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Register a new userFactorized Three-body S-Matrix Restrained by Yang-Baxter Equation and Quantum Entanglements
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This paper investigates the physical effects of Yang-Baxter equation (YBE) to quantum entanglements through the 3-body S-matrix in entangling parameter space. The explicit form of 3-body S-matrix $breve{R}_{123}(theta,varphi)$ based on the 2-body S-matrices is given due to the factorization condition of YBE. The corresponding chain Hamiltonian has been obtained and diagonalized, also the Berry phase for 3-body system is given. It turns out that by choosing different spectral parameters the $breve{R}(theta,varphi)$-matrix gives GHZ and W state respectively. The extended 1-D Kitaev toy model has been derived. Examples of the role of the model in entanglement transfer are discussed.
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Usually the $ell_2$-norm plays vital roles in quantum physics, acting as the probability of states. In this paper, we show the important roles of $ell_1$-norm in Yang-Baxter quantum system, in connection with both the braid matrix and quantum entanglements. Concretely, we choose the 2-body and 3-body S-matrices, constrained by Yang-Baxter equation. It has been shown that for 2-body case, the extreme values of $ell_1$-norm lead to two types of braid matrices and 2-qubit Bell states. Here we show that for the 3-body case, due to the constraint of YBE, the extreme values of $ell_1$-norm lead to both 3-qubit $|GHZrangle$ (local maximum) and $|Wrangle$ (local minimum) states, which cover all 3-qubit genuine entanglements for pure states under SLOCC. This is a more convincing proof for the roles of $ell_1$-norm in quantum mechanics.
In this paper, several proposals of optically simulating Yang-Baxter equations have been presented. Motivated by the recent development of anyon theory, we apply Temperley-Lieb algebra as a bridge to recast four-dimentional Yang-Baxter equation into its two-dimensional counterpart. In accordance with both representations, we find the corresponding linear-optical simulations, based on the highly efficient optical elements. Both the freedom degrees of photon polarization and location are utilized as the qubit basis, in which the unitary Yang-Baxter matrices are decomposed into combination of actions of basic optical elements.
Several aspects of relations between braces and non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation are discussed and many consequences are derived. In particular, for each positive integer $n$ a finite square-free multipermutation solution of the Yang-Baxter equation with multipermutation level $n$ and an abelian involutive Yang-Baxter group is constructed. This answers a problem of Gateva-Ivanova and Cameron. It is also proved that finite non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation whose associated involutive Yang-Baxter group is abelian are retractable in the sense of Etingof, Schedler and Soloviev. Earlier the authors proved this with the additional square-free hypothesis on the solutions. Retractability of solutions is also proved for finite square-free non-degenerate involutive set-theoretic solutions associated to a left brace.
Spin interaction Hamiltonians are obtained from the unitary Yang--Baxter $breve{R}$-matrix. Based on which, we study Berry phase and quantum criticality in the Yang--Baxter systems.
We construct the 1D $mathbb{Z}_3$ parafermionic model based on the solution of Yang-Baxter equation and express the model by three types of fermions. It is shown that the $mathbb{Z}_3$ parafermionic chain possesses both triple degenerate ground states and non-trivial topological winding number. Hence, the $mathbb{Z}_3$ parafermionic model is a direct generalization of 1D $mathbb{Z}_2$ Kitaev model. Both the $mathbb{Z}_2$ and $mathbb{Z}_3$ model can be obtained from Yang-Baxter equation. On the other hand, to show the algebra of parafermionic tripling intuitively, we define a new 3-body Hamiltonian $hat{H}_{123}$ based on Yang-Baxter equation. Different from the Majorana doubling, the $hat{H}_{123}$ holds triple degeneracy at each of energy levels. The triple degeneracy is protected by two symmetry operators of the system, $omega$-parity $P$($omega=e^{{textrm{i}frac{2pi}{3}}}$) and emergent parafermionic operator $Gamma$, which are the generalizations of parity $P_{M}$ and emergent Majorana operator in Lee-Wilczek model, respectively. Both the $mathbb{Z}_3$ parafermionic model and $hat{H}_{123}$ can be viewed as SU(3) models in color space. In comparison with the Majorana models for SU(2), it turns out that the SU(3) models are truly the generalization of Majorana models resultant from Yang-Baxter equation.
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