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Journal of Physics A Mathematical and Theoretical

The role of the ℓ1-norm in quantum information theory and two types of the Yang–Baxter equation

Kai Niu, Kang Xue, Qing Bo Zhao, Mo‐Lin Ge

Published Jun 6, 2011• 10 citations
Publisher

Abstract

The role of the l1-norm in the Yang–Baxter system has been studied through Wigner's D-functions, where l1-norm means ∑i|Ci| for |Ψ = ∑iCi|ψi with |ψi being the orthonormal basis. It is shown that the existing two types of braiding matrices, which can be viewed as particular solutions of the Yang–Baxter equation (YBE) with different spectral parameters can be unified in the 2D YBE. We prove that the maximum of the l1-norm is connected with the maximally entangled states and topological quantum field theory with two-component anyons, while the minimum leads to the deformed permutation related to the familiar integrable models.

DOI
10.1088/1751-8113/44/26/265304

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The role of the l1-norm in the Yang–Baxter system has been studied through Wigner's D-functions, where l1-norm means ∑i|Ci| for |Ψ = ∑iCi|ψi with |ψi being the orthonormal basis. It is shown that the existing two types of braiding matrices, which can be viewed as particular solutions of the Yang–Baxter equation (YBE) with different spectral parameters can be unified in the 2D YBE. We prove that the maximum of the l1-norm is connected with the maximally entangled states and topological quantum field theory with two-component anyons, while the minimum leads to the deformed permutation related to the familiar integrable models.
The ℓ1\ell_{1}ℓ1​-norm in quantum information via the approach of Yang-Baxter Equation
The role of ℓ1\ell_{1}ℓ1​-norm in Quantum Mechanics (QM) has been studied through Wigner's D-functions where ℓ1\ell_{1}ℓ1​-norm means ∑i∣Ci∣\sum_{i}\left|C_{i}\right|∑i​∣Ci​∣ for $\left|Ψ\right\rangle =\sum_{i}C_{i}\left|ψ_{i}\right\rangle $ if $\left|ψ_{i}\right\rangle $ are uni-orthogonal and normalized basis. It was shown that the present two types of transformation matrix acting on the natural basis in physics consist in an unified braiding matrix, which can be viewed as a particular solution of the Yang-Baxter equation (YBE). The maximum of the ℓ1\ell_{1}ℓ1​-norm is connected with the maximally entangled states and topological quantum field theory (TQFT) with two-component anyons while the minimum leads to the permutation for fermions or bosons.
The solutions of Yang–Baxter equation (YBE) associated with quantum information have been reviewed with new progress. The additivity rule for two particles is shown to obey Lorentzian type other than the Gallilean. Acting the braiding operations on the topological basis, it turns out that the [Formula: see text]-dimensional representations obeying YBE are nothing but the Wigner’s [Formula: see text]-functions. The connection between the extremization of [Formula: see text]-norm as well as of von Neumann entropy and the reduction of YBE to braid relation is also discussed.
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 ℓ 1 -norm leads to the maximum of the entanglement by taking the extreme value and shows that it is related to the Wigner's 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 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 rather than Galilean rule ( u + v ). Most possibly, for the type-II solutions of YBE there may not exist RTT relation. Further more, for 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 .
In this paper, we review some results obtained in our works with extension. The new type of braiding matrices called type-II has been extended to the solutions of Yang–Baxter equations (YBE). It is related to quantum entangling states. The four-dimensional realization is connected with Majorana fermions and the two-dimensional realization for the topological basis is the wigner DJ-function with J = 1/2. The YBE leads to the conclusion that the additivity rule of velocities for two anyons obeys Lorentz rather than Galilean. The discussion is extended to three body S-matrix formed by three two-body S-matrices. The three-qubit entanglement based on two-qubit entanglements is discussed.
The topological basis associated with Birman-Wenzl-Murakami algebra (BWMA) is constructed and the three dimensional forms of braiding matrices S have been found for both S+=SS^+=SS+=S and S+=S−1S^+=S^{-1}S+=S−1. A familiar spin-1 model related to braiding matrix associated with BWMA is discussed. The extreme points (θ=±π/2(θ=\pmπ/2(θ=±π/2 and ±π)\pmπ)±π) of L1-norm and von Neumann entropy are shown to be connected to each other. Through the general discussion and examples we then point out that the L1-norm describes quantum entanglement.
The Yang-Baxter Equation (YBE) plays a crucial role for studying integrable many-body quantum systems. Many known YBE solutions provide various examples ranging from quantum spin chains to superconducting systems. Models of solvable statistical mechanics and their avatars are also based on YBE. Therefore, new solutions of the YBE could be used to construct new interesting 1D quantum or 2D classical systems with many other far-reaching applications. In this work, we attempt to find (almost) exhaustive set of solutions for the YBE in the lowest dimensions corresponding to a two-qubit case. We develop an algorithm, which can potentially be used for generating new higher-dimensional solutions of the YBE.