Entanglement sharing among quantum particles with more than two orthogonal states

Consider a system consisting of n d-dimensional quantum particles (qudits), and suppose that we want to optimize the entanglement between each pair. One can ask the following basic question regarding the sharing of entanglement: what is the largest possible value E-max(n,d) of the minimum entanglement between any two particles in the system? (Here we take the entanglement of formation as our measure of entanglement.) For n = 3 and d=2, that is, for a system of three qubits, the answer is known: E-max(3,2) = 0.550. In this paper we consider first a system of d qudits and show that E-max(d,d)greater than or equal to1. We then consider a system of three particles, with three different values of d. Our results for the three-particle case suggest that as the dimension d increases, the particles can share a greater fraction of their entanglement capacity.