Perfect state transfer of a qudit over underlying networks of group associationschemes

Abstract

As generalizations of results of Christandl et al (2004 Phys. Rev. Lett. 92 187902; 2005Phys. Rev. A 71 032312) and Facer et al (2008 Phys. Rev. A 77 012334), Bernasconi et al (2008 0808.0510 [quant-ph]; 2008 0806.2074 [math.CO]) studied perfect state transfer(PST) between two particles in quantum networks modeled by a large class of cubelikegraphs (e.g. the hypercube) which are the Cayley graphs of the elementary Abelian groupZ2n. In Jafarizadeh and Sufiani (2008 Phys. Rev. A 77 022315) Jafarizadehet al (2008 J. Phys. A: Math. Theor. 41 475302) respectively, PST of a qubitover distance regular spin networks and optimal state transfer (ST) of ad-level quantum state (qudit) over pseudo-distance regular networks were discussed, wherethe networks considered there were not, in general, related to a certain finite group. In thispaper, PST of a qudit over antipodes of more general networks, called underlying networksof association schemes, is investigated. In particular, we consider the underlying networksof group association schemes in order to employ the group properties (such as irreduciblecharacters) and use the algebraic structure of these networks (such as Bose–Mesneralgebra) in order to give an explicit analytical formula for coupling constants in theHamiltonians so that the state of a particular qudit initially encoded on one site willperfectly evolve to the opposite site without any dynamical control. It is shownthat the only necessary condition in order for PST over these networks to beachieved is that the centers of the corresponding groups be non-trivial. Therefore,PST over the underlying networks of the group association schemes over all thegroups with non-trivial centers such as the Abelian groups, the dihedral groupD2n with evenn, the Cliffordgroup CL(n) andall of the p-groups can be achieved.