Study of two-spin entanglement in singlet states

We study the entanglement properties of two-spin subsystems in spin-singlet states. The average entanglement between two spins is maximized in a single valence-bond (VB) state. On the other hand, $E_v^2$ (the average entanglement between a subsystem of two spins and the rest of the system) can be maximized through a homogenized superposition of the VB states. The maximal $E_v^2$ rapidly increases with system size and saturates at its maximum allowed value. We adopt two ways of obtaining maximal $E_v^2$ states: (1) imposing homogeneity on singlet states; and (2) generating isotropy in a general homogeneous state. By using these two approaches, we construct explicitly four-spin and six-spin highly entangled states that are both isotropic and homogeneous. Our maximal $E^2_v$ states represent a new class of resonating-valence-bond states which we show to be the ground states of the infinite-range Heisenberg model.