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Communications in Theoretical Physics

Application of Y ( sl (2)) Algebra for Entanglement of Two-Qubit System

Li-Jun Tian, Qin Li-Guo, Jiang Ying, Hongbiao Zhang, Kang Xue

Published Jun 1, 2010• 0 citations
Publisher

Abstract

We construct the transition operators in terms of the generators of the general Yangian and the reduced Yangian. By acting these operators on a two-qubit pure state, we find that the entanglement degrees of the states are all decreased from the certain values to zero for the reduced Yangian algebra, which makes the state disentangled. This result sheds new light on the physical meaning of Y ( sl (2)) in quantum information.

DOI
10.1088/0253-6102/53/6/08

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We construct the transition operators in terms of the generators of the general Yangian and the reduced Yangian. By acting these operators on a two-qubit pure state, we find that the entanglement degrees of the states are all decreased from the certain values to zero for the reduced Yangian algebra, which makes the state disentangled. This result sheds new light on the physical meaning of Y ( sl (2)) in quantum information.
The applications of the general and reduced Yangian Y(sl(2)) and Y(su(3)) algebras are discussed. By taking a special constraint, the representation of Y(sl(2)) and Y(su(3)) can be divided into two 2 \times 2 and three 3 \times 3 blocks diagonal respectively. The general and reduced Yangian Y(sl(2)) and Y(su(3)) are applied to the bi-qubit system and the mixed light pseudoscalar meson state, respectively. We can find that the general ones are not able to make the initial states disentangled by acting on the initial states, however the reduced ones are able to make the initial state disentangled. In addition, we show the effects of Y(su(3)) generators on the the decay channel.
Yangian, as an algebra beyond the Lie algebra, is an infinite dimensional algebra and a powerful mathematical method for inVestigating the new symmetry of quantum systems which are nonlinear and integrable. Based on the su(3) symmetry of the quarK-flaVor in the meson states and the transition property of the generators in Yangian algebra, we study the applications of Yangian algebra Y(su(3)) in the decay of three mixed meson states(, K and Ki0) composed of the three positiVe and negatiVe meson states (, K, K0 and K0). As the transition operators, the eight generators (Ī, Ŭ, V, Ī3 and Ī8) of Yangian algebra Y(su(3)) are acting on the three mixed meson states, respectiVely. Then, the possible decay channels and the changes of the entanglement are studied. Results show: (i) Under the effects of Ī3 and Ī8 within the Lie algebra on the three mixed meson states, the compositions of the final states after decays of the three mixed meson states are not changed as compared with the initial state. The entanglement is not changed for the decay of the mixed meson state with the effect of Ī^8, and the others are changed. (ii) Under the effects of the other six generators (Ī, Ŭ and V) beyond the Lie algebra on the three mixed meson states, the compositions of the final states after the decay are changed compared with the initial state. In the six possible decay channels, the two final states become single states without entanglement; two decay channels are absent; and the entanglements of the final states in the remaining two decays are changed. In addition, the entanglement of the final meson states in the possible six decay channels of the two types K mixed meson states, the charged (K+, K-) and neutral (K0, K0) meson states, are the same two by two. (iii) The three mixed meson states can be circularly transferred by the operators Ī, Ŭ and V, implying the obVious symmetry. In this paper the Yangian method is presented to study the possible decay channels of the mixed meson states and may be used to present a possible interpretation of the new unKnown or Known particle in the decay of the mixed meson.
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.