Method of constructing braid group representation and entanglement in a Yang-Baxter sysytem

TL;DR

This paper constructs new braid group representations on tensor spaces, derives a specific unitary R-matrix, and investigates its entanglement properties, enabling controlled entanglement generation in two-qutrit systems.

Contribution

It introduces a reducible n^2 braid group representation, constructs a 9x9 unitary R-matrix from a braiding matrix, and explores its entanglement capabilities.

Findings

Constructed a reducible braid group representation on tensor spaces.

Derived a 9x9 unitary R-matrix satisfying braid relations.

Demonstrated controllable entanglement generation in two-qutrit states.

Abstract

In this paper we present reducible representation of the $n^{2}$ braid group representation which is constructed on the tensor product of n-dimensional spaces. By some combining methods we can construct more arbitrary $n^{2}$ dimensional braiding matrix S which satisfy the braid relations, and we get some useful braiding matrix S. By Yang-Baxteraition approach, we derive a $ 9\times9 $ unitary $ \breve{R}$ according to a $ 9\times9 $ braiding S-matrix we have constructed. The entanglement properties of $ \breve{R}$-matrix is investigated, and the arbitrary degree of entanglement for two-qutrit entangled states can be generated via $ \breve{R}(\theta, \phi_{1},\phi_{2})$-matrix acting on the standard basis.