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 $E_{\max }\mathrm{}(n,d)\mathrm{}$ of the minimum entanglement between any two particles in the system? (Here we take the entanglement of formation as our measure of entanglement.) For $n=3\mathrm{}$ and $d=2,$ that is, for a system of three qubits, the answer is known: $E_{\max }\mathrm{}(3,2)=\mathrm{0.550.}\mathrm{}$ In this paper we consider first a system of d qudits and show that $E_{\max }\mathrm{}(d,d)\mathrm{}>~1.$ 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.