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arXiv:1301.0850 · Journal of Mathematical Physics

Generalized Bell states and principal realization of the Yangian $Y(\mathfrak {sl}_{N})$Y(slN)

Ming Liu, Chengming Bai, Mo‐Lin Ge, Naihuan Jing

Published Feb 1, 2013• 9 citations
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Abstract

We prove that the action of the Yangian algebra $Y(\mathfrak {sl}_N)$Y(slN) is better described by the principal generators on the tensor product of the fundamental representation and its dual. The generalized Bell states or maximally entangled states are permuted by the principal generators in a dramatically simple manner on the tensor product. Under the Yangian symmetry the new quantum number $\bf J^2$J2 is also explicitly computed, which gives an explanation for these maximally entangled states.

DOI
10.1063/1.4789317

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Generalized Bell states and principal realization of the Yangian Y(slN)Y(\mathfrak {sl}_{N})Y(slN​)Y(slN)
We prove that the action of the Yangian algebra Y(slN)Y(\mathfrak {sl}_N)Y(slN​)Y(slN) is better described by the principal generators on the tensor product of the fundamental representation and its dual. The generalized Bell states or maximally entangled states are permuted by the principal generators in a dramatically simple manner on the tensor product. Under the Yangian symmetry the new quantum number J2\bf J^2J2J2 is also explicitly computed, which gives an explanation for these maximally entangled states.
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We study the rank of a general tensor uuu in a tensor product $H_1\ot...\ot H_k$. The rank of uuu is the minimal number ppp of pure states v1,...,vpv_1,...,v_pv1​,...,vp​ such that uuu is a linear combination of the vjv_jvj​'s. This rank is an algebraic measure of the degree of entanglement of uuu. Motivated by quantum computation, we completely describe the rank of an arbitrary tensor in $(\C^2)^{\ot 3}$ and give normal forms for tensor states up to local unitary transformations. We also obtain partial results for $(\C^2)^{\ot 4}$; in particular, we show that the maximal rank of a tensor in $(\C^2)^{\ot 4}$ is equal to 4.
In this chapter we will reveal another piece of the algebraic structure of the Hubbard model. As was first observed by Uglov and Korepin [462], the Hubbard Hamiltonian on the infinite line is invariant under the action of the direct sum of two so-called Yangian quantum groups, extending the rotational and the η-pairing su(2) symmetries we encountered earlier. Following [172] we shall address the issue in a more general context. We present two pairs of fermionic representations of the Y(su(2)) Yangian quantum group which commute with the trigonometric [162] and hyperbolic [40, 41] versions of a Hubbard Hamiltonian with non-nearest-neighbour hopping. In both cases the two representations are also mutually commuting, hence can be combined into a representation of Y(su(2))⊕Y(su(2)). The generators of the Yangian symmetry of the ordinary Hubbard model (with nearest-neighbour hopping) and of a number of other interesting models like the Haldane-Shastry spin-chain [194, 394] are obtained as special cases of our general result. Introduction Quantum groups were introduced by Drinfeld [107, 109]. His original intention was to put what we called the Yang-Baxter algebra into the mathematically more conventional context of Hopf algebras. The Yangians are special quantum groups. Their representation theory [80, 81] is intimately related to the classification of integrable quantum systems with rational R -matrices.
Bell states and quantum teleportation play crucial roles in quantum information and computation. However, a comprehensive theoretical study of both topics remains to be carried out. This work aims to investigate key algebraic properties of generalized Bell states and explore the topological features of quantum teleportation. First, the basis theorem and the basis group are introduced to show that the extension of a generalized Bell basis by a unitary matrix still forms an orthonormal basis. Then, a twist operator is defined to establish a connection between a generalized multi-qubit Bell state and a tensor product of two-qubit Bell states. In addition, the Temperley--Lieb algebra, the braid group relations, and the Yang--Baxter equation are employed to provide a topological description of generalized Bell states and quantum teleportation. The results demonstrate that our approach not only offers a clear illustration of relevant quantum information protocols but also reveals the topological nature of quantum entanglement and teleportation.