Multi-qudit states generated by unitary braid quantum gates based on Temperley-Lieb algebra

TL;DR

This paper constructs braid quantum gates using Temperley-Lieb algebra to generate entangled multi-qudit states, including maximally entangled GHZ states, directly from the gates without additional transformations.

Contribution

It introduces a novel method to generate entangled multi-qudit states using braid quantum gates based on Temperley-Lieb algebra, enabling direct creation of GHZ states.

Findings

Braid quantum gates can produce maximally entangled GHZ states directly.

Multiple unitary representations of braid group generators are constructed.

A connection exists between multi-qudit states and braid gates based on coefficient norms.

Abstract

Using a braid group representation based on the Temperley-Lieb algebra, we construct braid quantum gates that could generate entangled $n$-partite $D$-level qudit states. $D$ different sets of $D^n\times D^n$ unitary representation of the braid group generators are presented. With these generators the desired braid quantum gates are obtained. We show that the generalized GHZ states, which are maximally entangled states, can be obtained directly from these braid quantum gates without resorting to further local unitary transformations. We also point out an interesting observation, namely for a general multi-qudit state there exists a unitary braid quantum gate based on the Temperley-Lieb algebra that connects it from one of its component basis states, if the coefficient of the component state is such that the square of its norm is no less than $1/4$.