Geometric phase for entangled spin pairs

The entanglement dependence of the noncyclic geometric phase is analyzed. A pair of noninteracting spin-$\frac{1}{2}$ particles prepared in an arbitrarily entangled state and precessing in an external time-independent uniform magnetic field is considered. It is shown that the geometric phase reduces to a sum of one-particle geometric phases for product states and takes on the two values corresponding to the phase factors $\pm 1$ for maximally entangled states. If only one of the particles is affected by the magnetic field it is demonstrated that the influence of entanglement on the geometric phase may be interpreted as an effective reduction of the degree of polarization of the affected particle. The generalization to more than two precessing spin-$\frac{1}{2}$ particles, in the particular case where Schmidt decompositions exists, is briefly outlined. The geometric phase for a pair of spin-$\frac{1}{2}$ particles with a spin-spin interaction is calculated. In this model we show that the noncyclic geometric phase for a certain class of states may be interpreted solely in terms of the solid angle enclosed by the geodesically closed curve on a two-sphere parametrized by the evolving Schmidt coefficients. This suggests a geometric interpretation of Schmidt decompositions for spin-$\frac{1}{2}$ pairs analogous to that of the Poincaré sphere for a single spin $\frac{1}{2}.$