Citation: Shi, Bowen, Kohtaro Kato, and Isaac H. Kim. "Fusion rules from entanglement." Annals of Physics 418 (2020): 168164.
Web: https://arxiv.org/abs/1906.09376
Tags: MTC-reconstruction, Foundational
In this paper, the authors initiate the entanglement bootstrap programme. In this programme, one begins with a single reduced density matrix of the ground state of (2+1)D topological order on a disk, and deduce the entire algebraic theory of that topological order.
Of course, not every reduced density matrix has an algebraic theory associated with it. Only the reduced density matrices of (2+1)D topological orders do. So, what conditions are put on the state? Axioms A0 and A1 - two entropic conditions which roughly state that distant regions have zero correlation and consistent ground states can be uniquely merged. These axioms are entirely local - they are only presumed to hold on bounded radius balls, and the fact that they hold on all larger balls is a corollary. The axioms state that certain linear combinations of entanglement entropies will vanish.
This is really a magnificent programme. The core tool that it is based on is the information convex. It was shown in
> Shi, Bowen, and Yuan-Ming Lu. "Characterizing topological order by the information convex." Physical Review B 99.3 (2019): 035112.
that anyons in a topological order can be characterized by the structure of the information convex. In this paper, the authors show that A0 and A1 are enough to imply that the information convex has the conjectured structure and hence that all of the axioms of a fusion ring can be derived from the information convex.
This paper goes into all sorts of wonderful details, and really puts this programme on solid footing. They also formally introduce the programme (and the term "entanglement bootstrap") in their four-page preprint:
> Shi, Bowen, Kohtaro Kato, and Isaac H. Kim. "Entanglement bootstrap program." (2021).
The term is first used in an operational way in
> Shi, Bowen, and Isaac H. Kim. "Entanglement bootstrap approach for gapped domain walls." Physical Review B 103.11 (2021): 115150.
An early version of the programme was outline in Isaac Kim's slides,
> Kim, Isaac H. "Conservation laws from entanglement." Sydney Quantum Information Theory Workshop. 2015.
An important point is that this programme does NOT yet apply to most topologically ordered states. Generically, topological states will have finite correlation length meaning they will only satisfy the axioms A0 and A1 approximately. Writing an approximate version of the present paper is a major open problem.
This paper has all sorts of appendices and redundancies, making it a great place for new people to get started learning in this field.