A Special Yangian Realization And Its Application To Quantum Entangled State Transitions

Yang-Baxter equation first appeared in the related work of Professor C.N.Yang and Baxter.On this basis,the Leningrad School represented by Faddeev proposed quantum inverse scattering method,and RTT relation is the core content of this kind of method.With the help of RTT relation,Drinfeld proposed the concept of quantum groups,in-cluding quantum algebra and Yangian.The Yangian algebra plays an important role in the study of nonlinear quantum integrable systems,which is mainly reflected in two aspects:first,it is used to describe the symmetry of the physical system;second,the Yangian generator can act as a transition operator to realize transitions between quantum states,and can also constitute the rise and fall operator of energy spectrum.In recent years,Yang-Baxter equation has been introduced into the field of quan-tum information and quantum computing.Kauffman and Lomonaco found that there is a very close relationship between the solution of unitary Yang-Baxter equation and Bell basis,thus inspiring a series of in-depth studies on the application of Yang-Baxter equa-tion in quantum information.In the field of quantum information,a special class of base vectors has important research value,such Bell basis and the extended high-dimensional Bell basis have the most entangled states.In order to describe the symmetries of this class of quantum entangled states by means of Yangian algebra,a new class of Yangian is studied in this paper,and the application of this new Yangian realization in quantum entangled state transitions is discussed.For a given integer or half integer j,this paper introduces a set of completely or-thogonal basis into(2j+1)~2-dimensional Hilbert space,and constructs a new sum of Yangian generators(?)and(?)with this set of orthogonal basis.The commutation relation between the components of Yangian generators(?)and(?)satisfies the Y(sl(2))algebraic relation.Then,this new set of Yangian generators is realized through the spin system,so that the matrix representation of the Yangian generators(?)and(?)and each compo-nent can be obtained.Then,with the help of(?)and(?),the Hamiltonian with Yangian symmetry is constructed,so as to eliminate the degenerativity of space and get the Bell basis and the extended high-dimensional Bell basis.The newly constructed Yangian generators can act as transition operators to realize the transition between the Bell ba-sis and the extended high-dimensional Bell basis.Through comparison,it is found that the newly constructed Yangian and the traditional Yangian follow a different spatial de-composition mode.The traditional Yangian follows angular momentum addition,while the newly constructed Yangian decomposes the Hilbert space into several subspaces of the same dimension.Combined with the theory of Frank Wilczek and A.Zee,the gauge potential related to this group of Yangian algebras is discussed.If we employ(?)parameterization,the gauge potential of each subspace obtained is the same and does not depend on the quantum number of the subspace.However,if we employ(?)param-eterization,the gauge potential of the subspace obtained is not the same,depending on the quantum number of the subspace.The results show that the gauge potential of this kind of algebra can also be divided into several subspaces with the same properties.