**Abstract**
In the first part of this talk, we address the non-binary coupling of identical angular momenta based upon the representation theory for the symmetric group. We provide a detailed analysis of the coupling of three and four spin-1/2 systems and discuss a symmetric coupling of four spin-1/2 systems.

In the second part of this talk, we study a lifetime of decoherence-free (DF) qubit made up of three spin-1/2 atoms. The DF qubit stays coherent perfectly as long as the spins of the three atoms are affected homogeneously. The inhomogeneous evolution of the atoms causes decoherence, but this decoherence can be suppressed efficiently by applying a bias magnetic field of modest strength perpendicular to the plane of the atoms. We compare this result with other DF qubits made up of two and four spin-1/2 atoms, and show that the resulting lifetime of the DF qubit from three atoms can be extremely long.

References:

R. Han, N. Lorch, J. Suzuki, and B.-G. Englert, Phys. Rev. A 84, 012322 (2011).

J. Suzuki and B.-G. Englert, J. Phys. A: Math. Theor., 45, 255301 (2012).