  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) is greater than
or equal to 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.
