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A new realization of doubling degeneracy based on emergent Majorana operator $Gamma$ presented by Lee-Wilczek has been made. The Hamiltonian can be obtained through the new type of solution of Yang-Baxter equation, i.e. $breve{R}(theta)$-matrix. For 2-body interaction, $breve{R}(theta)$ gives the superconducting chain that is the same as 1D Kitaev chain model. The 3-body Hamiltonian commuting with $Gamma$ is derived by 3-body $breve{R}_{123}$-matrix, we thus show that the essence of the doubling degeneracy is due to $[breve{R}(theta), Gamma]=0$. We also show that the extended $Gamma$-operator is an invariant of braid group $B_N$ for odd $N$. Moreover, with the extended $Gamma$-operator, we construct the high dimensional matrix representation of solution to Yang-Baxter equation and find its application in constructing $2N$-qubit Greenberger-Horne-Zeilinger state for odd $N$.
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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.
The relationships between quantum entangled states and braid matrices have been well studied in recent years. However, most of the results are based on qubits. In this paper, We investigate the applications of 2-qutrit entanglement in the braiding associated with $mathbb{Z}_3$ parafermion. The 2-qutrit entangled state $|Psi(theta)rangle$, generated by acting the localized unitary solution $breve{R}(theta)$ of YBE on 2-qutrit natural basis, achieves its maximal $ell_1$-norm and maximal von Neumann entropy simultaneously at $theta=pi/3$. Meanwhile, at $theta=pi/3$, the solutions of YBE reduces braid matrices, which implies the role of $ell_1$-norm and entropy plays in determining real physical quantities. On the other hand, we give a new realization of 4-anyon topological basis by qutrit entangled states, then the $9times9$ localized braid representation in 4-qutrit tensor product space $(mathbb{C}^3)^{otimes 4}$ are reduced to Jones representation of braiding in the 4-anyon topological basis. Hence, we conclude that the entangled states are powerful tools in analysing the characteristics of braiding and $breve{R}$-matrix.
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.
Any solution to the Yang-Baxter equation yields a family of representations of braid groups. Under certain conditions, identified by Turaev, the appropriately normalized trace of these representations yields a link invariant. Any Yang-Baxter solution can be interpreted as a two-qudit quantum gate. Here we show that if this gate is non-entangling, then the resulting invariant of knots is trivial. We thus obtain a general connection between topological entanglement and quantum entanglement, as suggested by Kauffman et al.
The Yang-Baxter equation is an important tool in theoretical physics, with many applications in different domains that span from condensed matter to string theory. Recently, the interest on the equation has increased due to its connection to quantum information processing. It has been shown that the Yang-Baxter equation is closely related to quantum entanglement and quantum computation. Therefore, owing to the broad relevance of this equation, besides theoretical studies, it also became significant to pursue its experimental implementation. Here, we show an experimental realization of the Yang-Baxter equation and verify its validity through a Nuclear Magnetic Resonance (NMR) interferometric setup. Our experiment was performed on a liquid state Iodotrifluoroethylene sample which contains molecules with three qubits. We use Controlled-transfer gates that allow us to build a pseudo-pure state from which we are able to apply a quantum information protocol that implements the Yang-Baxter equation.
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