home | literature reviews

## "Topological entanglement entropy", Alexei Kitaev, John Preskill, 2006

*Reviewed January 14, 2024*

*Citation:* Kitaev, Alexei, and John Preskill. "Topological entanglement entropy." Physical review letters 96.11 (2006): 110404.

*Web:* https://arxiv.org/abs/hep-th/0510092

*Tags:* Physical, Foundational

This is the paper that introduced the key idea of topological entanglement entropy.
In the words of the authors: "While it is clear that these unusual properties emerge
because the ground state is profoundly entangled, up until now no firm connection has been established between
topological order and any quantitative measure of entanglement. In this paper we provide such a connection
by relating topological order to von Neumann entropy,
which quantifies the entanglement of a bipartite pure state".

The idea of topological entanglement entropy
had already been sitting in the literature, and had
been computed implicitly for the toric code:

> Hamma, Alioscia, Radu Ionicioiu, and Paolo Zanardi. "Bipartite entanglement and entropic boundary law in lattice spin systems." Physical Review A 71.2 (2005): 022315.

After defining the topological entanglement entropy,
they give an amazing and elegant computation of this entropy
for any topological phase. The result, it turns out,
is the natural logarithm of the global dimension.

One nice feature about topological entanglement entropy
is that it is very amenable to observation in the laboratory.
One major protocol for measuring this entropy is via a thermodynamic Maxwell's equation:

> Cooper, N. R., and Ady Stern. "Observable bulk signatures of non-Abelian quantum Hall states." Physical review letters 102.17 (2009): 176807.

> Sankar, Sarath, Eran Sela, and Cheolhee Han. "Measuring topological entanglement entropy using Maxwell relations." Physical Review Letters 131.1 (2023): 016601.

Measuring the topological entanglement entropy in interesting regions
can tell you a large amount about your phase. For instance, specialized
topological entanglement entropy measurements can allow one to deduce both
the fusion and braiding statistics:

> Shi, Bowen, Kohtaro Kato, and Isaac H. Kim. "Fusion rules from entanglement." Annals of Physics 418 (2020): 168164.

> Zhang, Yi, et al. "Quasiparticle statistics and braiding from ground-state entanglement." Physical Review B 85.23 (2012): 235151.