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 Eₘₐₓ(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: Eₘₐₓ(3,2)=0.550. In this paper we consider first a system of d qudits and show that Eₘₐₓ(d,d)>~1. 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.