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arXiv:0904.3621 · Journal of Mathematical Physics

Berry phase and entanglement of three qubits in a new Yang–Baxter system

Taotao Hu, Kang Xue, Chunfeng Wu

Published Aug 1, 2009• 6 citations
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Abstract

In this paper we construct a new 8×8 M matrix from the 4×4 M matrix, where M/M is the image of the braid group representation. The 8×8 M matrix and the 4×4 M matrix both satisfy extraspecial 2-group algebra relations. By Yang–Baxteration approach, we derive a unitary R̆(θ,φ) matrix from the M matrix with parameters φ and θ. Three-qubit entangled states can be generated by using the R̆(θ,φ) matrix. A Hamiltonian for three qubits is constructed from the unitary R̆(θ,φ) matrix. We then study the entanglement and Berry phase of the Yang–Baxter system.

DOI
10.1063/1.3177295

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Entanglement and Berry Phase in a 9×99\times 99×9 Yang-Baxter system
A M-matrix which satisfies the Hecke algebraic relations is presented. Via the Yang-Baxterization approach, we obtain a unitary solution R˘(θ,φ1,φ2)\breve{R}(θ,φ_{1},φ_{2})R˘(θ,φ1​,φ2​) of Yang-Baxter Equation. It is shown that any pure two-qutrit entangled states can be generated via the universal R˘\breve{R}R˘-matrix assisted by local unitary transformations. A Hamiltonian is constructed from the R˘\breve{R}R˘-matrix, and Berry phase of the Yang-Baxter system is investigated. Specifically, for $ φ_{1}=φ_{2}$, the Hamiltonian can be represented based on three sets of SU(2) operators, and three oscillator Hamiltonians can be obtained. Under this framework, the Berry phase can be interpreted.
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In this paper we describe connections among extraspecial 2-groups, unitary representations of the braid group and multi-qubit braiding quantum gates. We first construct new representations of extraspecial 2-groups. Extending the latter by the symmetric group, we construct new unitary braid representations, which are solutions to generalized Yang-Baxter equations and use them to realize new braiding quantum gates. These gates generate the GHZ (Greenberger-Horne-Zeilinger) states, for an arbitrary (particularly an \emph{odd}) number of qubits, from the product basis. We also discuss the Yang-Baxterization of the new braid group representations, which describes unitary evolution of the GHZ states. Our study suggests that through their connection with braiding gates, extraspecial 2-groups and the GHZ states may play an important role in quantum error correction and topological quantum computing.