The quantum Perron–Frobenius space

Abstract

We introduce the concept of \mathfrak{F} -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.