nLab entanglement entropy

Contents

Context

Measure and probability theory

measure theory

probability theory

Contents

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

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

References

General

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

In AQFT:

Topological entanglement entropy

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

Review:

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)