  We introduce $\mathfrak{F}$-positive elements in planar algebras. We
establish the Perron-Frobenius theorem for $\mathfrak{F}$-positive elements. We
study the existence and uniqueness of the Perron-Frobenius eigenspace. When it
is not one-dimensional, we characterize its multiplicative structure. Moreover,
we consider the Perron-Frobenius eigenspace as the space of logical qubits for
mixed states in quantum information. We describe the relation between these
mathematical results and quantum error correction, especially to the
Knill-Laflamme theorem.
