell's theorem

References

General

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

In AQFT:

• Roberto Longo, Feng Xu, Von Neumann Entropy in QFT (arxiv:1911.09390)

See also

• Wikipedia, Entropy of entanglement

Topological entanglement entropy

On entanglement entropy in arithmetic Chern-Simons theory:

• Hee-Joong Chung, Dohyeong Kim, Minhyong Kim, Jeehoon Park, Hwajong Yoo, Entanglement entropies in the abelian arithmetic Chern-Simons theory [arXiv:2312.17138]

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:

• Alexei Kitaev, John Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96 (2006) 110404 (arXiv:hep-th/0510092)

• Michael Levin, Xiao-Gang Wen, Detecting topological order in a ground state wave function, Phys. Rev. Lett., 96, 110405 (2006) $[$arXiv:cond-mat/0510613, doi:10.1103/PhysRevLett.96.110405$]$

  (in view of string-net models)

Review:

• Shunsuke Furukawa, Entanglement Entropy in Conventional and Topological Orders, talk at Topological Aspects of Solid State Physics 2008 (pdf, pdf)

• Tarun Grover, Entanglement entropy and strongly correlated topological matter, Modern Physics Letters A 28 05 (2013) 1330001 $[$doi:10.1142/S0217732313300012$]$

• Bei Zeng, Xie Chen, Duan-Lu Zhou, Xiao-Gang Wen:

  Sec. 5 of: Quantum Information Meets Quantum Matter – From Quantum Entanglement to Topological Phases of Many-Body Systems, Quantum Science and Technology (QST), Springer (2019) $[$arXiv:1508.02595, doi:10.1007/978-1-4939-9084-9$]$

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)

and in the example of Chern-Simons theory:

• Aditya Dwivedi, Siddharth Dwivedi, Bhabani Prasad Mandal, Pichai Ramadevi, Vivek Kumar Singh, Topological entanglement and hyperbolic volume, J. High Energy Phys. 21021 172 (2021) 172 [arXiv:2106.03396, doi:10.1007/JHEP10(2021)172]

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)

See also:

• 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

Characterizing topological order

On characterizing anyon braiding / modular transformations on topologically ordered ground states by analysis of (topological) entanglement entropy of subregions:

• Yi Zhang, Tarun Grover, Ari M. Turner, Masaki Oshikawa, Ashvin Vishwanath, Quasiparticle statistics and braiding from ground-state entanglement, Phys. Rev. B 85 (2012) 235151 $[$doi:10.1103/PhysRevB.85.235151$]$

• Yi Zhang, Tarun Grover, Ashvin Vishwanath, General procedure for determining braiding and statistics of anyons using entanglement interferometry, Phys. Rev. B 91 (2015) 035127 $[$arXiv:1412.0677, doi:10.1103/PhysRevB.91.035127$]$

• Zhuan Li, Roger S. K. Mong, Detecting topological order from modular transformations of ground states on the torus, Phys. Rev. B 106 (2022) 235115 [doi:10.1103/PhysRevB.106.235115, arXiv:2203.04329]

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:

• Isabelle Dumé, Long-range quantum entanglement measured at last, PhysicsWorld (Jan 2022)