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arXiv:1211.6178

Topological Basis Associated with BWMA, Extremes of L1-norm in Quantum Information and Applications in Physics

Qing Bo Zhao, Ruo-Yang Zhang, Kang Xue, Mo‐Lin Ge

Published Nov 27, 2012• 4 citations
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Abstract

The topological basis associated with Birman-Wenzl-Murakami algebra (BWMA) is constructed and the three dimensional forms of braiding matrices S have been found for both $S^+=S$ and $S^+=S^{-1}$. A familiar spin-1 model related to braiding matrix associated with BWMA is discussed. The extreme points $(θ=\pmπ/2$ and $\pmπ)$ of L1-norm and von Neumann entropy are shown to be connected to each other. Through the general discussion and examples we then point out that the L1-norm describes quantum entanglement.

DOI
10.48550/arxiv.1211.6178

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The topological basis associated with Birman-Wenzl-Murakami algebra (BWMA) is constructed and the three dimensional forms of braiding matrices S have been found for both S+=SS^+=SS+=S and S+=S−1S^+=S^{-1}S+=S−1. A familiar spin-1 model related to braiding matrix associated with BWMA is discussed. The extreme points (θ=±π/2(θ=\pmπ/2(θ=±π/2 and ±π)\pmπ)±π) of L1-norm and von Neumann entropy are shown to be connected to each other. Through the general discussion and examples we then point out that the L1-norm describes quantum entanglement.
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Important developments in fault-tolerant quantum computation using the braiding of anyons have placed the theory of braid groups at the very foundation of topological quantum computing. Furthermore, the realization by Kauffman and Lomonaco that a specific braiding operator from the solution of the Yang-Baxter equation, namely the Bell matrix, is universal implies that in principle all quantum gates can be constructed from braiding operators together with single qubit gates. In this paper we present a new class of braiding operators from the Temperley-Lieb algebra that generalizes the Bell matrix to multi-qubit systems, thus unifying the Hadamard and Bell matrices within the same framework. Unlike previous braiding operators, these new operators generate directly from separable basis states important entangled states such as the generalized Greenberger-Horne-Zeilinger states, cluster-like states, and other states with varying degrees of entanglement.