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Quantum Information and Computation

Gaussian (N, z)-generalized Yang-Baxter operators

Eric C. Rowell

Published Jan 1, 2016• 2 citations
Publisher

Abstract

We find unitary matrix solutions R˜(a) to the (multiplicative parameter-dependent) (N, z)-generalized Yang-Baxter equation that carry the standard measurement basis to m-level N-partite entangled states that generalize the 2-level bipartite entangled Bell states. This is achieved by a careful study of solutions to the Yang-Baxter equation discovered by Fateev and Zamolodchikov in 1982.

DOI
10.26421/qic16.1-2-7

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We find unitary solutions R~(a)\tilde{R}(a)R~(a) to the (multipicative parameter-dependent) (z,N)(z,N)(z,N)-generalized Yang-Baxter equation that carry the standard measurement basis to mmm-level NNN-partite states that generalize the Bell states corresponding to R~(0)\tilde{R}(0)R~(0) in the case m=N=2m=N=2m=N=2. This is achieved by a careful study of solutions to the Yang-Baxter equation discovered by Fateev and Zamolodchikov in 1982.
Entangled states, such as the Bell and GHZ states, are generated from separable states using matrices known to satisfy the Yang-Baxter equation and its generalization. This remarkable fact hints at the possibility of using braiding operators as quantum entanglers, and is part of a larger speculated connection between topological and quantum entanglement. We push the analysis of this connection forward, by showing that supersymmetry algebras can be used to construct large families of solutions of the spectral parameter-dependent generalized Yang-Baxter equation. We present a number of explicit examples and outline a general algorithm for arbitrary numbers of qubits. The operators we obtain produce, in turn, all the entangled states in a multi-qubit system classified by the Stochastic Local Operations and Classical Communication protocol introduced in quantum information theory.
Recent study suggests that there are natural connections between quantum information theory and the Yang--Baxter equation. In this paper, in terms of the generalized almost-complex structure and with the help of its algebra, we define the generalized Bell matrix to yield all the GHZ states from the product base, prove it to form a unitary braid representation and present a new type of solution of the quantum Yang--Baxter equation. We also study Yang-Baxterization, Hamiltonian, projectors, diagonalization, noncommutative geometry, quantum algebra and FRT dual algebra associated with this generalized Bell matrix.
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