nLab
entanglement entropy

Idea

entropy induced by entanglement in quantum physics, really 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 $\bar A$, the its entanglement entropy is

$$S_A \;=\; - Tr_{A}\big( \rho_A \ln \rho_A \big)$$

for

$$\rho_A \;\coloneqq\; Tr_{\bar A}(\rho) \,.$$

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

    • qbit, dimer, tensor network state

    • space of states (in geometric quantization)

    • state on a star-algebra, quasi-state

    • probability amplitude, quantum fluctuation

    • pure state

      • wave function

      • bra-ket

    • superposition, quantum interference

    • quantum measurement

      wave function collapse

      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

  • 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

Identification of a non-vanishing contribution to the (entanglement-)entropy at absolute zero, due to topological order/topological phase (“topological entropy”):

• 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)

Review:

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

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)

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)

See also:

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