Planar algebras, quantum information theory and subfactors

TL;DR

This paper explores the connection between planar algebras and quantum information theory by defining biunitary elements and demonstrating their role in representing key quantum objects within the spin planar algebra framework.

Contribution

It introduces generalized biunitary elements in planar algebras and links them to quantum information structures, establishing new subfactor planar algebras associated with these elements.

Findings

Hadamard matrices and quantum Latin squares are biunitary elements in the spin planar algebra.

Biunitary elements correspond to natural subfactor planar algebras.

The framework unifies quantum information objects within the theory of planar algebras.

Abstract

We define generalised notions of biunitary elements in planar algebras and show that objects arising in quantum information theory such as Hadamard matrices, quantum latin squares and unitary error bases are all given by biunitary elements in the spin planar algebra. We show that there are natural subfactor planar algebras associated with biunitary elements.