# Unitary Braid Matrices: Bridge between Topological and Quantum Entanglements

## TL;DR

This paper explores how parametrized unitary braiding matrices in all dimensions can generate and analyze quantum entanglements, linking topological braid structures with quantum entanglement measures.

## Contribution

It introduces two classes of unitary braid matrices that explicitly generate and evaluate multi-party entanglements, bridging topological braiding and quantum entanglement analysis.

## Key findings

- Unitary braiding operators generate entangled superpositions of four terms.
- Explicit evaluation of 3-body and 2-body entanglements, including tangles.
- Parametrized matrices can be tuned to explore entanglement domains.

## Abstract

Braiding operators corresponding to the third Reidemeister move in the theory of knots and links are realized in terms of parametrized unitary matrices for all dimensions. Two distinct classes are considered. Their (non-local) unitary actions on separable pure product states of three identical subsystems (say, the spin projections of three particles) are explicitly evaluated for all dimensions. This, for our classes, is shown to generate entangled superposition of four terms in the base space. The 3-body and 2-body entanglements (in three 2-body subsystems), the 3-tangles and 2-tangles are explicitly evaluated for each class. For our matrices, these are parametrized. Varying parameters they can be made to sweep over the domain (0,1).Thus braiding operators corresponding to over- and under-crossings of three braids and, on closing ends, to topologically entangled Borromean rings are shown, in another context, to generate quantum entanglements. For higher dimensions, starting with different initial triplets one can entangle by turns, each state with all the rest. A specific coupling of three angular momenta is briefly discussed to throw more light on three body entanglements.