  An important problem in quantum information theory is the quantification of
entanglement in multipartite mixed quantum states. In this work, a connection
between the geometric measure of entanglement and a distance measure of
entanglement is established. We present a new expression for the geometric
measure of entanglement in terms of the maximal fidelity with a separable
state. A direct application of this result provides a closed expression for the
Bures measure of entanglement of two qubits. We also prove that the number of
elements in an optimal decomposition w.r.t. the geometric measure of
entanglement is bounded from above by the Caratheodory bound, and we find
necessary conditions for the structure of an optimal decomposition.
