De Finetti Theorems for Braided Parafermions

TL;DR

This paper extends the de Finetti theorem to states invariant under braid group actions on parafermion algebras, revealing their structure and extremality conditions relevant for quantum information.

Contribution

It introduces a new de Finetti theorem for braid-invariant states on parafermion algebras, including explicit characterizations and distinctions based on algebra order.

Findings

Braid-invariant states are extremal iff they are product states.

Explicit characterization of braid-invariant states on parafermion algebra.

Distinction in state structure depending on whether the algebra order is square free.

Abstract

The classical de Finetti theorem in probability theory relates symmetry under the permutation group with the independence of random variables. This result has application in quantum information. Here we study states that are invariant with respect to a natural action of the braid group, and we emphasize the pictorial formulation and interpretation of our results. We prove a new type of de Finetti theorem for the four-string, double-braid group acting on the parafermion algebra to braid qudits, a natural symmetry in the quon language for quantum information. We prove that a braid-invariant state is extremal if and only if it is a product state. Furthermore, we provide an explicit characterization of braid-invariant states on the parafermion algebra, including finding a distinction that depends on whether the order of the parafermion algebra is square free. We characterize the extremal nature of product states (an inverse de Finetti theorem).