De Finetti theorems for braided parafermions

Kaifeng Bu Harvard University, Cambridge, MA, 02138, USA; Zhejiang University, Hangzhou, China Arthur Jaffe Harvard University, Cambridge, MA, 02138, USA Zhengwei Liu Harvard University, Cambridge, MA, 02138, USA; Tsinghua University, Beijing, China Jinsong Wu Harvard University, Cambridge, MA, 02138, USA; Harbin Institute of Technology, Harbin, China

Functional Analysis Mathematical Physics Probability Spectral Theory and Operator Algebra mathscidoc:2207.12005

Communications in Mathematical Physics, 373, 435–456, 2019.10
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).
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@inproceedings{kaifeng2019de,
  title={De Finetti theorems for braided parafermions},
  author={Kaifeng Bu, Arthur Jaffe, Zhengwei Liu, and Jinsong Wu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220707153815043588566},
  booktitle={Communications in Mathematical Physics},
  volume={373},
  pages={435–456},
  year={2019},
}
Kaifeng Bu, Arthur Jaffe, Zhengwei Liu, and Jinsong Wu. De Finetti theorems for braided parafermions. 2019. Vol. 373. In Communications in Mathematical Physics. pp.435–456. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220707153815043588566.
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