The $\ell_{1}$-norm in quantum information via the approach of Yang-Baxter Equation

TL;DR

This paper explores the role of the $$-norm in quantum mechanics through the lens of Yang-Baxter equations, linking it to entanglement, topological quantum field theory, and particle statistics.

Contribution

It introduces a unified braiding matrix as a solution to the Yang-Baxter equation, connecting $$-norm extremal points to quantum entanglement and particle types.

Findings

Maximum $$-norm correlates with maximally entangled states.

Minimum $$-norm corresponds to permutation states for fermions or bosons.

Unified braiding matrix acts on the natural basis in quantum systems.

Abstract

The role of $\ell_{1}$-norm in Quantum Mechanics (QM) has been studied through Wigner's D-functions where $\ell_{1}$-norm means $\sum_{i}\left|C_{i}\right|$ for $\left|\Psi\right\rangle =\sum_{i}C_{i}\left|\psi_{i}\right\rangle $ if $\left|\psi_{i}\right\rangle $ are uni-orthogonal and normalized basis. It was shown that the present two types of transformation matrix acting on the natural basis in physics consist in an unified braiding matrix, which can be viewed as a particular solution of the Yang-Baxter equation (YBE). The maximum of the $\ell_{1}$-norm is connected with the maximally entangled states and topological quantum field theory (TQFT) with two-component anyons while the minimum leads to the permutation for fermions or bosons.