The Jones index classifies how one von Neumann algebra embeds inside another — a measure of “how much bigger” the containing algebra is. In the context of conformal field theory (CFT), it captures topological data about the theory: the number and type of superselection sectors, which physically correspond to distinct types of anyonic excitations.
Ares et al. (arXiv:2603.13013) extract the Jones index from Rényi entropies in the Ising CFT. Rényi entropies are computable from the reduced density matrix of a subregion — they're entanglement measures. The Jones index is an algebraic invariant — it counts sectors. The connection between them is not obvious.
The bridge: in a CFT with a finite number of superselection sectors, the entanglement entropy of an interval contains a universal constant term (the topological entanglement entropy) that depends on the Jones index. Different Rényi parameters (α = 2, 3, 4, ...) probe different aspects of the entanglement spectrum, and the Jones index can be extracted from how the Rényi entropies vary with α.
For the Ising CFT specifically — the simplest non-trivial example with three sectors (identity, fermion, Ising anyon) — the computation is explicit. The Jones index is 2 (reflecting the Z₂ symmetry), and the Rényi entropies at different orders confirm this value through a pattern of corrections to the area law.
The result connects two mathematical structures that developed independently. Jones indices emerged from operator algebras in the 1980s. Rényi entropies became central to quantum information in the 2000s. That they contain the same information — that entanglement measures encode algebraic invariants — is a manifestation of the deep relationship between quantum correlations and the structure of the algebras that describe them.