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The quantum Perron–Frobenius space

  • Linzhe Huang

    Beijing Institute of Mathematical Sciences and Applications, P. R. China
    ORCID

  • Arthur Jaffe

    Harvard University, Cambridge, USA
    ORCID

  • Zhengwei Liu

    Tsinghua University, Beijing, P. R. China; Beijing Institute of Mathematical Sciences and Applications, P. R. China
    ORCID

  • Jinsong Wu

    Beijing Institute of Mathematical Sciences and Applications, P. R. China
The quantum Perron–Frobenius space cover
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Abstract

We introduce the concept of -positive elements in planar algebras and establish a Perron–Frobenius theorem for such elements. We investigate the existence and uniqueness of the Perron–Frobenius eigenspace. When the eigenspace is not one dimensional, we characterize its multiplicative structure. Furthermore, we interpret the Perron–Frobenius eigenspace as the space of logical qubits for mixed states in quantum information theory. We also explore the connection between these mathematical results and quantum error correction, particularly in relation to the Knill–Laflamme theorem.

Cite this article

Linzhe Huang, Arthur Jaffe, Zhengwei Liu, Jinsong Wu, The quantum Perron–Frobenius space. J. Noncommut. Geom. (2025), published online first

DOI 10.4171/JNCG/631