Hard-core dimer aspects of the SU(2) singlet wave function

We demonstrate that any SU(2) singlet wave function can be characterized by a set of valence bond occupation numbers, testing dimer presence-vacancy on pairs of sites. This genuine quantum property of singlet states (i) shows that SU(2) singlets share some of the intuitive features of hard-core quantum dimers, (ii) gives rigorous basis for interesting albeit apparently ill-defined quantities introduced recently in the context of quantum magnetism or quantum information to measure, respectively, spin correlations and bipartite entanglement, and (iii) suggests a scheme to define consistently a wide family of quantities analogous to high order spin correlation. This result is demonstrated in the framework of a general functional mapping between the Hilbert space generated by an arbitrary number of spins and a set of algebraic functions found to be an efficient analytical tool for the description of quantum spins or qubits systems.