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Register a new user$ell_1$-norm in three-qubit quantum entanglement constrained by Yang-Baxter equation
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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.
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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.
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.
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.
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.
Entanglement is believed to be crucial in macroscopic physical systems for understanding the collective quantum phenomena such as quantum phase transitions. We start from and solve exactly a novel Yang-Baxter spin-1/2 chain model with inhomogeneous and anisotropic short-range interactions. For the ground state, we show the behavior of neighboring entanglement in the parameter space and find that the inhomogeneous coupling strengths affect entanglement in a distinctive way from the homogeneous case, but this would not affect the coincidence between entanglement and quantum criticality.
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