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Register a new user$ell_1$-norm and entanglement in screening out braiding from Yang-Baxter equation associated with $mathbb{Z}_3$ parafermion
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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.
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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.
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.
Starting from the Kauffman-Lomonaco braiding matrix transforming the natural basis to Bell states, the spectral parameter describing the entanglement is introduced through Yang-Baxterization. It gives rise to a new type of solutions for Yang-Baxter equation, called the type-II that differs from the familiar solution called type-I of YBE associated with the usual chain models. The Majorana fermionic version of type-II yields the Kitaev Hamiltonian. The introduced $ell_1$ -norm leads to the maximum of the entanglement by taking the extreme value and shows that it is related to the Wigners D-function. Based on the Yang-Baxter equation the 3-body S-Matrix for type-II is explicitly given. Different from the type-I solution, the type-II solution of YBE should be considered in describing quantum information. The idea is further extended to $mathbb{Z}_3$ parafermion model based on $SU(3)$ principal representation. The type-II is in difference from the familiar type-I in many respects. For example, the quantities corresponding to velocity in the chain models obey the Lorentzian additivity $frac{u+v}{1+uv}$ rather than Galilean rule $(u+v)$. Most possibly, for the type-II solutions of YBE there may not exist RTT relation. Further more, for $mathbb{Z}_3$ parafermion model we only need the rational Yang-Baxterization, which seems like trigonometric. Similar discussions are also made in terms of generalized Yang-Baxter equation with three spin spaces ${1,frac{1}{2},frac{1}{2}}$.
The action of a Backlund-Darboux transformation on a spectral problem associated with a known integrable system can define a new discrete spectral problem. In this paper, we interpret a slightly generalized version of the binary Backlund-Darboux (or Zakharov-Shabat dressing) transformation for the nonlinear Schrodinger (NLS) hierarchy as a discrete spectral problem, wherein the two intermediate potentials appearing in the Darboux matrix are considered as a pair of new dependent variables. Then, we associate the discrete spectral problem with a suitable isospectral time-evolution equation, which forms the Lax-pair representation for a space-discrete NLS system. This formulation is valid for the most general case where the two dependent variables take values in (rectangular) matrices. In contrast to the matrix generalization of the Ablowitz-Ladik lattice, our discretization has a rational nonlinearity and admits a Hermitian conjugation reduction between the two dependent variables. Thus, a new proper space-discretization of the vector/matrix NLS equation is obtained; by changing the time part of the Lax pair, we also obtain an integrable space-discretization of the vector/matrix modified KdV (mKdV) equation. Because Backlund-Darboux transformations are permutable, we can increase the number of discrete independent variables in a multi-dimensionally consistent way. By solving the consistency condition on the two-dimensional lattice, we obtain a new Yang-Baxter map of the NLS type, which can be considered as a fully discrete analog of the principal chiral model for projection matrices.
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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